Abstract: This discussion focuses on the application of Baez' and Bunn's infinitesimal expression1 for the Einstein equation at the core of general relativity to a pair of simple, expansionary models of the universe to estimate the universe's present scale. The analysis is guided by current insights into the famous, “big bang”38,40,41,43,44 cosmological model of Lemaître43,44 offered by modern, (WMAP) astronomical data. A simple proof begins the analysis by comparing two equations (the Baez and Bunn1 infinitesimal Einstein equation expression and an Einstein equation expression associated with the Hubbleflow2) that have been referenced by Baez and Bunn (and S. Carroll2) to establish the basis fortheir equality.
The content is inspired by a much more detailed discussion of the related conceptualization of Einstein's equation for infinitesimal application written by Baez and Bunn1 themselves. Anyone who has not yet encountered general relativity via tensor calculus will likely read the introduction to relativity as presented by Baez and Bunn in their paper with great pleasure. What is offered here is intended principally as encouragement to do so.
A very simple, exponential, “big bang” model is first produced in the course of this discussionthat considers, in a conveniently crude manner,the effect of a cosmological constant 27, 36 within a homogeneous and isotropic universe that evolved to an essentially “flat” state relative to space-time curvature early in its long history. This is offered to demonstrate the intriguing nature of the infinitesimal Einstein equation conceptualized by Baez and Bunn with regard to its capacity to provide basic
insight into cosmology18 and gravity when considering vast scales of time. The “Baez andBunn” Einstein equation is applied to the data set of the derived exponential expansionary model that is developed here to illustrate this. (An exponential model is one of the first, crude models produced.)
A direct, numerical integration model employing only the Einstein equation of Baez and Bunn over fifteen billion years is also compared to the predictions of the widely accepted Friedmann-Walker-Robertson equation in terms of computing the expansion of the universe from the “big bang” to the present time. The source data is substantially derived from the WMAP survey data and conclusions drawn from it by others.83, 84
This discussion is unlikely to provide new perspective to anyone with an advanced grasp of relativity. Dense fields of equations summarizing a new insight and intended for experts in the field are not present. Instead, considerable effort is made to explain what is done. Although simple integration is used, tensor mathematics and differential equations are not. In other words, what follows is meant to be fun and easy to read.
The content may also help, in a very general sense, to illustrate the usefulness of “first approximation” models and a “state-space” approach when seeking solutions associated with physical systems via a method that does not balk at the possibility that one should “think crude” in the initial stages of pursuit of a scientific or mathematical model in a mannercapable of offering a sense of basic understanding by recognizing that the opportunity to refine the solution in the future
Special relativity is now more than a century old. General relativity followed as an extensionof special relativity but was originally introduced by Einstein via a differential equation that lacked a closed form solution. It was not until a Prussian artillery officer produced his namesake Schwarzschild (in very rough, phonetic terms, SHVARTS-SHIELD) solution for Einstein's equation as it pertains to a stationary mass during World War One that Einstein's differential field equation relating energy density to changes in the curvature of space-time as asserted in his theory of general relativity had a solution expressed in four, Euclidean dimensions that was applicable to a general problem in physics.10
The Einstein equation is usually explored using tensor variables.11 Tensor mathematics is not a common element of undergraduate engineering curriculums (which suggests one prospective audience for this discussion). Non-physics (or mathematics) majors may shrink from their firstencounter with Einstein's most famous insight when presented in the context of its tensorial roots. They may then seek a more straightforward presentation (as did the author),such as the Einstein equation of Baez and Bunn.1
A great deal of patience can overcome this mathematical obstacle if one is willing to delve into tensor calculus, and many textbooks seem to confirm that nothing will fully correct for want of a solid foundation in tensor mathematics if one wishes to consider the shapeof space-time in detail relative to matters
germane to cosmology. (An alternative path to understanding general relativity, focused on LaGrangian mechanics34, has been previously presented elsewhere. LaGrangian mechanics was a favorite domain of the famous, twentieth century physicist, Richard Feynman, and he haswritten an introduction to the subject for undergraduates, which may appeal to those with suitable curiosity.)
The obstacle erected by tensor mathematics is usually sufficient to limit undergraduate engineering students to momentary contact withthe subject of relativity. This may occur within a broader, statistical and modern physics coursein which only special relativity is briefly discussed, perhaps, too often, while being perceived by the instructor as a subject that a particular section of engineering undergraduates will rarely if ever find to be applicable to their future work.
Such an attitude is not historically surprising. Einstein himself noted that general relativity was a field that was not given a great deal of emphasis in college curriculums, perhaps in part due to its specialized application to astronomy and cosmology before the introduction of global positioning system based navigation and analysis of the decay rates of muons79, which, together, add only very narrow, relevant applications in engineering andparticle physics. Of course, Einstein was in favor of correcting the want of attention regarding general relativity and authored a book, The Meaning of Relativity, to enhance such possibilities.
Other books, such as Introduction to the Theoryof Relativity, by Peter Gabriel Bergmann, (an older text), and Gravity, An Introduction to Einstein's General Relativity, by James B. Hartle, have acquired some renown among past students and some teachers of the subject for individuals seeking more accessible treatments available in paperback. Even with such resources, relativity is not commonly
Even quantum theory was somehow originally conceived without the need to integrate Einstein's older, general relativity. With relativity commonly perceived as limited in relevance to somewhat obscure cosmological and astronomical questions focused on gravitational interactions beyond the expectations of the laws of Newton, the founders of quantum physics originally turned simply to the clock on the mantle instead of Einstein's relativistic theory of space and time.24
We should recall that in the long march of human history both relativity and quantum theory were new to the first half of the twentieth century. Near the middle of the nineteenth century many physicists had come tobelieve that all that remained for science to discover could be achieved through the measurement of a few more significant digits associated with the constants employed by classical physical theories then held sacrosanct by government, industry, and teaching institutions, whose funding they substantially controlled.60
The black body radiation spectrum and inabilityto resolve other conundrums via classical physical theories, including problems such as the so-called, “occultation of Mercury”, (leading to the assertion of an unobserved and non-existent planet, given the name, “Vulcan”,88 to explain Mercury's orbital irregularities through Newton's gravitational theory) suggested the need for scientific advances, but acceptance of movement away from classical models as the ultimate expressions of natural theory came slowly in a
world of experts known specifically for their prowess with those same theories. Determined attempts to apply classical perspectives to seemingly unsolvable scientific problems continued (unsuccessfully relative to both of thepuzzles just described) into the twentieth century.
It was then that Albert Einstein, as an undergraduate in a state run college intended to produce Switzerland's next generation of sub-college level scientific teachers, chose to seek to define the profession of a “theoretical physicist” and, by working resolutely toward that goal, overcame the strict notion of science as no more than the practical servant of industryand, in certain instances, the chosen field of peculiar aristocrats, in a world in which everything was perceived to be very old, and thus a place in which everything must already have been very well defined. With bold new scientific perspectives rising to confront the new problems that much older ideas were proving powerless to address, there arose a corresponding potential for eager young “theoretical physicists” to be perceived as an annoyance or simply as misguided by elder scientists, whose reputations had been built upon their mastery of classical theories.
In the ensuing conflict between the old and the new there arose not a whisper of the specter of the harsh lesson learned by Galileo (1564-1642)61 when presenting his evidence in favor of displacing the earth from the center of the universe before the foot soldiers of a disapproving religious power and under the influence of governments controlled by princes,who claimed to rule by the will of the favored deity, and who had grown fond of having their crowns bestowed in publicly acclaimed ordinations by revered, religious elites to impress those they ruled in a world in which thelives of many, amid the ravages of disease and war, were sufficiently harsh that heaven could easily seem their sole hope for happiness. Galileo's ideas did not suit the intellectual
The apparent link between the infinitesimal statement of the Einstein equation of Baez and Bunn and a variation on that equation focused on the Hubble flow that is published by Carroll is initially considered in the context of a proof to clearly establish the mathematical connectionbetween one version of the Einstein equation described in terms of volume and another expressed relative to “scale factor”, which, as previously indicated, is a term closely linked to the concepts of Hubble and spatial expansion via an outward flow. With the connection between the two equations scrutinized in the early segment of this discussion via the referenced proof to plainly establish equivalence and to further explore the theoretical foundations of the Einstein equation of Baez and Bunn before we apply it, a simple mathematical problem is next pursued in a very crude manner via an exponential (“EXP”) model of expansion to test the concepts related to Baez' and Bunn's Einstein equation in the context of that equation's ability to predict fifteen billion years of history via an instant of relativity.
A second model is next considered that seeks toapply the Baez and Bunn (“B-B”) model to the “big bang” expansion of the universe via a numerical integration (using an amusing, “time jump” technique). The difference between the two approaches developed here is in use of the “Baez and Bunn” Einstein equation only with
regard to an instant of time at the beginning of the universe (the “big bang”) in the exponential (“EXP”) model, but the application of the Einstein equation of Baez and Bunn (without the pressure term) over the entire interval of expansion from the “big bang” to the present via a “time jump” numerical integration.
Warning Summary
Warnings have already been issued to note that the first part of the discussion that follows amounts to no more than a crude proof of the identical nature of two, superficially different Einstein equations published on the web sites ofBaez1 and Carroll2. (As has been described, one is simply expressed in terms of volume, and the other in terms of scale factor.) This may seem like a mechanical means of beginning a discussion that attempts to be “fun”, but it leads to a conclusion that establishes equality and thus provides grounds for the Baez and Bunn (“B-B”) model to later be applied to “big bang” expansion in a numerical integration.
The second element of the discussion is an illustration of the application of the Einstein equation of Baez and Bunn to the “big bang” theory of Lemaître via the development of a crude, bifurcated, state-space, exponential (“EXP”) model that has already been referenced. The Friedmann-Walker-Robertson (“F-W-R”) model56 of cosmological expansion is then considered to provide a basis for comparison to the results of the immensely crude, exponential model produced here (that relies upon general relativity only for acceleration data at the first instant of the “big bang”) and the “time jump” numerical integration results.
No new scientific insights should be expected from any of what has been proposed. Those with some interest in the prospect of seeing space-time considered by means of a simple, geometrical equation based upon no more than
Those who truly want to have a little fun can stop reading here and revert to the paper by Baez and Bunn1 (referenced in this document's bibliography and found on the website of Doctor Baez) if they have not yet experienced it. (Readers with well-developed tensor math skills seeking a review of general relativity focused on past experience with it might find the relativity notes or text of S. Carroll or the internet videos of L. Susskind more useful to them if some or all of these resources have not vanished from the internet or become too restricted to easily access over time.)
(A Few, Crude Notes on the Einstein Equation of Baez and Bunn for Those Disinclined to Find Their Own Copy of the Paper by Baez and Bunn Describing in Detail the Version of the Einstein Equation Used Here)
This paper was never intended to introduce the reader to the theoretical foundations of Einstein's Theory of Relativity. There are manyfine discussions of that topic, and the paper by Baez and Bunn is the most relevant to this effort to extend the ideas contained within that introduction to Einstein's universe with a glimpse of the “big bang”.
Einstein's basic idea is commonly associated with the notion of simultaneity. Keep in mind that he studied in Berne, Switzerland at a time when trains were the principle means of traveling over long distances, and worked there in the patent office. The train stations at that time, when linked to a mode of travel that took long periods of time, were subject to reasonableinterest by passengers and those operating the train system with regard to whether two train station clocks could remain simultaneously
synchronized within the train system over long distances.
