Einstein’s Theory of Special Relativity

1Lynn Umbarger 04/28/2005

Einstein’s Theory of Special Relativity

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Topics (46 slides)Topics (46 slides)

Einstein’s Thought ExperimentsEinstein’s Thought Experiments Reference FramesReference Frames The State of Classical Physics in 1900The State of Classical Physics in 1900 The ProblemThe Problem The SolutionThe Solution The Effects of the SolutionThe Effects of the Solution SimultaneitySimultaneity GammaGamma Time DilationTime Dilation Length ContractionLength Contraction The Lorentz TransformationThe Lorentz Transformation The Addition of VelocitiesThe Addition of Velocities Relativistic MassRelativistic Mass Mass and EnergyMass and Energy General Relativity (13 additional slides, time permitting)General Relativity (13 additional slides, time permitting)

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Einstein’s Thought Einstein’s Thought ExperimentsExperiments

At the turn of the 20th century Einstein asked the At the turn of the 20th century Einstein asked the questions:questions:– If I dropped a pebble from the window of a train If I dropped a pebble from the window of a train

carriage, I would see the stone accelerate toward the carriage, I would see the stone accelerate toward the moving ground 4 ft. beneath my window in a straight moving ground 4 ft. beneath my window in a straight line, then what would the person sitting on the line, then what would the person sitting on the embankment next to the tracks see? Would they not embankment next to the tracks see? Would they not see it travel more than 4 ft. and in a parabolic see it travel more than 4 ft. and in a parabolic trajectory? Whose right?trajectory? Whose right?

– If I ran at the speed of light and looked into a mirror at If I ran at the speed of light and looked into a mirror at my face, would I see my reflection?my face, would I see my reflection?

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What is a Reference Frame?What is a Reference Frame?

A place to perform physical measurementsA place to perform physical measurements Could be thought of as a grid-work of meter-rods and Could be thought of as a grid-work of meter-rods and

clocks so that trajectories and timings can be clocks so that trajectories and timings can be performedperformed

Your reference frame always moves with youYour reference frame always moves with you When someone or something is at rest relative to you, When someone or something is at rest relative to you,

then you are both in the same “inertial” reference then you are both in the same “inertial” reference frameframe

When someone or something is not at rest relative to When someone or something is not at rest relative to you, then they are in a different reference frameyou, then they are in a different reference frame

Reference frames in Special Relativity are said to be Reference frames in Special Relativity are said to be “inertial” because they are moving at constant “inertial” because they are moving at constant velocity; no acceleration, no rotation.velocity; no acceleration, no rotation.

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The reference frame O is at rest to the reference The reference frame O is at rest to the reference frame O’ which is in motion at a velocity of v and in frame O’ which is in motion at a velocity of v and in the direction of the x – axis of both reference framesthe direction of the x – axis of both reference frames

Not shown (yet) are the dimensions of time t and t’Not shown (yet) are the dimensions of time t and t’

What is a Reference Frame?What is a Reference Frame?

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The state of physics up to the The state of physics up to the turn of the 20th centuryturn of the 20th century

Aristotle (349 BC)Aristotle (349 BC)– The universe was geocentricThe universe was geocentric– Everything moved on concentric spheresEverything moved on concentric spheres– The Earth was a very special placeThe Earth was a very special place– Ptolemy (140 AD) added: The planets moved, at times, in Ptolemy (140 AD) added: The planets moved, at times, in

tiny perfect circles to explain retrogradetiny perfect circles to explain retrograde

Copernicus (1543)Copernicus (1543)– The universe was heliocentricThe universe was heliocentric– But everything moved in perfect circlesBut everything moved in perfect circles

Brahe/Kepler (c. 1600)Brahe/Kepler (c. 1600)– The known planets were heliocentricThe known planets were heliocentric– The planets moved in ellipsesThe planets moved in ellipses– The universe was not necessarily a perfect placeThe universe was not necessarily a perfect place

