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Lynn Umbarger 04/28/2005
Einsteins Theory of Special
Relativity
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Topics (46 slides)
Einsteins Thought Experiments Reference Frames The State of Classical Physics in 1900 The Problem
The Solution The Effects of the Solution Simultaneity Gamma Time Dilation Length Contraction
The Lorentz Transformation The Addition of Velocities Relativistic Mass Mass and Energy General Relativity (13 additional slides, time permitting)
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Einsteins Thought Experiments
At the turn of the 20th century Einstein asked thequestions:
If I dropped a pebble from the window of a train
carriage, I would see the stone accelerate toward themoving ground 4 ft. beneath my window in a straightline, then what would the person sitting on theembankment next to the tracks see? Would they not
see it travel more than 4 ft. and in a parabolictrajectory? Whose right?
If I ran at the speed of light and looked into a mirror at
my face, would I see my reflection?
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What is a Reference Frame?
A place to perform physical measurements
Could be thought of as a grid-work of meter-rodsand clocks so that trajectories and timings can be
performed Your reference frame always moves with you
When someone or something is at rest relative toyou, then you are both in the same inertial
reference frame When someone or something is not at rest relative
to you, then they are in a different reference frame
Reference frames in Special Relativity are said to beinertial because they are moving at constant
velocity; no acceleration, no rotation.
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The reference frame O is at rest to the referenceframe O which is in motion at a velocity of v and in
the direction of the xaxis of both reference frames
Not shown (yet) are the dimensions of time t and t
What is a Reference Frame?
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The state of physics up to theturn of the 20th century
Aristotle (349 BC) The universe was geocentric Everything moved on concentric spheres The Earth was a very special place
Ptolemy (140 AD) added: The planets moved, at times, in tinyperfect circles to explain retrograde
Copernicus (1543) The universe was heliocentric But everything moved in perfect circles
Brahe/Kepler (c. 1600) The known planets were heliocentric The planets moved in ellipses The universe was not necessarily a perfect place
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Galileo (c. 1630) The solar system was heliocentric (got him in trouble) It was a non-perfect universe (I.e. Sunspots, Jupiter had moons,
Venus was actually a crescent)
The natural state of motion is in a straight line until acted upon bya force (inertia) One cannot tell if they are at rest or if in non-accelerated motion There is no absolute rest frame of reference
Newton (c. 1680)
The laws of motion (mechanics) are the same for everyoneprovided that they are in uniform motion
Absolute Rest and Absolute Motion are meaningless unless theyare relative to something (Galilean/Newtonian Relativity)
He also implied with his rotating bucket experiment, that thereexisted a frame of reference at absolute rest
The state of physics up to theturn of the 20th century
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Maxwell (1860)
Unifies electricity and magnetism intoElectromagnetism with 4 (beautiful) equations
Electromagnetic waves move at the speed of light(effectively unifying optics with electromagnetism)
The speed of light was at that time already known to bearound 186,00 miles per sec (~300,000 km/sec)
But to what was the speed of light relative?
The state of physics up to theturn of the 20th century
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The ther (ether) was then proposed
A flexible substance enough to penetrate everything, yetrigid enough to be a medium for the high speed of light
How do we find the existence of the ether?
In 1887, the Michaelson-Morley experiment had a null-result
An explanation
Lorentz proposed that space shrinks (or contracts) in thedirection of travel through the ether by a factor of:
The state of physics up to theturn of the 20th century
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The Problem(at the turn of the century)
There may exist a reference frame atabsolute rest, relative to which, light is at a
constant velocity of c
If motion (mechanics) is relative toparticular reference frames, then why isnt
light?