For Einstein, what are commonly presented as his youthful “thought experiments” focused upon an effort to consider what it would be like to attempt to catch up to and travel along with aphoton of light may well have been unconsciously enriched by the train station clocks of the Swiss. (The problem was eventually solved using the technology of the telegraph.)
Einstein's ultimate insight, perhaps not surprising given his studies in a land well known for watch making, was based on the impression that the photon was the ideal timepiece of the universe, always traveling the same distance in the same amount of time. Even today we use the photon as a measuring stick and describe the “light year” as the distance that light travels in a year.
It was this knowledge of the constancy of the velocity of light that caused Einstein to recognize that there must be something more at work in the universe than classical physics was prepared to predict. (Actually, light, over very short distances, is not constant in speed, as Feynman has observed in his QED: The Strange Theory of Light and Matter, varying above and below the so-called speed of light along its path ever so slightly. This paper takes the perspective that light is closely approximatein its average speed around “c”, the so-called “speed of light”, so the “speed of light” is approximately a constant. In this perspective, this paper is, perhaps, outmoded. Nevertheless,it is consistent with how the subject is approached.)
If light must always be perceived as traveling atthe same speed, what does that tell us about the observer? No matter how fast an observer is moving (below the speed of light), the velocity of light is always the same. The observer cannot ever catch up with the photon of light, or
From Einstein's perspective, the only reasonable conclusion was that observers in an accelerated reference frame cannot experience the passage of time in the same manner as observers in a stationary reference frame. If that were not true, an accelerated observer would measure the speed of a photon traveling on the observer's path in terms of the sum or difference of the observer's velocity and the speed of light. This does not occur in reality. The notion of absolute simultaneity of clocks traveling at different velocities (and as Einstein ultimately established, over local terrain with different mass densities) was thus rendered an impossibility.
We know that if an observer is traveling at nearly the speed of light, the photon must still appear to travel at the speed of light to that observer. The same must be true for an observer whose velocity is zero. That could only occur if something had happened to the accelerated observer to cause the passage of time in his reference frame to shift and make the observation of the velocity of light the sameas for a person who was not moving at all.
The near light speed observer's clock must thus slow dramatically compared to the stationary observer's clock to make this possible. How could this be? If the near light speed observer'sclock were running slower than the stationary observer's clock as perceived by the stationary observer due to slower passage of time in the reference frame of the near light speed observer(see “twin paradox” in discussions of relativity that use two twin's as biological clocks), but both observers' clocks were operating perfectly,the only possible conclusion would be that the near light speed observer was truly experiencing a different rate of passage of time itself. The near light speed observer had somehow fallen behind the stationary observer'sreference for the passage of time.
This presentation, although crude, is not excessively removed from the fundamental ideas of the theory of relativity as initially conceived by Einstein. The concept of a four dimensional, space-time “continuum” as a model for our universe was asserted only after the introduction of relativity and was conceivedby Hermann Minkowski in 1908 rather than Einstein himself.87 (Einstein initially received the idea of a four-dimensional space-time continuum as an idea that amounted to so much“excessive egg-headedness”, until he realized itwas necessary for his next step, general relativity.)
What Einstein's theory of relativity asserts, when expressed mathematically in the form of the equation of Baez and Bunn, is a straightforward link between the rate at which acertain volume of space defining a body of mass (and cosmological constant) is expanding or contracting and the types and magnitudes of energy contained within that volume of interest.The mass density term is associated with inducement of a contraction of volume. The “cosmological constant” term provides for a corresponding expansionary term, contrary to gravity.
The pressure terms are the least consistent with a simple, “gut feeling” interpretation of the model. We have to embrace the concept that if mass is moving through space-time, a pressure is produced that has the same effect as additional mass with regard to expansion or contraction of a spatial volume. In short, the pressure terms correlate to relativistic mass or a“flux of momentum”. (Today we might be inclined to consider a Higg's field.) We will eventually find that the pressure terms are proportional to the square of the velocity of a particle with mass.
If we link all of the terms on the right of the Baez and Bunn Einstein equation (Equation 1.0) to a specific object comprised of a
Pressure Terms...in an Einstein Equation...in the Vacuum of Space?
As has been observed, the pressure terms of the Einstein equation of Baez and Bunn are probably the most shocking element of the equation for those who first encounter it. As human beings we are familiar with life on what we perceive to be a giant planet and the effect of the mass of that planet in the context of gravity. “Cosmological constant” sounds like an odd notion, but it isn't too difficult to move beyond the initial inclination to raise an eyebrow and recognize that the energy of a “cosmological constant” has the opposite effect on gravity as the energy of the mass of the planet with which we are so familiar. The pressure terms aren't so easily embraced. After all, what does pressure have to do with gravity?
We know that the pressure of the weight of the overhanging atmosphere is the cause of the air pressure on the surface of the earth. We know that the weight of the water in the oceans is the basis for the crushing pressures in the depths of the seas. What seems odd is the prospect of matter moving in empty space producing a pressure. At least it seems odd when considered solely in the context of the Einstein equation of Baez and Bunn, because being new to relativity, we are considering it from a perspective focused on our prior, classical concept of gravity.
We'll continue our crude consideration of the Einstein equation of Baez and Bunn with regardto the pressure terms by asking ourselves how we might have dealt with the notion of the constancy of the velocity of light when observed by a party traveling at any velocity when developing this equation. We've recognized that the Einstein equation of Baez and Bunn contains only terms related to mass, cosmological constant, and pressure.
We know that the value of the mass term associated with mass density of what we take tobe a spherical volume containing mass and cosmological constant energy does not change in a specific model of a massive object according to the Einstein equation of Baez and Bunn. We know the same is true for the cosmological constant term. How can we accommodate our need to maintain a constant speed of light regardless of the velocity of an observer if we have only constant terms for restmass density and cosmological constant? The answer is simple. We can't.
We could speak in terms of “geodesics” and “shortest paths”, and perhaps even provide sufficient misunderstanding and apprehension in the process to leave the reader feeling desperate, though potentially entirely unprepared, for a course in LaGrangian mechanics. Instead, we'll consider the Einstein equation.
On with the Show: The Einstein Equation of Baez and Bunn
Baez' and Bunn's statement of Einstein's field equation1 can be written as follows.
Evaluated solely at t = 0, with all elements of a spherical system at rest with regard to each other:
Equation 1.0 – Baez' and Bunn's Version of Einstein's Field Equation with Speed of Light (“c”) NOT Normalized
where:
Vtt = the second derivative of the rate of changeof the volume of a sphere (of particles representing the system of interest) in space-time, (or the “acceleration” of the volume.)
V = the volume of the sphere (of particles comprising the system of interest) in space-time.
G = Newton's gravitational constant = 6.67384 x 10-11 m 3 / (kg s 2). 21
ρ = the density of the matter in the spherical system of interest.
Px = pressure component in the x axis direction on the sphere's surface.
Py = pressure component in the y axis direction on the sphere's surface.
Pz = pressure component in the z axis direction on the sphere's surface.
c = speed of light = 300,000 km/s.
The result, as the left side of Equation 1.0 indicates, is in terms of per unit time (as a change in a quantity “per unit time” divided by the quantity).
Carroll (and Baez and Bunn) have published a slightly different statement2 of the equation, representing it alternately as:
At t
A=
−4 π G3
( ρ+3 P ) .
Equation 2.0 – A Different Presentation of the Einstein Equation
where:
Att = the acceleration of a “scale factor” associated with the universe (per the concept of Hubble).
A = the scale factor associated with the universe.
(Note: 58 A(t) = R(t) / R0 (or, here, A = R / R0), where “R” is the radius being characterized, and “R0” is the radius at a reference time (such as the present age of the universe). Given this, “A” is a unit-less or “per-unit” value expressed as the multiple of a base value measured in units of length, such as meters, light years, or parsecs, and “Att” is in units of “per square second” as a second derivative of a per-unit value, although it too could be rendered entirelyunit-less through the imposition of a base value of time, should it ever suit specific calculations.The “per-unit” nature of scale factor serves to explain the conversion necessary to compute a volume for the universe (in cubic meters) when we later apply the Friedmann-Walker-Robertson (“F-W-R”) equation to evaluate the results of a model that we will construct in a computer program.)
G = gravitational constant.
ρ = the density of matter in the sphere associated with the scale factor.
P = pressure in the spherical system of interest.
One fundamental difference between the two Einstein equations (infinitesimal and Hubble flow/scale factor based) now under consideration is easily resolved if one recognizes that Carroll states3 in his published notes that setting the velocity of light equal to one (normalization of “c”) is fundamental in theanalysis of Einstein's Equation. This is purely amatter of units, but the effect must be considered to produce equality between two
We should also note that according to a second statement of the Einstein equation by Baez and Bunn, the pressure term of our spherical, Lemaître “cosmic egg” model in the first instant of time must be multiplied by three where the pressure in the system is the same everywhere within the volume of interest, as it would be (on the largest scale) for the universalsystem we are considering.48 This is easily explained based upon the concept of isotropy and homogeneity, or a universe with properties that are the same everywhere and in all directions (in the context of a model of the entire universe).
This renders fundamentally relevant the second statement regarding the pressure terms by Baez and Bunn in their paper presenting their versionof the Einstein equation, which asserts the need to multiply the pressure in any system in which the pressure is “the same everywhere” by a factor of three. This is true because the pressure must be “the same everywhere” withinour homogenous and isotropic model, whether considered as a massive volume of space comprising our modern universe, or a single, infinitely small, geometric point particle representing a Lemaître “cosmic egg”. Where spatial expansion is the only basis for increasing the size of the universe, and where spatial expansion produces no particle motion, there can also be no pressure variation in the universe in our grand scale model. This rendersthe basis for use of a multiple of three in the pressure term clear with regard to the Einstein equation of Baez and Bunn as compared to the Einstein equation expressed in terms of “scale factor”.
The use of a scale factor based upon radius and applied to a sphere and the rate at which the scale factor of the sphere changes (as the radiusat any time in the history of the universe divided by the radius of the universe), per Hubble, is based on the concept of the velocity
of expansion of the volume of the sphere and the geometric link between volume and radius. The tie between the two versions of the Einsteinequation already presented (one in terms of volume, Equation 1.0, and the other focused on scale factor, Equation 2.0) is less obvious and is the focus of the first part of what follows.
Equivalence
Given:
A=RR0
.
At=R t
R0
.
At t=Rt t
R0
.
The basis for the following relationships are now clear:
R=R0 A .
Rt=R0 At.
Rt t=R0 At t.
Proof by Example
First, we'll consider a cube 100 centimeters on aside. That cube is is 1,000,000 cubic centimeters in volume. If we increase each sideby one percent in some time, the length of each side is 101 centimeters, and the volume is then 1,030,301 cubic centimeters. The volume thus increases by 3%.
For our purposes, this will suffice as proof of equivalence.
Vtt and Rtt are calculated literally as infinitesimals. In this computation, they are notinfinitesimals. This produces the error. (We must wonder whether to divide the number by avalue on one end or the other of the volume. Ina true infinitesimal calculation, this would not matter.)
Illustrating this Technique
Although spatial expansion associated with the universe and the “big bang” theory does not produce velocity for particles with mass, a version of this “time jump” numerical integration technique is later introduced and presented in computer code in this discussion toillustrate the numerical technique. The results are compared to those of the Friedmann-Walker-Robertson equation. (Use of a small time step is no issue when one need merely estimate the initial slope of a value before initiating a “time jump”, and when the initial slope does not substantially change even over many intervals of a time step, or in a radical manner even over the course of a well selected interval for a “time jump”.)
We would expect the “time jump” numerical method being described to underestimate the reality when modeling spherical expansion withan accelerating value of slope, which the acceleration of the universe is said by astronomers to possess, because the initial estimate of the slope of acceleration of volume
in a brief interval at the beginning of any “time jump” interval would tend to underestimate the value of the actual slope over the majority of each “time jump” interval. This is clearly apparent, because the estimate of slope is determined at the beginning of the “time jump” interval, before acceleration relative to the value of interest has occurred.
The results of the “time jump” technique will clearly be affected by the accuracy of the estimate produced at the beginning of each “time jump”. The principle goal of the “time jump” method just described in its use here relative to the “big bang” and expanding space-time is simply to illustrate the “time jump” numerical technique and to evaluate the results produced through application of the Einstein equation of Baez and Bunn over the longest conceivable time to the present without substantial computational delay using a novel method, to which we will now return.
Analysis: Applying the Einstein Equation of Baez and Bunn to a Fundamental, Cosmological Problem
At first sight Equation 1.0 may well appear to offer only limited facility for application in the most broadly conceivable realm of relativistic applications, even if it is advertised by Baez and Bunn as conveying the entire intent 1 of Einstein's equation in terms of the relationship between space and time in an instant. There area number of reasons for this.
1.Any spherical system analyzed must be comprised of particles that have no initial velocities relative to each other.
2.The Einstein equation presented by Baez and Bunn is indicated by its developers to only be valid in the first instant after time equal to zero in a simulation (before particulate components acquire unique velocities), which precludes anydetailed analysis of complex motion and interactions over time due to the equation being
3.The Einstein equation of Baez and Bunn doesnot consider such effects as gravitational radiation. This is not particularly relevant to consideration of the “big bang”, where gravitational radiation is not an element of focus (as it might be with massive objects, suchas black holes in some mutual orbit), but gravitational radiation is a significant area of application of the Einstein equation and generalrelativity.
At this point, given the limitations that seem to be imposed on the application of the equation, and particularly if one has not read the related paper by Baez and Bunn1, one might begin to feel suspicious of the range of facility of the referenced variation on the Einstein equation, so it is worth considering whether this version of the Einstein field equation can be applied meaningfully to some small but not entirely insignificant problem without further delay to test the level of insight that the Einstein equation of Baez and Bunn might have the power to bestow. One could select from a number of gravitational alternatives in the context of specific systems, some of which are discussed in a most insightful manner elsewhere in the paper by Baez and Bunn in a highly logical development and with impressiveresults.
Rather than attempting to retrace such familiar ground in what would inevitably prove to be an inferior effort, a single, modest problem in the field of gravitation and cosmology will be considered to help to allay any apprehension that may have arisen regarding the importance of the Einstein equation of Baez and Bunn due
to its limitations. The problem of interest here can be simply stated.
Can the Einstein equation of Baez and Bunn be used to produce a valid, order of magnitude estimate of the scale of the modern universe approximately fifteen billion years after the “big bang”? We shall make an effort to apply the Einstein equation of Baez and Bunn at the moment of the “big bang”, and hope for the best (with some help from modern astronomy) in estimating the resulting expansion of the universe. Having agreed to pursue this common goal, if we succeed, how could anyonestill be inclined to chaff when considering an equation in a form that easily offers useful, cosmological insight into the fundamental origin and future history of everything in the cosmos after billions of years of evolution?
A Simplified Analysis of the Cosmological Evolution of the Universe Via the Einstein Equation of Baez and Bunn
A Cosmological Equation
We should recall that Einstein's field equation isessentially a classical equation. In fact, its classical nature, disconnected from quantum physics, is recognized to pose substantial challenges for physicists working on problems where powerful gravitational forces interact with the world of the very small.19
Because of our need for observational guidancein th proposed calculations, we suddenly find ourselves requiring astronomical data and related, published insights relevant to our computational aspirations. What has the universe been doing over the past fifteen billionyears (or so)?
The Universe with a Cosmological Constant
The most likely statistical range for the cosmological constant has been considered elsewhere.14, 15 Recently published data
As we'll see later in the Friedman-Walker-Robertson (“F-W-R”) simulation, “slipped” may be too subtle a word. Gravity's capacity torestrict the expansion of the universe based upon modern data relative to its component energy densities seems to have always been tenuous, a fact concealed only by observational limitations and what may have simply been biasin favor of a perfectly balanced, and, perhaps, intellectually (or philosophically) satisfying, “Einstein universe”, possessing a very unique, steady-state future. (WMAP analysis suggests that we may be closer to this “Einstein universe” result in terms of the balance betweenmatter and cosmological constant energies than modern headlines might be taken to suggest, but a graph of the expansion of the universe from the “big bang” to the present that results from the asserted energy balance and the Friedmann-Walker-Robertson equation seems more determined to support consistent expansion.)
Einstein's original basis for the cosmological constant was founded in an attempt to produce a non-expansive, steady-state universe in whichthe gravitational impact of mass and positive energy on space-time curvature would be balanced by cosmological constant.40,43 His tenuously balanced, “steady-state” (or “Einsteinuniverse”) model corresponds to an expansion that slowly decelerates after a “big bang” originto achieve a constant rather than an ever expanding or subsequently imploding volume. (As mentioned previously, some have presentedthis simply as a by-product of the limits of astronomical observations at the time and not
an attempt by prominent scientists to introduce unique characteristics to the universe occupied by man.)
Such an “Einstein” or “steady-state” universe would never renew itself with a “big crunch” orsuffer endless “evaporation” via re-accelerated expansion after slipping past the steady-state, matter density limit fueled by “cosmological constant” roughly seven billion years ago47 in the manner consistent with astronomers' current(as of 199859) model for our universe. 22, 35, 50 The “Einstein universe” model is the first approach to modeling the universe over its entire, approximately fifteen billion year life to (the present time) that we will consider using the Einstein equation of Baez and Bunn (via thefirst pass at developing the “EXP” model). (A present estimate of the age of the universe is thirteen point four billion years. It is rounded to the nearest five billion years in the math that is performed here. A minor adjustment to one of the “F-W-R” model's parameters is made to compensate for the approximation and produce the thirteen point four billion year estimate of volume in fifteen billion years.)
It is interesting to observe that Einstein may have been somewhat uncomfortable with the notion that one need merely add a cosmologicalconstant to his equation for general relativity to produce an equation that would consider an “anti-gravity” force at work in the universe. The notion that so pervasive a force as “anti-gravity”, particularly given present day model's that grant cosmological constant claim to seventy percent of the energy of the universe, would affect only one term of the Einstein equation expressed merely as an addend may still empower a sense of subtle surprise or unease.
Modern observations have produced widespread belief in an exponentially expanding universe35,9, which appears to have had enough energy associated with a cosmological constant to overcome the effect of
Our purpose in seeking a model based upon thisinitial, immensely “crude” perspective is not cognizant denial, merely the simplification of the mathematical modeling approach used here in a manner that is taken to an extreme and designed to produce a speedy, ball-park, order of magnitude estimate of the volume of the universe in the present day. If the order of magnitude predicted through this course of action is absurd, the concepts associated with an exponential model and the Einstein equation of Baez and Bunn should guide a quick “refinement” or correction to improve the resultbased upon astronomers' modern perspectives regarding the history of the expansion of the universe.
If we err in representing some constant or the equation for some other key value, we won't have spent our time incorporating the error into a complex mathematical relationship designed to represent a final, more complex model that we find, in the end, also will not work in the context of producing a valid, order of magnitude estimate of the current volume of theuniverse. Simply producing a more complex looking equation does not necessarily assure better results.
This “crude” initial modeling approach helps tobreak the development into smaller pieces that can be easily evaluated to determine if an error exists within them. If nothing else, subsequent improvement, if necessary, given our order of magnitude goal, should help to demonstrate the underlying concept of making a “crude”, mathematical approximation then refining it further if it is not sufficiently accurate for a particular model's purpose.
The Einstein Equation of Baez and Bunn at the Instant of Creation
Figure 3.0 illustrates the curve most relevant tothe mathematical modeling approach that we have selected for our first attempt at producing an order of magnitude estimate of the present volume of the universe consistent with modern astronomical theory by using an “Einstein universe” model. If we accept that it is reasonably valid in the context of the universe expanding exponentially9 toward a final volumenear the end of fifteen billion years (with a five billion year time constant) in the context of imposition of a “steady-state” conclusion to thatexpansion in our first pass at developing a solution, then we need only follow a straightforward mathematical approach based upon the acceleration predicted by the equation of Baez and Bunn at the moment of creation to apply the exponential curve of the first half of Figure 1.0 (or all of the curve of Figure 3.0) to the time that followed creation (“the big bang”).
Perhaps we should evaluate the basic nature of what astronomers have proposed before we proceed to any mathematical model in order to insure that we have shed potential mis-conceptions. The “Einstein universe” (or steady-state universe) model is not the preferredimage of the evolution of the universe according to modern cosmologists. The relevant picture changed abruptly in the late nineteen nineties and what had been presented with a great deal of uncertainty began to seem clear with some help from the Hubble Space
The “Einstein Universe” and the “Big Bang” as Perceived Today
Most astronomers and cosmologists today embrace the conclusion that our universe has too much cosmological constant to be closed. Such an assertion is contrary to modern astronomical observations. That means that our“Einstein universe” model is known to us to be wrong from the very beginning.. It is importantto realize that no argument is made here to counter that fact. We're simply seeking a simple mathematical model to produce an orderof magnitude estimate of the current volume of the universe, and a steady-state universe model produced using an exponential equation seems to offer some potential in this regard.
Cosmologists predict a curve associated with the size of our universe over time that may be classified via some variation on the plot of Figure 1.0. The “variations” are generated using different combinations of mass and cosmological energy in the Einstein equation. No argument is made here to resist that perspective, but we are seeking to apply the equation of Baez and Bunn to produce a reasonable estimate over the entire age of the universe to the present day, and with that equation limited merely to the first instant of time (if one is intent on producing accurate results per the limitations of the Ricci tensor, aspreviously outlined), we must find another mathematical equation to “fill in the blanks” relative to the shape of the curve for the volumeof the universe from the “big bang” to the present day.
(Note: The path toward acceptance of the concept of a cosmological constant over time by astronomers is discussed in greater detail in an exposition on the subject of modern, cosmological theory by an award winning author that may be read or downloaded via the internet.35 NASA also offers discussions of the
topic.50)
Figure 1.0 – Conceptualization of One Perspective of the Expansion of the Universe50
Figure 1.0 is meant to initially describe an increase in the volume of the universe to a magnitude that occurs over one era during a period of slowing expansion, followed by an endless period in which universal expansion accelerates. What is proposed here in the context of an “Einstein universe” as a first pass model is to very crudely ignore the later, re-acceleration of the expansionary phase (that began some seven billion years ago) and simplyseek an order of magnitude estimate for the present volume of the universe based upon a steady-state conclusion to expansion as reflected in the plot of Figure 3.0.
A Diversion to Consider Expansion and the Nature of our Exponential Model's Guiding, Astronomical Data
We might presume that analytical computationsbased upon redshift serve to identify the rate of expansion of our universe at various times in the past from data culled from increasingly distant sources providing ever more ancient light and related insight into the rates of expansion in the past, but if we accept astronomers' claims of faster than light expansion in the most distant regions of the
Astronomers have always gazed into the past. This exposed them to light from objects that were at shorter distances from us long ago in a universe in which the rate of expansion was much smaller for the most distant objects. Using light that departed from those astronomical objects billions of years ago in a past in which the universe was a smaller, less rapidly expanding place empowers modern perceptions of depth of field.