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The Problem(at the turn of the century)
Newton, who created the Inertial Reference Frame(constant velocity), said it extended indefinitely,across the universe
The only difference between two different inertialreference frames, would be a change in constantvelocity: Once you knew one inertial reference frame,then you knew them all
Therefore, when one changes inertial referenceframes, one should measure a different velocity in thespeed of light
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Dispense with the concept of an ether
There are no reference frames at absolute rest
Einsteins two 1905 postulates:
All reference frames moving in uniform (non-accelerating),translational (non-rotating), motion; are perfectly valid forperforming all types of physics experiments, including
experiments with light (optics)
The speed of light is constant in any reference frame nomatter what its speed
Einsteins solution in 1905(On The Electrodynamics of Moving Bodies)
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Einsteins solution in 1905(On The Electrodynamics of Moving Bodies)
Einstein didnt have a problem with the physicaldescriptions of matter and radiation (light)
He did have an issue with how it was measured; inparticular he objected to the classical view of what weresimultaneous events, or Simultaneity
Einsteins two postulates could be rewritten to say: All the laws of physics are the same in every inertial
reference frame (positive statement) No test of the laws of physics can distinguish one inertial
reference frame from another (negative statement)
(As a consequence)
The measured value for the speed of light must be thesame for all of observers
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The Effects of Einsteins Solution
Clocks run slower in the reference frame of amoving object relative to the clocks of areference frame at rest to the first
Clocks slow to zero time as its referenceframe, relative to one at rest, approaches thethe speed of light
The dimensions of an object shrinks (or
contracts) in its direction of travel An object flattens to a plane as its reference
frame, relative to one at rest, approaches thespeed of light
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The Effects of Einsteins Solution
Time and space are now variable depending onones velocity
Time and space are now connected in a newmetric called: Space-Time
Whereas space and time may vary, intervals of
Space-Time are invariant (like light)
The speed of light has become a cosmicconversion factor
Si lt it
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Simultaneity
To the track-side observer in the middle of the top picture, bothlighting strikes occurred simultaneously
To the observer on the middle of the train, in the middlepicture; the front lighting strike occurred first
http://astro.physics.sc.edu/selfpacedunits/Unit56.html
Si lt it
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Simultaneity
In fact, between the on-board observers and the track-sideobservers, there is a general disagreement as to what time thelighting strikes occurred
Their clocks are now desynchronized as well
http://astro.physics.sc.edu/selfpacedunits/Unit56.html
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In order to properly measure something, one must do themeasurement at the same time
Observers in the moving reference frame will not with agree
the time, at which, the resting observers performed themeasurement
This is because:
Synchronization of clocks is frame dependent. Different
inertial frame observers will disagree about propersynchronization
Simultaneity is a frame dependent concept. Differentinertial frame observers will disagree about the simultaneity
of events separated in space
Simultaneity
http://astro.physics.sc.edu/selfpacedunits/Unit56.html
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The importance of the relativisticfactor (Gamma)
Gamma appears as a velocity based variable
throughout Special Relativity (recall Lorentz)
It is the key mathematical solution for telling us by
how much does time slow down (dilate) and spaceshrinks (contracts)
=
Gamma grows to infinity as the v approaches thespeed of light, and shrinks to unity when oneapproaches rest (see next slide)
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The importance of the relativisticfactor (Gamma)
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The Lorentz Transformation
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When we are rest, we are actually traveling inthe time dimension at the speed of light
When we divert that some of that speed overthe three dimensions of space, i.e. we go intomotion; then we travel through less time
The amount that time slows is a factor of onesvelocity relative to a reference frame at rest
How does the speed of lightaffect our experience with time?
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If t is the time in the moving reference
frame, then the amount by which timeappears to dilate is t, shown by thefollowing formula:
t=t/
How does the speed of lightaffect our experience with time?
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When the two reference frames are restrelative to each other, their time dimensionsare parallel to each other and perpendiculartheir respective space dimensions (orthogonal)
When one of the reference frames goes intomotion, it begins to rotate with respect thereference frame at rest while its timedimension must stay orthogonal to its spacedimensions
This causes the measuring rods ends todesynchronize with the measuring rod at restcausing a visible foreshortening
How does the speed of lightaffect our experience with space?
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If x is the length of a measuring rod inthe moving reference frame, then the
amount by which length appears tocontract is x, shown by the followingformula:
x=x/
How does the speed of lightaffect our experience with space?
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The Lorentz Contraction on Time andSpace
Space-Time Diagrams are a graphical tool to show the effects of
the Lorentz Contraction on space and on time. These diagramsrepresent a frame of reference at rest, there is no motion yet.
The vertical axis which is time, is labeled ct so that the speedof light can be shown as a 45-degree angle (slope=1)
Only the x-axis is shown for simplicity; y and z are suppressed,so that all motion continues down the x-axis
Th L t C t ti Ti d
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Diagram A shows the original reference frame at rest(un-primed), and a new one in motion (primed)
Try not to think of ct-axis and x-axis as contracting in towardthe c-line, but rather rotating about it.