Astronomers plot acceleration curves with distance and look for patterns pointing to increasing velocities of recession with time and related distance (after they have identified a standard source of light that will produce a set of reference wavelengths at the distances of interest to use to evaluate redshifts for all sources). Ancient light (“radiation”) from distant sources traveling through expanding space-time also suffers a redshift both due to the recession of the source and the expansion ofthe space-time through which the light travels.
Expanding space-time significantly reduces the energy associated with radiation subject to sucheffects over cosmologically relevant time intervals.49 This reduces the impact of radiationon space-time and cosmological expansion overtime.
If the universe seems to be spherical, a rate of expansion via a red shift curve over distance (and time) consistent with faster than light expansion for the most distant objects would present the effect described in Figure 2.0. (When considering astronomical data we must always keep in mind that astronomers today useancient light to photograph distant objects, eventhough the light that such objects may be emitting at this very moment may never reach the earth, stopped by its passage through too rapidly expanding space-time.)
Figure 2.0 – Observable Limit Due to Faster than Light Expansion at Some Distance From Earth if FTL Expansion of the Universe at Some Distance Is Valid (Not to Any Scale)
Back to An Extremely Crude Assumption, our “Einstein Universe”, a “First-Pass” Model
We have chosen to presume that Figure 3.0 is avaguely reasonable candidate for a model of theexpansion of the universe to the present day in the context of the universe starting with rapid, slowing, exponential9 expansion, and aggressively ignoring the later initiation of a more rapid expansion.23,35,40,43,44 We've also
15
Faster Than LightExpansion Radial Observational Limit
Earth
Space-time Surrounding EarthExpanding at a Constant Rateat Any Instant in Time
Figure 3.0 shows two things. As the plot marked by squares, the exponential plot, increases toward its final value of one for the function shown (with the magnitude of 1.0 representing 100% of the final value), most of the increase in magnitude occurs, as measured in thirds of the time required to reach ninety-nine percent of its final value, in the first third of the time required to reach that final value. Infact, in any natural phenomenon that can be
Figure 3.0 – How an Increasing Exponential Equation Changes on the Way to a Final Value
described by the exponential plot shown in Figure 3.0, one would be reasonably close to the final value if one considered only the magnitude of the plot after the first third of the time required to reach ninety-nine percent of the final value. The plotted result would have achieved roughly two-thirds (63%) of the final value after one third of the time (one “time constant”) required to very nearly achieve that final value.
The graph in Figure 3.0 shows that the exponential plot can be approximated by two lines, which share the value of the exponential
plot at one third of the time required for the exponential plot to reach its final value. The lower time valued linear plot has a slope less than the initial slope of the exponential plot, so if we plotted a line using the initial slope of the exponential curve, we'd overshoot our final value (of 1.0, here) considerably. (This is why we won't linearize our model given only data inthe first instant of the “big bang”.)
Figure 3.0 suggests that we might at least attempt to make a crude, order of magnitude estimate of the final volume if we were to applythese concepts and use the initial rate of changeof a process presumed to be exponentially varying (to which the acceleration of the expansion of the universe could be grossly assigned over much of its past 9) to estimate thefinal value using a time change equal to one third of the time required to reach 99% of the final value as the time constant.
The prescribed model is admittedly imperfect and likely to substantially underestimate the modern scale of the universe given that astronomers believe that the universe has moved past its phase of slowing expansion and shifted into a new era of accelerating growth (assuggested by Figure 3.0). The goal here is simply to employ the inherent simplicity of an exponentially slowing expansion model to advantage for an order of magnitude calculation. We are, after all, “thinking crude”. Next we must consider how this model will be built?
Building a Mathematical Model for the Expansion of the Universe Based on the Einstein Equation of Baez and Bunn
The Einstein equation of Baez and Bunn is capable of producing an estimate for the acceleration of the expansion of the universe only at the first instant of time. It has no powerto establish a time constant for an exponential expansion over time, because the expansion of a presumably spherical universe places particles
For the order of magnitude computation in which we hope to engage, we could conclude that the acceleration will occur over fifteen billion years in an exponential manner like that shown in Figure 3.0 to produce the current volume of the universe (knowing that we are ignoring the later, re-acceleration phase of expansion described by astronomers in this firstpass model). If we have some vague belief in a universe that behaved like an exponentially slowing process relative to expansion before theexpansion began to speed back up, and if we aren't too particular with regard to accuracy, that may not be entirely unrealistic, given that we are only seeking an order of magnitude estimate of volume. (We will characterize it here simply as a crudely computed, minimum, “ball park” estimate for the present size of the universe.)
We've shown that exponentially slowing processes moving toward steady-state (Figure 3.0) reach (roughly) two thirds of their final value after roughly one third of the total time required to approximate that final value. (The one third of total time to approximate final value interval is the time constant for the exponential process.) For our purposes, one third of the total time to the present size of the universe is one third of approximately fifteen billion years, or five billion years.
One could, at this point, assert the wisdom of employing an exponentially slowing equation for outward acceleration (Vtt) based on the first moment of acceleration as:
V t t( t )=V t t i n i t (e(−t / δ )) .
where:
δ = time constant.
Vtt(t) = the acceleration of the volume at any point in time.
Vttinit = the initial acceleration of volume per theequation of Baez and Bunn.
After the first integration of this equation, the velocity of volume equation is given by:
V t ( t )=−δ V t t i n i t e(−t /δ )
+V t 0.
Since the velocity of expansion must be zero at time equals zero (Vt(0) = 0), the value of Vt0 is prescribed to render the right side of the preceding equation zero at time equals zero:
V t ( t )=−δ V t t i n i t e(−t /δ )
+δV t t i n i t.
A second integration of the preceding produces the following equation for volume, which must be evaluated between two points in time:
V ( t )=δ2 V t t i n i t e(−t /δ )
+δ t V t t i n i t.
Equation 3.0 – Volume for Universe, Precise
With the closed form of this integral over time always ranging between time equal to zero and three time constants (“3δ”, or three times five billion years for our universe), or, equivalently, the present, we can produce a closed form version of the preceding equation to the present age of the universe:
V u=δ 2V t t i n i t e(−3 )+3 δ2 V t t i n i t−( δ2
)V t t i n i t.
The first term on the left of the right side of thisequation can be eliminated within a reasonable approximation, since it is much smaller than the
Equation 4.0 – Approximating the Volume ofthe Universe After Three Time Constants (roughly to the present, with “δ”, the time constant, equal to five billion years) for an “Einstein Universe” Crude Model Based on Acceleration at First Instant of “Big Bang” with Time Constant Given by One Third the Present Age of the Universe.
The approximation of Equation 4.0 will be counted as sufficient for use here to determine if we have finished completing our exponential model, as the “first step” in the process. The selection of a universe with an initial size of one cubic meter in volume at time equal to zeromay trouble some fond of the concept of a quantum point (or “singularity”) at the “big bang” expanding into a massive universe, but a smaller initial volume renders the mathematics related to volume less straightforward, because quantum point particles have no volume. (We've discussed this in the context of a geometric argument favoring zero initial velocity of expansion for the Einstein equation of Baez and Bunn in terms of volume if it is to be equivalent to the Einstein equation in terms of scale factor.)
An atomic or smaller initial volume for the universe would increase the density of cosmological constant in the computation, and increase the initial acceleration of volume (as suggested in Table 1.0). A larger initial volumefor the universe slows the expansion rate for time equal to zero with less concentrated cosmological constant energy. Since the goal here is to compute the size of the universe fifteen billion years after the “big bang”, one cubic meter of initial size for the universe at time equal to zero is a convenient measure for use in Equation 4.0, and is certainly not so large as to likely introduce a significant temporal error over the interval of interest due
to the time associated with the expansion from a quantum scale object to an object one cubic meter in volume in the context of a “big bang” origin based on the present intent of that origin theory.
Getting Pressure Data from Density and Velocity
In a model of the expansion of the universe due to the expansion of space-time we don't need to consider the velocity of mass particles. Particles have no velocity in a universe in which only space-time is expanding. If we consider the motion of mass particles in a spherical volume under the influence of a gravitational field, we do need a means of considering the “flux of momentum” of the particles, which in the case of the Einstein equation of Baez and Bunn, requires a pressure term. That is the purpose of the development that immediately follows, although adjunct to our needs.
Newton assures us that:
F = ma,
Equation 5.0 – Newtons Law of Force
where:
F = force.
m = mass.
a = acceleration of the mass due to force.
To work with the Einstein equation of Baez andBunn we must consider the impact of the spherical volume in which we are interested expanding into a field of test particles with density, “ρ”. These test particles, imagined as the “flux of momentum” induced by velocity ofmass particles, strike the surface of our spherical, mass volume and induce a pressure, “P” on that surface as it expands. We begin
A = surface area of spherical Volume representing the mass in which we are interested (that of the universe).
R = radius of the cross-sectional area of the spherical volume in which we are interested (the radius of the universe).
If we take the field of test particles striking the surface of our sphere of particles under gravitational influence to have a mass density, “ρ”, and the volume that is being struck by the moving cloud of external test particles to move with a radially oriented velocity given by “vradial”, the velocity of a point on the sphere's surface expanding outward orthogonally to the surface, then the encompassing test particle mass that is coming into contact with the surface area of this spherical system at any timeinterval, “Δt”, is:
m = ρ 4πR2 vradial Δt,
with all variables as previously defined.
Notice that this equation has two components that make its relevance here apparent. The “4πR2” component is the surface area of the sphere. The component represented by “vradial Δt” is the distance traveled by the surface of thesphere in the context of outward expansion perpendicular to the surface of the sphere at every point. It is the increase in the radius of the sphere due to expansion in the time “Δt”.
This means that the product of the “4πR2” and “vradial Δt” terms is the additional volume acquired by the sphere due to expansion in the time “Δt”, described in the MKS system in terms of cubic meters. If this volume is multiplied by a mass density, the result is the
mass of the test particles encountered by the surface of the sphere, with the test particles assigned the mass density of the sphere.
What is the acceleration given to each of these particles?
a = Δvradial / Δt.
This makes it possible to compute the force dueto the test particles on the area of the surface of the universe/sphere perpendicular to the direction of motion. We know the equation for force according to Newton in terms of mass andacceleration:
F = ma.
We can thus simply expand the equation by substituting known terms for mass and acceleration.
Given that the test particle mass being encountered by the surface of the sphere, under the assumption that “Δt” is infinitely small, so that there is no significant change in the densityassociated with the spherical system during thattime interval, is given by:
m = ρ 4πR2vradial Δt,
with:
ρ = density of test particles (or mass particles inspherical system).
R = radius of sphere
Vradial = radial velocity of point on surface of sphere.
If the radius of the object at the moment of interest is fixed, so “R” does not vary (as with aplanet), the force is purely a function of the change in velocity, so the cumulative force is given by:
F=∫ ρ 4π R2 vr a d i a l d vr a d i a l
F = 2ρπR2 (vradial)2 .
The surface area of the entire spherical volume is given by:
Asphere = 4πR2..
If pressure is force divided by area, the average pressure on one hemisphere of the sphere in thedirection of motion is:
P = F/A = 2ρπR2 vradial2 / (2πR2).
P = (ρ vradial2).
We could directly substitute the result obtained for pressure into Equation 1.0 as Equation 5.5:
V t t
V=−4 π G( ρ+
3 ρ v2
c2 −2 Λ) .
Equation 5.5 – Einstein Equation of Baez and Bunn with Pressure Term Associated with Mass with Velocity Directly Inserted
Is An Expanding Universe the Same as A Spherical Mass Expanding Due to Internal Forces Not Related to Space-Time Expansion?