Say the that ct-axis is lifting off the slide towards you as thex-axis is rotating away from you beneath the plane of the slide
Diagram B shows a faster moving frame of reference
Rotated more about the c-line
The Lorentz Contraction on Time andSpace
The Lorentz Contraction on Time and
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This is the Lorentz Transformation at work
Say an event (A) like a pulse of light was heading away from
the origin of both reference frames Diagram A shows how the un-primed frame would measure it
Diagram B shows how the frame in motion would measure it
Important to note: The ct and x-axis are still at right-angles to
each other; so are the measurement lines out to Event A
The Lorentz Contraction on Time andSpace
The Lorentz Contraction on Time and
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In both reference frames is one measuring rod atdifferent times and at rest with respect to its frame (itonly travels in the time dimension)
Even though in B, the reference frame is in motion
Note how the rod must always stay parallel to the x or
x-axis
The Lorentz Contraction on Time andSpace
Th L t C t ti Ti d S
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We wish to compare the length of the moving rod with the oneat rest at time ct1
During this time both the right and left ends of the moving rodwill be seen at different times in the resting reference frame
In B, we catch the moving rod at ct1 when its left end is
aligned with the left end of the rod at rest
The Lorentz Contraction on Time and Space
The Lorentz Contraction on Time and Space
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The Lorentz Contraction on Time and Space
Because the observer at rest can only measure parallel to his x-axis at time ct, the extent of his measurement can only go tothe right ends trajectory path (Diagram A)
He then measures from there straight down (or parallel to histime axis) to his x-axis (Diagram B)
We now see the rod in motion as foreshortened
h d
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At ct2, a moment later, the moving rods rightend aligns with the resting rods right end
But the moving rod is still foreshortened
The Lorentz Contraction on Time and Space
The Lorentz Contraction on Time and Space
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The same measurement of time shows the aspects of Time
Dilation Even though the clocks were synchronized at the start they
continue to see each other as running slower because of therequirement to measure parallel to their own x-axises
Ct3 sees ct2 as running slower and ct2 sees ct2 as running
slower
The Lorentz Contraction on Time and Space
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On board an all-glass bus moving at .75c, a(strong) person throws a ball from the back ofthe bus towards the front at a velocity of .75c
relative to the bus How fast would this ball appear to go relative
to an observer at the bus stop (at rest)?
Would they see it travel at 1.5c?
No, actually they would see it move at 24/25c(or .96c)
In fact, no matter how fast the bus or the ballwas traveling, you will never see an object hitor exceed the speed of light
The Addition of Velocities
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Because of the addition of relativisticvelocities, you can only approach the speed oflight
Einstein used the following formula to describethis effect; if v1 was the velocity of the busand v2 was the velocity of the ball on board,then V would be the observed velocity:
V=
The Addition of Velocities
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The reason for what the resting observer saw:
The observer would see a foreshortened bus
The clocks at the back and front of the bus would beobserved as very much out of synch with each other,
and more importantly, out synch with the observers
The observer would never agree, given the above
conditions, that the ball was traveling as fast as theperson that threw it believed it was going
The Addition of Velocities
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Heres the space-time diagram representationof the addition of velocities
The Addition of Velocities
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Say two cars of identical mass, each travelingat .75c, hit each other head on
According to the classical laws of theconservation of momentum and energy, thewreckage would come to a complete halt infront of an Observer A
A
Relative Mass(Einstein runs into trouble)
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Now say an Observer B was traveling along with theleft-vehicle (in its inertial rest frame)
He would see the right-vehicle coming at him at a speed
of .96c (Addition of Velocities) At the moment of impact one would assume that
Observer B would see the wreckage go by at half theclosing speed of the two vehicles, or at .48c
AB
Relative Mass(Einstein runs into trouble)
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How could Observer B pass the wreckage at .48c and yet
pass Observer A at .75c when Observer A was at rest to
the wreckage?
Was Einsteins addition of velocities wrong, or was
classical physics off (again) at relativistic speeds?
A B
Relative Mass(Einstein runs into trouble)
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How about using Gamma again?