We need to consider the model that is represented by Equation 5.5. The developmentof Equation 5.5 considered a sphere expandinginto space-time. We might think of a cartoon model of a round ball with a fuse that has been lit. If the bomb goes off, in an ideal fashion, the mass elements of the device will expand radially outward assuming the explosive is spherically packed with constant density.
This has absolutely nothing to do with the expansion of space-time over fifteen billion years since the “big bang”. A bomb explodes asthe result of a (typically) chemical reaction. Equation 5.5 and its development would NOT apply to that situation. Any moving mass expanding outward from a sphere that had just exploded would acquire real velocity. No velocity is obtained here. Space-time, in the context of expansion of the universe, is actuallyincreasing in volume in between the bits and pieces of the universe driven by cosmological constant (or “dark energy”).
If we presume that mass does not acquire velocity or “flux of momentum” due to expansion of the universe because space-time in between massive objects is the only element that is expanding, so the mass is not generating a “flux of momentum” in space-time due to its own motion, how might we write an equation for the expansion of the universe based upon Equation 5.5 while incorporating the cosmological constant? We might consider Equation 5.6 to be a logical conclusion:
V t t
V=−4 π G( ρ−2 Λ).
Equation 5.6 – Einstein Equation Assuming No Pressure Term is Required Because Expansion of the Universe Does not Produce Any Real Velocity of Objects in the Universe Due the Motion of Objects Because of Cosmological Constant's Effect Inducing Expansion of the Fabric of Space-Time Between Such Objects
Accumulating Data for Computations – Not Entirely Trivial in an Era of Rapidly Expanding Astronomical Knowledge
We need to try out Equation 4.0 (developed forour exponential model) and Equation 5.6 to determine if the results that they produce when they are employed together are credible. We need some readily accessible data to do that. The value of the speed of light, “c”, is 299,792,458 m / s. The gravitational constant, “G”, is 6.674 × 10-11 m3 kg-1 s -2.
The value of the cosmological constant is not asconsistently established as that of the gravitational constant or the speed of light in scientific literature of the past several decades. Astronomical observations provide a basis for its determination.
Warning! - The Only Data of Interest With Regard to the Mass of the Universe When Applying an Einstein Equation are “Density” and “Volume”
Density, not my density, or your density, but thedensity of the universe, is one of the two most critical pieces of information required for any Einstein universe model. The other informationthat must be relevant to determining the mass ofthe universe is its volume. One is unwise to assume that the “mass of the universe” posted on what appears to be a relevant internet web site in a respectable entry is necessarily always suitable for use with a specific volume of the universe.
As astronomers peer ever further into space, theknown volume of the universe changes. If the volume of the universe changes, and the typicaldensity estimate remains the same (as one might expect), then the total mass of the
universe must expand, or Einstein's equation won't work. The reason is straightforward. If the total mass data is not correct, the total cosmological constant data won't be right as a multiple of that mass, and if both are based upon a volume that understates the latest estimates for the latest “volume of the universe” (or “radius of the universe”), the results achieved using any version of the Einstein equation will be wrong. It's as simple as that.
Some Mass Data
At least one internet site proclaims the total mass of the Universe to be 1 x 1053 kilograms.32
One should not rely upon such an estimate in terms merely of kilograms with no corresponding volume data for the universe. Mass data is intrinsically based upon an estimate of the size of the universe. Neither may be up-to-date, and may not correlate. One must correlate the volume of the universe, the mass density of the universe, and the mass of the universe before applying this information inany combination.
Recent (Relative to This Writing), but Not Absolute Data Regarding the Mass and Volume of the Universe
One should take all of the following data with agrain of salt, and confirm the mass density and volume of the universe information that you employ before proceeding to apply it to an Einstein equation, or you may find that you've gotten the proverbial cart before the horse, and your results may be disappointing. Data on the volume or radius of the universe tends to change with time. In the past decade and a half,it has changed quite rapidly.
The density of the universe is given at a university web site (from the U.S.) as 3 x 10-30 g / cm3.80 This is 3 x 10-27 kg/m3. Should we trust this value? We can find an estimate for the total energy density of the universe on a
We might have encountered a 2004 news story focused on data derived from the Wilkinson Microwave Anisotropy Probe (WMAP) and data it gathered from the cosmic microwave background radiation. The result proclaims the radius of the universe to be seventy-eight billion light years.81 That corresponds to a volume of 1.69 x 1081 cubic meters.
If we were in a hurry, we might simply multiplythis seemingly modern volume estimate by the density estimate and compute a mass for the universe. For better or worse, this is not the eighteen hundreds, and astronomy has been moving much faster than it did in that era, so we should probably strive to find some information that is a little more recent.
If we were professional astronomers, we might have seen something in the monthly notices of the Royal Astronomical Society from 201182 that would have given us some rapid insight into the matter of the most recent estimate of the volume of the universe in a paper from 2010, which also is based upon WMAP data.83 The 2010 estimate asserts the radius of the universe to be 27.9 gigaparsecs in a flat space-time model. This correlates to a volume for theuniverse of 2.67 x 1081 cubic meters. (That's a change of approximately 1 x 1081 cubic meters, corresponding to more than half of the 2004 estimate, only six years later.)
We'll use the larger estimate, since that seems tobe the prevalent direction in which estimates of
the volume of the universe tend to be going. With the 2010 estimate of the volume of the universe, the mass of the universe, based on thedensity data previously cited, is roughly 8 x 1054 kilograms. This is the estimate that we'll use in this discussion for the mass of the universe. Don't use it in any future analysis until you've researched the latest estimate for the density of the universe and its volume (or radius) and performed the simple calculation to produce a result.
Cosmological Constant and Mass Based on the Most Recent Ratio
If the most recent astronomical data is taken to establish that mass is thirty percent of the universe and cosmological constant is seventy percent of the universe (with radiation taken to have no relevant presence compared to these other two quantities effect), then the cosmological constant mass density would be two and one third times the magnitude of the normal mass density, or, based on the number just given for the mass of the universe, cosmological constant would have an equivalent mass of 18.67 x 1054 kilograms.
(As recently discussed, we could have easily arrived at the wrong conclusion regarding the magnitude of the mass and cosmological constant of the universe, because it is not difficult to find estimates of the volume of the universe on the internet that include 3.38 x 1080 cubic meters 28 and 1.2 x 1079 cubic meters29, suggesting one source that is out of date and another based only on the distance traveled by light since the estimated beginning of the universe, ignoring the expansion of space-time as required for consistency with an Einstein equation. Note that it is also not difficult to encounter misleading data regarding the mass density of the universe in the context of a 3x10-
28 kilogram per cubic meter estimate that was inconsistent with the WMAP published result by a factor of ten.32)
With the volume of the modern universe crudely approximated to be 2.67 x 1081 cubic meters, the density of the universe is then givenby:
ρ=8 x 1054 k g /2.67 x 1081c u b i c me t e r s .
ρ=3 x10−27 k g p e r c u b i c me t e r .
If the current matter and cosmological constant densities (30% and 70%, respectively) of our universe are estimated here, for an order of magnitude, crude approximation, to have originally been within a spherical region of onecubic meter volume at the “big bang”, the original density of the positive mass in the universe and the original density of the cosmological constant at the “big bang” would have been:
ρ=3 x10−27 k g pe r c u b i c me t e r .
Λ=7 x 10−27 k g p e r c u b i c me t e r .
These results are consistent with a “seventy-thirty” ratio of mass to cosmological constant.
The Result
The resulting rate of initial expansion acceleration, based upon Equation 5.6, is:
V t t
V=−4 π G( ρ−2 Λ).
where, at the “big bang”:
V = 1 cubic meter.
G is 6.674 × 10-11 m3 kg-1 s -2.
ρ = 8 x 1054 kg/m3 .
Λ = 7ρ/3 = 7/3(8 x 1054 kg/m3).
Λ = 17.66 x 1054 kg/m3.
The acceleration of the volume at the “big bang” is then given by:
V t t
V=−4 π G( ρ−2 Λ).
V t t=−( 4π )6.674×10−11
x (8 x1054−(2 )17.66 x 1054
)V .
V t t=2.29 x1046
cubic meters per square second.
Assuming a 15 billion year present age for the universe, with one time constant, “δ”, given by five billion years (or 5,000,000,000 years x 365¼ days/year x 24 hours/day x 60 minutes/hour x 60 seconds/minute = 157.788 x 1015 seconds),the current size of the universe would be on the order of 1.7 x 1079 cubic meters, produced via Equation 4.0:
V u≈2 (δ2)V t t i n i t
m3.
V u≈2 (157.788 x 1015)
2 2.29 x1046 m3.
2 x 2.29 x 1046 m3 /s2 x (157.788 x 1015 s)2 =
1.1 x 1081 m3.
This result is somewhat smaller than the approximately 2.67 x 1081 cubic meter volume estimate for our current universe produced by astronomers. Still, this conclusion is not bad for a crude estimate based only upon acceleration during the very first moment of creation computed via the Einstein equation of
We might question whether our one cubic meterinitial size for the universe produced a lucky result. Of course, the initial size of the universeaffects the initial density of the mass and cosmological constant. As has already been discussed, if we make the size of the universe atthe first instant smaller, we compress the cosmological constant energy density further. This accelerates the expansion, a factor countered by the accelerated expansion affecting a smaller initial volume, which can beunderstood via Equation 4.0.
Table 1.0 shows the effect of different initial sizes of the universe in the context of the predicted current volume of our universe for various initial sizes of the universe. Use of various initial sizes for the universe at the “big bang” in Table 1.0 permits us to examine our “lucky guess” concern regarding the initial volume that we chose of one cubic meter.
Table 1.0 – Variation in Size of Predicted Universe After Fifteen Billion Years Based Upon Initial Size of Universe at Big Bang Using Equation 4.0 and Equation 5.6
The “lucky guess” hypothesis relative to the useof one cubic meter as the initial size of universeat the “big bang” in the initial computation resented here is established to not be valid based on the results of Table 1.0. The one cubic meter scale was simply appealing because
it forced Equation 4.0 to produce results in terms of cubic meters per second squared, which made it a computationally convenient choice. It has been shown in Table 1.0 to not be necessary to select a one cubic meter initial volume for the universe at the “big bang” giventhe same equations employed in order to produce the same mathematical outcome.
The only conversion that was necessary to produce Table 1.0 was multiplication of the mass density and cosmological constant densityused for one cubic meter at the “big bang” by the multiplier produced by one cubic meter divided by the whole or fractional cubic meter volume indicated in the table. The result produces more or less concentrated levels of cosmological constant, which correctly modifies the initial acceleration of the expansion to produce the same, final volume for any initial volume choice.
There is no change in the results of Table 1.0 for different initial volumes at the “big bang”, as long as the initial matter and cosmological constant densities are suitably adjusted for the new, initial containment volume. Of course, wehaven't justified the level of cosmological constant used beyond the assumption that what is present in the universe today was present at the “big bang”, which isn't a perspective that is easily defended on its own.
“Zero point energy” is the lowest energy state of the vacuum and thus must demonstrate an increase in its net value in the universe with space-time volume expansion. As a result, net cosmological constant energy should increase with the expansion of the universe. We've modeled the mass and cosmological constant energy that is present today as being present at the “big bang”. The same is true for the Friedmann-Walker-Robertson equation based estimate that comes later.