Einstein use the equation: m= m
(m = relativistic mass, m = resting mass)
And the right-vehicle then had enough mass to push the
wreckage passed Observer B at .75c
Although this appears to only be an observational
phenomena, it is actually a measurable fact in particle-
colliders with hi h s eed electrons
AB
Relative Mass(Gamma to the rescue!)
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Mass and Energy
But where did the extra mass comefrom?
Einstein assumed it came from thekinetic energy (KE) that the right-vehicle had gained
Kinetic energy was related to therelativistic mass minus the restingmass, or: KE = m - m
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KE = m - m
KE is measured in units of joules or
kilograms times a meter per secondsquared
But seconds (time) and meters (length)
get varied at relativistic speeds Use the speed of light c, as a
conversion factor to get rid of these
units
Mass and Energy
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Mass and Energy
KE = (m - m)c
But when an object is at rest, it must
also have a resting energy E, and norelativistic mass m, or:
E = mc
2
2
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End of Special Relativity
Other effects of Special Relativity Relativistic Energy
Energy gains at higher velocities
Relativistic Momentum Momentum gains at higher velocities
Relativistic Aberration How the surrounding star field would appear at higher velocities
Causality
Cause precedes effect as a function of the speed of light
Light Cones Tool used to show causality and the limit of c
Minkowski Space A mathematical trick to make space-time coordinate manipulation a little
easier
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General RelativityThe Motivation
Einstein sought to extend Special Relativity to phenomenaincluding acceleration
He wondered if he could modify Newtonian gravity to fit into SR
But Newtonian gravity was (instantaneous) action-at-a-distanceand it was a force
And Galileo (and before) understood gravity to accelerate alldifferent masses at the same rate (Universality of Free Fall(UFF) 32 ft./sec sec)
Einstein thought if F=ma, and a is a constant when m varies,
then how can F vary identically with m in the case of gravity? Is it really that smart
Is it really that fast, exceeding the speed of light? Newton said if the Sun were to disappear in an instant, the
Earth would immediately fly (tangent) out of its orbit
Is gravity really a classical force?
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General RelativityThe Equivalence Principle
In 1908 Einstein had another break through via one of his thoughtexperiments:
Gravitational mass, the property of an object that couples it with agravitational field, and Inertial mass, the property of an object thathinders its acceleration, were identical to each other
A reference frame in free fall was indistinguishable from areference frame in the void of outer space (or in the absence of agravitational field)
A reference frame, in the void of outer space, being accelerated
up, was indistinguishable from a reference frame at rest on thesurface of the Earth
We can no longer tell the difference between being at rest or beingaccelerated
Einsteins new reference frames were now safe from effects of
acceleration and/or gravity (but they were no longer inertial and theyhad to be small)
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General RelativityIdentifying the Gravitational Field
Next step was to identify the gravitational field through fieldequations (but not as a force)
Since acceleration was motion, and motion affects time and
space, so must gravity affect time and space In 1912 Einstein realized the the Lorentz Transformation will
not apply to this generalized setting
He also realized that the gravitational field equations werebound to be non-linear and that the Equivalence Principlewould only hold locally
He said: If all accelerated systems are equivalent, thenEuclidean geometry cannot hold up in all of them
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General RelativityEinstein Revisits Geometry
With the help of his good friend Grossman,Einstein researches the works of: GaussTheory of surface geometry
Reimann - Manifold geometry
Ricci, Levi-CevitaTensor calculus and differentialgeometry
ChristoffelCovariant differentiation or
coordinate-free differential calculus Einstein realized that the foundations (and
newly developed aspects) of geometry have aphysical significance (in the theory of gravity)
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General RelativitySpace-Time is Curved
The paths of free-bodies define what we mean bystraight in 4-dimensional space-time
And if the observed free-bodies deviate from a
constant velocity, it must mean that space-time itself,in that locality, is non-linear or curved
In any and every locally Lorentz (inertial) frame, thelaws of SR must hold true
The only things which can define the geometricstructure of space-time are the paths of free-bodies(the Earth or an apple)
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General RelativityThe Consequences
Euclidean inertial reference frames are abandoned
Only a locally-inertial coordinate system for extremelysmall, tangent pieces of flat space-time (Minkowski)
can survive as a reference frame Reference frames are now in a free-fall