We are told that all that we require to explain the expansion of the universe is an environmentin which cosmological constant energy produced by the “big bang” can generate an expanding volume of space-time that establishes a universe that is big enough to conform to astronomical observations through the expansion of the space-time in which matteris trapped as space-time expands to increase theseparation between stars and galaxies. (Any particles traveling at light speed could traverse 1.4 x 1026 meters in fifteen billion years. If we use this value as the radius of a sphere, the universe would have a maximum, possible volume of 1.2 x 1079 cubic meters based upon light speed limited expansion, which explains the basis for one of the estimates of the volume of the universe previously cited.)
A few points associated with our results seem hard to deny. The “Einstein universe” model we employed produces a result that is roughly one order of magnitude less than the correct value for the volume of our universe after fifteen billion years. The time required to produce a one cubic meter volume universe from a quantum scale universe in “big bang” models is not a significant fraction of fifteen billion years (so we were not wrong to leave out the time of the related expansion phase to one cubic meter from a "cosmic egg" given the
very little time required for the universe to expand to one cubic meter according to common, cosmological theories relative to the fifteen billion year age of our universe and our own estimated value for the initial rate of expansion of the universe via the Einstein equation of Baez and Bunn).
The results achieved here, given the crude nature of the mathematical approach employed to produce them, although low, as one would expect for an expansion model that produces a steady-state outcome, seem surprisingly close to modern, scientific estimates of the volume ofthe universe relative to the “order of magnitude” nature of the exercise being attempted with an exponential approximation toan “Einstein universe” model and the acceleration of volume computed only at the first instant in time. In short, the result generated is not horrible in that it approaches the right order of magnitude for the volume of the universe of the present day in the right amount of time from a relatively small volume of space-time, but not so “good” in its too crudeconceptualization and low, ball-park estimate, combined with the steady-state, rather than expanding nature of the result.
Some may find the one meter initial scale of theuniverse to be particularly troubling. To delve down to the Planck scale “big bang” universe, we'd need to understand how the universe was changing, and how those changes might influence the physical model that we would employ. That is well beyond the wildest dreams of this discussion, which is only meant to demonstrate how the Einstein equation of Baez and Bunn might be used to approximate the expansion of the universe from the time of the “big bang” to the present. As long as we accept that the time required for the universe to expand from a "cosmic egg" to one cubic meterwas not significant relative to fifteen billion years, the approach is reasonable.
We can expand the model of Equation 5.0 to allow for re-acceleration of expansion after seven and a half billion years using Equation 6.0. With the time interval being considered lasting fifteen billion years, it is a small matter to create a temporally bifurcated model in which time is divided into two intervals, each lasting seven and a half billion years, in which Equation 4.0 dominates over the first seven and a half billion years, and Equation 6.0, withre-acceleration, dominates over the second seven and a half billion years. The time constant of either model becomes two and half billion years.
V u≈2 (δ2)V t t i n i t e3 .
Equation 6.0 – Exponential Model's Re-acceleration Equation for the Second Half of the Expansion of Our Universe (from Seven and a Half Billion Years to the Present)
Equation 4.0 and Equation 6.0 represent the volumes produced by expansion rates in the universe over two different eras. Since we havecrudely modeled the initial expansion to have lasted 7.5 billion years, and the re-accelerating expansionary era to have extended over the same, subsequent time interval, we can present the relevant time frames for the two equations modified to add data points based on the known
behavior of the exponential models. (The data points of the model correlate with equations described below.)
With “δ” crudely taken to be equal to 2.5 billionyears for our universe in both the initial expansion and second, re-acceleration phase of expansion, and with the assumption that for an exponential equation approaching a final value (during the initial expansion), the magnitude is sixty-three percent of the final value after one time constant, while for an exponential equation increasing exponentially to infinity, the value after two time constants is sixty-three percent of the value after three time constants, so we can compute a series of data points for a plot of the volume of the universe from the “bigbang” to the present.
For t = 2.5 billion years (one time constant afterthe “big bang”):
V u≈(0.63 )2(δ 2)V t t i n i t=1.26( δ2
)V t t i n i t
For t = 7.5 billion years:
V u≈2 (δ2)V t t i n i t
Now we compute values after the second exponential:
For t = 12.5 billion years:
V u≈0.63( 2(δ2)V t t i n i t )e3
=1.26(δ2V t t i ni t )e3
(Equal to the rough 63% of final value after twotime constants, plus the final value from the first equation.)
For t = 15 billion years:
V u≈( 2(δ 2)V t t i n i t)e3 .
(Equal to final value, plus the final value from the first equation.)
The initial value of the rate of expansion of the universe at the “big bang”, represented as “Vttinit”, is given by Equation 5.6, rewritten as
Vttinit = initial acceleration rate of volume of theuniverse.
Vinit = initial volume assumed for the universe at the time of the “big bang” (per volume data in preceding tables).
All other variables are as previously defined.
The preceding equations neatly divide the universe into two time periods (a bifurcated, state-space model) of seven and a half billion years each. The first is characterized as a decelerating, exponential expansion that ends atapproximately seven and a half billion years (inthis model) with the universe at the predicted volume. The second is a re-accelerating expansion to the present, fifteen billion year ageof the universe, which leads to the approximation of the present volume of the universe of Table 2.0.
We summarize the expansion values over time and plot the resulting expansion of the universe using the 2.67 x 1081 cubic meter estimate of the current size of the universe to compute matter and cosmological constant densities at the “big bang” as shown in Table 2.0 and Figure 4.0 using a simple, exponential (“EXP”)model:
Table 2.0 – Results of Attempt to Produce a Better Estimate of Expansion Using Equation 4.0 and Equation 6.0 (Data is
Plotted in Figure 4.0.)
Note that the results of Table 1.0, for an Einstein universe with a five billion year time constant, comes numerically closest (in terms of the exponential model results of Table 1.0 and Table 2.0) to the estimate of the current volume of the universe used here based on comparison of the final values after fifteen billion years of the pure Einstein universe exponential model and the temporally bifurcated exponential model with a two and half billion year time constant.
Figure 4.0 – Result of Attempt to Improve the Approximation of the Expansion of the Universe Using the Exponential Assumption Over Two Time Intervals.
The order of magnitude of the predicted volumeof the universe at present in Table 2.0 is 5.7 x 1081 cubic meters via the bifurcated “EXP” (exponential) model, which is on the same order of magnitude as astronomer's estimate of 2.7 x 1081 cubic meters 28 and is certainly reasonable as an order of magnitude approximation. The shape of the curve in Figure 4.0, controlled by the dual interval, state-space equations (one a decelerating exponential to seven and a half billion years, the other an accelerating exponential from seven and a half billion years to fifteen billion years) and the exponential assumptions that guided their development present a logical and
27
0.00E+000 1.00E+010 2.00E+0100.0E+000
1.0E+081
2.0E+081
3.0E+081
4.0E+081
5.0E+081
6.0E+081
7.0E+081
"EXP" Model of Expansion of the Universe
Volume (m^3)
Time, Years
Volu
me,
Cub
ic M
eter
s
Time (Yrs.) Volume (m^3)0.00E+000 1.0E+0002.50E+009 1.8E+0807.50E+009 2.9E+0801.25E+010 3.6E+0811.50E+010 5.7E+081
Cosmological Fuel and the Initial, Decelerating Expansion
If we take cosmological constant to be literally proportional to the volume of space-time, then its power to drive expansion vanishes with it in the vanishingly small volume of our current universe's space-time at the “big bang”. Equation 4.0 will not produce an adequate “bang” without cosmological fuel. We assume some source of energy existed to produce the “bang”.
Use of substantially less than the value of cosmological constant energy employed in our computation of the initial acceleration of the volume of the universe is not reasonable based on our attempt to reproduce astronomers' estimates of the current volume of the universe in the context of a decelerating, exponential expansion over roughly the first seven and a half billion years followed by a re-accelerating expansion over the subsequent seven and a half billion years.
Inflaton Theories and Colliding Membranes
It is likely that most astronomers inclined to take experimental data seriously, even where some level of uncertainty may be inherent due to the use of extremely distant objects to obtain the data, would see in the “big bang” an opportunity to seek to explain the expansion of space-time via modern theories of quantum physics. (This dates to Lemaître's “cosmic egg”.44)
That goal might be motivated by the simple question of how a “big bang” could drive expansion to produce a universe with its present, estimated volume and still retain the details produced through astronomical studies, such as cosmic background radiation with its very limited variation in temperature given low levels of “zero point energy” measured by
scientists and our “flat” space-time environment. Such theories support cosmological perspectives that do not rely uponsimply stuffing all of the energy of a cosmological constant after fifteen billion yearsinto a space vastly smaller than that of the modern universe and computing the expansion that the Einstein equation we derived from that of Baez and Bunn predicts using an exponentialmodeling assumption.
Advanced, modern cosmological theories may invoke an “inflaton” field that produces immense expansion very rapidly in a manner not consistent with an exponential model for expansion with a two and a half billion year time constant. Subsequent expansion to the present is on a much smaller scale. Past concepts in this regard have included invocation of a “super-heated Higg's field”6 as the basis for an “inflaton” field. Add to these insights offered by string theorists regarding colliding membranes as the origin of the “bang”in the “big bang”, and we may begin to wonder regarding both the nature of the forces driving the expansion initially and the scale of the initial expansion.
Simple application of “cosmological constant” within the Einstein equation in the context of a “big bang” and a rapid, exponential expansion20,30,9 to the present may seem less insightful upon encountering concepts such as the “inflaton” field and colliding membrane theories42 if the simple assertion of a correlationbetween cosmological constant levels and the volume of space-time in the universe is not sufficiently daunting when considering a quantum scale singularity as the origin of the universe.
The extent of the variation in modern “big bang” theories from a quantum scale singularityout of which all of matter, energy, and space-time develop41 (as envisioned by Lemaître44) to “banging” membranes that occasionally collide and create great bursts of energy amid massive
The Cosmologist's Friend – The Friedmann Equation – Child of the Einstein Equation
Cosmologists employ a mathematical formula when estimates associated with the expansion of the universe are required. It is called the Friedmann equation. It was produced from Einstein's equation and may be stated in variousways, including the Friedmann-Walker-Robertson (“F-W-R”) equation 56, 59 used for cosmological calculations:
1a0
d ad t
=d xd t
d xd t
=H 0√( Ω r x−2+Ωm x−1
+Ω λ x2+(1−Ω0))
Implemented Form of Friedman-Walker-Robertson Model of Cosmological Evolution Based Upon the Einstein Equation
With regard to the “F-W-R” model:
H0 = Hubble Constant Current Era of Universe. (Slightly modified here computationally to allow for extension of current age of universe toa crude approximation of fifteen billion years asthe present age of the universe, to remain consistent with the exponential model data.)
Ωr = fraction of energy in universe comprised of radiation. (Set to zero in model here.)
Ωm = fraction of energy in universe comprised of matter. (Set to 30% in model here.)
Ωλ = fraction of energy in universe comprised of “dark energy”. (Set to 70% in model here.)
Ω0 = valued at one (“1”) for our “flat space-time” model, but could be set to somewhat more or less than one, depending on anticipatedspace-time curvature.
dx = radius of universe. Used with current radius equal to 1.0 (per-unit, with the current radius of the universe as the basis).
dt = time interval for each step of numerical integration. (Set equal to one million years here.)
The Friedmann equation is the reason why estimating techniques such as those that have been presented here in the context of an exponential model or what might be perceived as an approximation based upon a “time jump” numerical solution (the “B-B” model) are not typically employed by cosmologists already familiar with general relativity. Remember thatthe derivation of Equation 4.0 was based on a crude model of the universe and recognition that over its history the universe has been becoming less energy dense.