Objects in a free-fall follow straight lines in 4-dspace-time known as Geodesics
In fact, the shortest distance between two events inspace-time is a geodesic, regardless of how curvedthe space-time is in between these two events
All measurements are done from these lines, but onlyfor small distances from them
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General RelativityUnderstanding Geodesics
A geodesic is the straightest line one can travel through space or across asurface
However in one dimension lower, this straight line (or its shadow) canappear to be curved
On curved or spherical surfaces, geodesics are part of a Great Circle
An airliner that departs from San Francisco for Tokyo, heads northwestin a straight path to get there. When this path is traced-out on a 2-dmap of the Pacific Ocean (or manifold), it appears as an arc or curve
When in an airliner heading west in a straight line through 3-d space,one can see its 2-d shadow deflect north and south across ridges and
valleys on the surface of the Earth; the airliners 3-d path is a geodesic So to, does the Earth travel in a geodesic through 4-d space-time
It appears to travel in a circle (or ellipse) in the lower 3-d space, but in4-d space-time it never completes a circuit because when it returns tothe same spot, one year in the time dimension has expired
All free bodies (unforced) in space travel in geodesics
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General RelativityTensors
Lorentz Transformations can no longer be used
In order to perform measurements now, one needs to paralleltransport vectors from free falling reference frames to otherreference frames, along geodesics
Tensors are the tool of choice to perform these translations Tensors are mathematical machines that take in one or
more vectors (say, tangent to an event in space-time) andput out one or more vectors at another event in space-time
If during translation, the vector(s) gets stretched, re-
directed or torsion is applied (twisted); then the tensor mustoutput this result (linearly) as: another vector, scalar, oreven another tensor
If one pokes a toy gyroscope in a linear fashion (torque); thegyro will eventually re-align itself in a different orientation thanbefore. The new orientation is linearly related to the original
one, but only a tensor can describe how it got there
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General RelativityEinsteins Tensors
Einsteins success in General Relativity was attributable to his use ofvarious tensors to describe his gravitational field equations. In addition tohis own, the Einstein Tensor, he used the following tensors:
Riemann Curvature Tensor, which was made up of:
Ricci Tensorwhich curls or curves up in the presence of energy/matter Weyl Tensor - which is similar to the the electromagnetic-field tensor and as a
result, it can be used in the Maxwell equations as medium to propagategravity as a wave (at the speed of light) across the voids of space. Also, thistensor only curls locally in the presence of a spinning mass (frame-dragging)
Stress-Energy (or Energy-Momentum) Tensor This tensor represents the source of gravity, the distribution and flow of
energy and its momentum
Metric Tensor Einsteins canvas on which these other tensors will interact. It is with this
tensor that the measurements of distance (space-time intervals) and angles
are performed. It also establishes boundary conditions which can be tricky.
l l
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General RelativityGravitational Field Equations
Einsteins Gravitational Field Equation:
The Ricci Tensor
The Ricci Scalar (these two define curvature)
The Metric Tensor Einsteins Cosmological Constant
The Coupling Constant containing Newtons Gravitational Constant G
The Stress-Energy Tensor (this defines matter)
G l l i i
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General RelativityGravitational Field Equations
The left side of equation tells us how space-timecurves (is also the same as the Einstein Tensor)
The right side tells us about the matter present
(in other words)
Matter (energy) tells space-time how much tocurve, and the curvature of space-time tellsmatter how to move
G l R l i i
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General RelativitySolutions to the Field Equations
The Schwarzschild Solution: For concentrated mass, give the radius of a
massive object as it becomes a black hole
The Friedman Solution Gives the solution for a homogenous, isotropic
universe which has an origin as well as a fate
Gravitational Waves Gravitational waves are a prediction just like
Maxwells field equations predictedelectromagnetic waves
G l R l ti it
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General RelativityOther Solutions and Proofs
1. Mercurys perihelion rotates 43 every century
2. Light at every frequency can be bent by gravity
3. Gravitational red shift can occur
4. Clocks run slower in a strong gravitational field
5. Gravitational Mass and Inertial Mass are identical
6. Black Holes exist
7. Gravity has its own form of radiation
8. Spinning bodies can rotate the space-time near them Frame-dragging
9. Spinning bodies can create an electrical like attractionGravito-magnetism
10. Space can stretch during the expansion of the universe
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Thank You
Questions and Answers
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