This corresponds to a “flat” space-time universepremised on the argument that if force is given by mass times acceleration, and curvature is associated with perception of gravitational force, low localized acceleration rates due to spatial curvature in the universe correlate to low, localized curvature. Over vast distances, cosmological constant clearly can create more rapid acceleration even with a relatively flat space-time in most locations. Distance can accelerate even such seemingly mild, local effects.
For example, imagine flicking your finger upward to a vertical position from the horizontal in something less than a second, thenextend an imaginary line segment along your finger outward three hundred million meters (the distance light travels in a second), and you'll find the end of that line segment tracing aquarter of the radius of a circle amounting to
We intrinsically incorporate the space-time “flatness” concept into the assumption that the universe would accelerate outward rapidly and distribute its energy to avoid excessive concentrations over most of its volume, so we used no specific “curvature” term in the development of the exponential equations. The model that produced Equation 4.0 still ignores the initiation of a new era of faster expansion over several billion years described by modern cosmologists,35 but this was remedied through the addition of Equation 6.0 and the resulting bifurcated state-space model that produced Figure 4.0.
“F-W-R” Model is Per-Unit
It is important to recognize that the value, “dx”,in the Friedman-Walker-Robertson (“F-W-R”) model is a “per-unit” value in the computational algorithm described here. It is added to or subtracted from the radius of the universe over time in a numerical integration, but the radius of the universe must also be in “per-unit” terms. We need a basis for our “per-unit” system of length. We choose the radius ofthe universe based on our estimate of its presentday volume. We then divide that same value byitself to render the radius of the universe “1.0” (per-unit) at the present time.
We then need to restore the radius to measurement in meter terms and compute the volume of the universe at any point in the past for our “F-W-R” time machine to produce results that can be compared to others calculated in cubic meters. The body of the code that performs the calculations and stores the results follows in “Einstein.h”. In that code one can find the restoration of a meter valued radius based on the per-unit valued radius base value in the numerical integration prior to
Computer Code from “Einstein.h” that Calculates Volume of Universe at Any Time Step Based Upon the Radius of the Present Day Universe, Rad_now
One would normally question the value of computer code in a paper focused on physics, but in this instance, the goal of the discussion isto enhance interest among non-physicists or those just beginning their studies. Such interestmay be expressed, in part, through a desire to produce meaningful simulations that generate quantitative results given that some of those considering the study of relativity may have an interest in constructing their own mathematical “time machine” according to one of various cosmological and relativistic models.
The “C++” (“see-plus-plus”) code to accomplish this task is divided into two segments. One is the core of the computer program, a C++ header file, titled “Einstein.h”, which contains the definition of a vector of structures by which data is passed to a routine (“CSV_writefile”) that acts, as one would expect, to store the results in a “comma separated value” file (with “.csv” as file name extension) for access via a spreadsheet. The Friedmann-Walker-Robertson solution routine is the principle element of this header file. The contents of “Einstein.h” are shown in Special Figure A.
//Friedmann-Walker-Robertson Implementation Followsvoid FWR(){/*BEGINNING OF CODE WRITTEN TO IMPLEMENTCOMPUTATIONVariable declaration and initiation:*/int counter=0,year=0,retval=0,count=0;double delta_t=1000000.;long double Vuni_now=2.67E81;double InputRad=double(Vuni_now*3./4/3.1415);double Rad_now=pow(InputRad,double(1./3.));double t0 = 15.E9;double Omega_0 = 1.;double Omega_r = 0;double Omega_lambda = 0.7;double Omega_matter = 1.-Omega_lambda;/*Modify H0 constant to conform to 15 billion yearexpansion to present based on value for 13 billion yearexpansion to present.*/double H0 = double(0.5*13./15./9.78E9);long double x = 1.;long double dx=0.;/*Following line of code creates a structured data variablecontaining radius and year for simulation,with years measured since "big bang".*/ SizeVector Universe;/*Size output structure to contain datafrom seven years over entire simulation time interval.*/Universe.resize(7);char const NameFile[]="D:\\Friedout.csv";for ( counter = 15000 ; counter >= 100 ;counter-- ){/*Begin implementation of Gaussian
approximation to expansionbased upon cosmological constant as dominant energy form."delta_t" is 15E9/15000, for crude, one million year resolution.*/dx=double(H0*-(delta_t)*sqrt(Omega_r/pow(x,2)+Omega_matter/x + Omega_lambda*pow(x,2) + (1-Omega_0)));x = double(x + dx);/*End Gaussian approximation to expansionSelection of years in which to divert to"Einstein.h" to store data follows.*/if (counter==100||counter==2500||counter==5000||counter==7500||counter==10000||counter==12500||counter==15000){/*This segment merely selects the valuesto be saved in the output filein a manner to be consistent with that used to produce spreadsheetgraphs via an alternate model.*/Universe[count].year =double(counter*1E6);Universe[count].V = double(4./3.*3.1415*pow(double(x*Rad_now),3.0));count = count++;}}/*Next call routine to write all data stored in structure to CSV fileto be loaded into spreadsheet.*/retval=CSV_filewrite(count, Universe, NameFile);//Check for errors writing file.if (retval ==-1){/*No specific notification required for error writing file.This is responsibility of operating system.*/return;}/*END OF CODE TO COMPUTE EXPANSION VALUESAND CALL FILE STORAGE ROUTINE*/}#endif
Special Figure A - “Einstein.H” C++ Header File (Computer Code). (This code creates a structure for data storage, and stores data in structure sent to it by “main.cpp” in a comma separated file.)
1.The reference to “pow(a,b)” is C++ computer code for a^b.2.The “*” is computer code for the multiplication symbol.3.The “double”, “long double”, and “int” references in the code are declarations of variable types controlling allocation of computer memory and accuracy with which related variables are stored.4.A “structure” declaration references a means of storing relateddata under a single variable type. For example, “color” and “hair_length” might be used as elements of a structure named “dog”, accessed as “dog.color” and “dog.hair_length”, with suitable values or names assigned according to how “color” and“hair_length” are declared.5.The semi-colon (“;”) defines the end of a line of code that is to be interpreted by the compiler.6.The “\\” marks the beginning of a line of code that is only a comment, and is not interpreted by the compiler.7.The “\*” followed later, or on a lower line, by “*\”, indicates what may be multiple lines of code that comprise comments of interest to humans, and are not interpreted by the compiler.8.“{” followed by “}” marks a subroutine, looping routine, suchas a “for-next” routine, or a Boolean (logical) test controlling some result or influencing program flow.9.White space is ignored by the compiler.10.This code was compiled under Visual C++ 2010 (R), Express Edition (R).
The “F-W-R” model employed here is based onflat space-time (Ω0 = 1) in the “F-W-R” model over the interval of interest, from the present, tothe limit to which our mathematical time machine will carry us here, two hundred fifty billion years after the “big bang”. Radiation is taken to have effectively no impact on the model. Cosmological constant is taken to be seventy percent of the energy in the model, and matter is associated with thirty percent of the energy of the model. The numerical results are compared via tabulation and a plot of the resultsfrom each simulation model in Table 3.0 and Figure 5.0, respectively.
Figure 5.0 – Comparison of Exponential (“EXP”) and Friedmann-Walker-Robertson
(“F-W-R”) Flat Space-time model of Expansion of the universe from the Present to Two Hundred Fifty Billion Years After the“Big Bang”
The exponential model's results clearly understate the predictions of the “F-W-R” model considerably over much of the lifespan of the universe. Both models do produce results within less than an order of magnitude for the universe's volume at fifteen billion years. The exponential model generates a moresevere deceleration in the early phase of the lifeof the cosmos, but rapidly re-accelerates to overshoot the final prediction of the “F-W-R” model.
The “foot race” between the “time machines” over the fifteen billion year approximate, current age of the universe, must be, in the end, judged in favor of the “F-W-R” model relative to intrinsic accuracy. Even so, the final volumeof the universe produced by the “EXP” model is surprisingly valid in an order of magnitude context while based only upon the acceleration of the volume of the universe in an initial instant of the “big bang” derived from the Baez and Bunn version of the Einstein equation.
Equation 4.0 and Equation 6.0 simply use the predicted initial acceleration of the universe produced by the Einstein equation of Baez and Bunn (expressed for our purposes in Equation 5.6) linked to two intervals of exponential expansion and guided by modern astronomical data to generate this result. Table 5.0 comparesthe “F-W-R” model results and the “EXP” model results.
Table 3.0 – Comparison of Values Produced by “F-W-R” (flat space-time) Model of Expansion of Universe and Exponential Model Presented Here Via Equation 5.0 (to seven and a half billion years) and Equation 6.0 (from seven and a half billion to fifteen billion years) in a Fifteen Billion Year Long Approximation of the Current Age of the Universe (Time Constant, “δ”, for Both Phases of the Crude, Bifurcated, ExponentialState-Space Model Described in Referenced Equations is Two and a Half Billion Years. Data is Plotted in Figure 5.0.)
Having achieved a successful, order of magnitude “EXP” model using a fifteen billion year life span for cosmos we might be tempted to end this discussion with a reasonable sense of satisfaction. The exponential model's plot in
Figure 6.0 is far from a perfect fit of the Friedmann-Walker-Robertson version of the expansion for a universe like our own, but given the exponential model's origin in but a moment of relativistic computation via the Baezand Bunn Einstein equation, the result seems surprisingly valid.
Beyond an Instant of Relativity – Applying the Baez and Bunn Model to a Simulation of the Expansion of the Universe from the “Big Bang” to the Present
Figure 6.0 is a conceptual representation of the numerical integration process that will initially be applied here, comprised of crawling over multiples of a very short time step, much much less than one second, to seek to eliminate errorsin the Baez and Bunn equation, followed by useof the resulting data provided relative to the slope of the expansion of the universe to “time jump” over billions of years.
Figure 6.0 – The “Time Jump” Integration Approach Employed To Adapt the Einstein Equation of Baez and Bunn to a Fifteen Billion Year Long Numerical Integration to Estimate the Size of the Universe Based Upon the Concept of a “Big Bang” Expansion with a Billion Year “Time Jump”
The fundamental assumption associated with the numerical integration approach illustrated
33
VERY BRIEFLY CRAWL TO ESTIMATE SLOPE
(RUNGA-
KUTTA
4TH ORDER)
"TIME J UMP"
- BILLIONS OF
YEARS)
(NOT TO SCALE)
(Exponential Model Vs. Friedmann, Walker Robertson Model)
Table 4.0 provides an estimate of the volume ofthe universe based upon the rate of acceleration of the volume in the first instant of time after fifteen billion years.
Table 4.0 – Estimate of Current Volume of the Universe Based On “B-B” Method Computed Rate of Expansion in First Moments of Time after “Big Bang” Using a Five Billion Year “Time Jump” for “B-B” Model.
The results of Table 4.0 are plotted in Figure 7.0. The result proves the order of magnitude consistency of the “B-B” model with the fifteenbillion year result of the “F-W-R” model at fifteen billion years. Figure 7.0 shows the dataproduced by the “B-B” model with a one billionyear time jump. The “B-B” model seems to closely shadow the “F-W-R” model to produce a nearly identical result for the current volume of the universe.Figure 7.0 – “B-B” (Baez and Bunn EquationWithout Pressure Term) Model with Five Billion Year Time Jump Compared to “F-W-R” (Friedmann-Walker-Robertson) Model
The five billion year time jump used with the “B-B” model anticipates approximately the same size universe as the “F-W-R” model. It should be observed that both models are based upon the presumption of a 70% to 30% ratio of
mass to cosmological constant energies at all times in the simulation. The constant cosmological constant may seem strange, giventhat a “zero point” model would produce cosmological constant with space-time vacuum as space-time expands. Cosmological constant should then increase in magnitude with time and spatial volume. The computer code that produced the Baez and Bunn (“B-B”) model results follows.
Computer Code for “B-B” (Baez and Bunn Equation Without Pressure Term) Model with Five Billion Year Time Jump
//Modified Baez and Bunn Implementation,5E9 Yr. Step Follows
void MBB5(){/*BEGINNING OF CODE WRITTEN TO IMPLEMENTCOMPUTATIONVariable declaration and initiation:*/
int retval=0,count=0;double year=0;double Omega_lambda = 0.7;double Omega_matter = 1.-Omega_lambda;double Mass_Common=8.*pow(10.,54.);//Assign Universe one cubic meter volumeat Big Bang as approximation.
double Vol_uni = 1.0;//Set up Densities of Mass and Lambda at
34
0.00E+0001.00E+010
2.00E+010
0
5.00E+080
1.00E+081
1.50E+081
2.00E+081
2.50E+081
3.00E+081
Expansion of Universe
B-B (m^3) (5E9 Yr. Time Jump)
F-W-R (m^3)
Time (Years)
Volu
me
(Cub
ic M
eter
s)
Expansion of Universe Models
Year B-B (m^3) (5E9 Yr. Time Jump) F-W-R (m^3)1.50E+010 2.8E+081 2.7E+0811.00E+010 1.2E+081 1.3E+0815.00E+009 3.1E+080 5.6E+080
0 0.0E+000 0.0E+000
(Baez-Bunn, 5E9 Time Jump Vs. Friedmann-Walker-Robertson Model)
double Dens_Matter=Mass_Common/Vol_uni;double Dens_Lambda=Dens_Matter*Omega_lambda/Omega_matter;//The gravitational constant, “G”, is 6.674 × 10^-11 m3 kg-1 s -2.
double G =6.674*pow(10.,-11.);double Runi =pow(Vol_uni/(4./3.*3.1415),double(1./3.));
double Vtt = 4.*3.1415*G*((-Dens_Matter +2*Dens_Lambda));//Compute Value of Rtt based upon Att and volume of sphere//based upon its radius.
double Rtt = Vtt/Vol_uni*Runi/3.;double time =0.;
double TIME_JUMP = 5.E9;//Set delta_t equal to magnitude based upon time jump size//and number of time jumps required for fifteen billion//year estimate
double nu = pow(15.E9/TIME_JUMP,3.);double delta_t = sqrt(abs(Runi/(2.*Rtt)))/nu;//Use of a multiplier of ten is not //specifically required for estimation //interval duration relative to time step//duration.
double EndTime=delta_t*10;double VttCount=0.;/*Following line of code creates a structured data variablecontaining radius and year for simulation,with years measured since "big bang".*/
SizeVector Universe;/*Size output structure to contain data from seven years over entire simulation time interval.*/
//If brief time frame used to estimate acceleration at beginningof
//"Time Jump" interval has NOT been reached, simply increment
//the "year" and "time" variables to proceed through brief estimating
//interval at beginning of "Time Jump".
time = time + delta_t; } //Selection of years in which to
store data follows.
if ((int)(year/1.E9)==5&&count<1||(int)(year/1.E9)==10&&count<2||(int)(year/1.E9)==15&&count<3)
{ /*This segment merely selects
the values to be saved in the output file
in a manner to be consistent with that used to produce spreadsheet
graphs via an alternate model.*/
Universe[count].year =year; Universe[count].V = Vol_uni; count = count++; } } /*Next call routine to write all
data stored in structure to CSV file
to be loaded into spreadsheet via output function "CSV_filewrite".*/
retval=CSV_filewrite(count, Universe, NameFile);
//Check for errors writing file.
if (retval ==-1) { /*No specific notification
required for error writing file.
This is responsibility of operating system.*/
return; }/*END OF CODE TO COMPUTE EXPANSION VALUESVIA MODIFIED BAEZ AND BUNN METHODAND CALL TO FILE STORAGE ROUTINE*/}
Special Figure B – “BB5()” Function C++ Code to Computer “B-B” Model Result with Five Billion Year Time Jump
The acceleration of the expansion of the universe is visibly apparent in the plot of Figure 7.0. Once again, the fifteen billion year estimates of the volume of the universe are wellwithin the order of magnitude that we initially sought and is nearly identical to the “F-W-R” model. Acceleration of the rate of change of the volume of the universe should render the “B-B” model somewhat conservative in its predictions given that the slope that is used for each time jump of billions of years in the “B-B”simulations should underestimate the true, average value of the slope over the entire interval. (This is because the slope is computedat the beginning of the “time jump” interval, before the rate of change of the acceleration increases during the interval of the subsequent jump.) Any effect upon the Baez and Bunn Einstein equation that would produce a more rapid expansion in the first time interval (or anysubsequent interval), including any error component, would be amplified more with a five billion year time jump than with a one billion year time jump.
Gravity and Membranes?
Advanced, theoretical models do not rule out the potential for effects not apparent from a terrestrial perspective in the context of
The concept of gravitons adrift between membranes has also been presented given the graviton's theorized capacity75 to travel betweenmembranes in string or membrane theory. Sucha model would impact gravitational influence and space-time curvature over the vast distances between interstellar and intergalactic objects due to masses on membranes other than our own.
Conclusions
At a time when the relativistic concept of the “flux of momentum” associated with matter accelerated through space-time for a single universe begins to sound particularly classical in light of modern discussions of the “Higg's Boson”, with the former references taking on the charm of an early, oaken laboratory from the dawn of the 20th century, physics is still evaluating concepts that are directly related to iton the grandest scale imaginable. The rise of quantum models that seem to begin to bestow mass and that seek to explain momentum, the investigation of such theories in powerful particle accelerators, combined with the capacity to measure the velocities of distant, planetary masses circling far away suns and tiny particles traversing intergalactic distances have all created a different perspective on the universe than what preceded it.
At the dawn of the 20th century, in a universe that seemed much smaller then, science informed us that human civilization existed within a single, known galaxy, able to perceive
only hazy, unexplained “nebulae” beyond the detailed reach of the telescopes of that time. Even as recently as the final decade of the twentieth century many astronomers and physicists were surprised (in 1998) to find that observational data supported belief in faster than light recession of extremely distant objects, pointing to an open universe, and new perspectives focused on “dark matter” and “dark energy”.
Today scientific exploration is producing discoveries that challenge17 accepted perspectives from the past, including Wheeler's famous concept of “quantum foam”, with its “zero point” energy37 concepts linking it directly to the notion of “cosmological constant”.36,26,27 We will continue to wonder about the nature of such fundamental ideas and their place in the history of the cosmos as NASA probes of distant blazar17,25 objects raise questions regarding the omnipresence of quantum foam in the intergalactic vacuum of space amid serious discrepancies between observed and theoretically predicted levels of virtual particles.16,53
If one accepts the assertion that seventy percentof the universe is “dark energy”, twenty-five percent “dark matter”, and only five percent thedetected particles of matter on which the standard model is principally based,16 then the foundations of physics may superficially appearto lack a certain dimension of meaningful insight based only upon the focused data sets gathered from the interiors of massive, modern,particle accelerators and given the very low levels of observed virtual particles reported by physicists (10120 times40,57 too small to conformto theory).
Theoretical insights backed by observation may, of course, provide for greater understanding over the long term. Today cosmologists and physicists speak of “vacuum energy” associated with Wheeler's “quantum foam” merely as one candidate for the “dark
Concurrently theorists speak of the possibility of energy stored in the vibrations of tiny “strings” in “hidden” dimensions55 of space thatmay number eleven or more (with figures as great as 10500 being mentioned 52), with all but four of those dimensions either “rolled up” on ascale small enough to frustrate the most determined Machian investigator, or simply beyond our reach as denizens of a lower dimensional string membrane.55 Could the darkenergies of modern physics reside within the hidden confines of higher dimensions so inaccessible as to be beyond the reach of modern science, where string vibrational modesconceptually redefine the very foundations of paired, virtual particles? There are even some who wonder if hidden within the “darkness” of the newly theorized “dark matter” and “dark energy” there may exist grounds to overturn elements of relativity itself.
Whatever the physical expression of the model that is considered, Einstein's equation and general relativity have long had the power to inspire interest in the most fundamental questions hidden from direct observation and temporally removed from our vision in the first instant of the origin the universe, in addition to the theoretical origin and nature of “cosmological constant”8 and “dark energy”.36,
37 The Einstein equation of Baez and Bunn is a practical means of introducing oneself to more fundamental concepts associated with gravitation. This is the basis for the idea of Baez' and Bunn's equation as the seed for our consideration of “an instant of relativity”.
Limitations imposed on knowledge of the origin and the definitive origin of cosmological constant at levels capable of creating a “big
bang” with rapid, initial expansion that slows after eight billion years (or so) and related matters regarding initial conditions in terms of volume did not prevent the Einstein equation ofBaez and Bunn from being used here to easily peer back to the first moment of cosmological history to illustrate, from an immensely crude perspective, the best known of modern cosmic origin predictions based upon Einstein's field theory relative to the “big bang” model without tensor calculus but with the help of some gross approximations and a little imagination to estimate how the universe would expand over fifteen billion years given some helpful insightsfrom more recent astronomical observations considered within the context of a little fun.35
The results do conform to rational interpretationrelative to the modern, “big bang” theory (as supported by Figure 5.0 and Figure 7.0) in the sense of predicting expansion of the universe after fifteen billion years to within an order of magnitude of commonly generated results in the cases of both the “EXP” and the “BB” model. This is demonstrated by comparison of the fifteen billion year estimates of volume produced by the exponential model using only an instant of the predictive power of the Baez and Bunn Einstein equation near the “big bang”and the Baez and Bunn Einstein equation “time jump” numerical model to the flat space-time expansionary results computed via the Friedmann-Walker-Robertson equation, which is itself a direct variation on Einstein's general theory of relativity and a worthy candidate for use as a simple, mathematical “time machine”.
The “B-B” model does predict acceleration and expansion of the universe with reasonable accuracy via a direct application of the Einstein equation of Baez and Bunn for a universe that is seventy percent cosmological constant and thirty percent matter. The “B-B” model is actually a fairly close fit of the “F-W-R” model results over the entire, fifteen billion year interval.
The final results of the “BB” and “F-W-R” models compare well with the experimental estimates of 2.67 x 1081 cubic meters83 as the volume of the universe. (See page 22 for discussion.) This is pleasant accuracy given the crudeness of the data as applied to this model. The actual size of the observable universe will of course vary with the equations as the mass and amount of cosmological constant changes.
The Baez and Bunn “time jump” numerical (“B-B”) model exceeds the quality of results of the “EXP” model by a notable degree and closely approximates the Friedmann-Walker-Robertson result at fifteen billion years after the“big bang”. From this perspective, it is plain that even with our view of the “big bang” obscured by nearly fifteen billion years, the power of the Einstein equation of Baez and Bunn to render the complex mathematics of classical, general relativity accessible (if only inthe first moment of a gravitational interaction between particles initially at rest) has substantial potential to increase understanding of the underlying, classical principles of generalrelativity for those not versed in tensor mathematics though eager to satisfy their interest in space-time with a glimpse of an
instant of relativity.
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Biography
Del John Ventruella completed a BSEE from The Rose-Hulman Institute of Technology in Terre Haute, Indiana and an MSEE from The University of Alabama at Birmingham. His principle, professional focus has been in the area of large industrial power systems analysis and design. This includes focused under-graduate and graduate study and many years solving power system related problems for Fortune 500 corporations.