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Galileo (c. 1630)Galileo (c. 1630)– The solar system was heliocentric (got him in trouble)The solar system was heliocentric (got him in trouble)– It was a non-perfect universe (I.e. Sunspots, Jupiter had It was a non-perfect universe (I.e. Sunspots, Jupiter had

moons, Venus was actually a crescent)moons, Venus was actually a crescent)– The natural state of motion is in a straight line until acted upon The natural state of motion is in a straight line until acted upon

by a force (inertia)by a force (inertia)– One cannot tell if they are at rest or if in non-accelerated One cannot tell if they are at rest or if in non-accelerated

motionmotion– There is no absolute rest frame of referenceThere is no absolute rest frame of reference

Newton (c. 1680)Newton (c. 1680)– The laws of motion (mechanics) are the same for everyone The laws of motion (mechanics) are the same for everyone

provided that they are in uniform motionprovided that they are in uniform motion– ““Absolute Rest” and “Absolute Motion” are meaningless unless Absolute Rest” and “Absolute Motion” are meaningless unless

they are relative to something (Galilean/Newtonian Relativity)they are relative to something (Galilean/Newtonian Relativity)– He also implied with his rotating bucket experiment, that there He also implied with his rotating bucket experiment, that there

existed a frame of reference at absolute restexisted a frame of reference at absolute rest


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Maxwell (1860)Maxwell (1860)– Unifies electricity and magnetism into Unifies electricity and magnetism into

“Electromagnetism” with 4 (beautiful) equations“Electromagnetism” with 4 (beautiful) equations

– Electromagnetic waves move at the speed of light Electromagnetic waves move at the speed of light (effectively unifying optics with (effectively unifying optics with electromagnetism)electromagnetism)

– The speed of light was at that time already known The speed of light was at that time already known to be around 186,00 miles per sec (~300,000 to be around 186,00 miles per sec (~300,000 km/sec)km/sec)

But But to whatto what was the speed of light relative? was the speed of light relative?


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The “Æther” (ether) was then proposedThe “Æther” (ether) was then proposed– A flexible substance enough to penetrate A flexible substance enough to penetrate

everything, yet rigid enough to be a medium for the everything, yet rigid enough to be a medium for the high speed of lighthigh speed of light

How do we find the existence of the ether?How do we find the existence of the ether?– In 1887, the Michaelson-Morley experiment had a In 1887, the Michaelson-Morley experiment had a

null-resultnull-result

An explanationAn explanation– Lorentz proposed that space shrinks (or contracts) Lorentz proposed that space shrinks (or contracts)

in the direction of travel through the ether by a in the direction of travel through the ether by a factor of:factor of:


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The Problem The Problem (at the turn of the century)(at the turn of the century)

There may exist a reference frame at There may exist a reference frame at absolute rest, relative to which, light is absolute rest, relative to which, light is at a constant velocity of ‘c’at a constant velocity of ‘c’

If motion (mechanics) is relative to If motion (mechanics) is relative to particular reference frames, then why particular reference frames, then why isn’t light?isn’t light?

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The Problem The Problem (at the turn of the century)(at the turn of the century)

Newton, who created the Inertial Reference Frame (constant Newton, who created the Inertial Reference Frame (constant velocity), said it extended indefinitely, across the universevelocity), said it extended indefinitely, across the universe

The only difference between two different inertial reference The only difference between two different inertial reference frames, would be a change in constant velocity: Once you frames, would be a change in constant velocity: Once you knew one inertial reference frame, then you knew them allknew one inertial reference frame, then you knew them all

Therefore, when one changes inertial reference frames, one Therefore, when one changes inertial reference frames, one should measure a different velocity in the speed of lightshould measure a different velocity in the speed of light

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Dispense with the concept of an etherDispense with the concept of an ether There are no reference frames at absolute There are no reference frames at absolute

restrest Einstein’s two 1905 postulates:Einstein’s two 1905 postulates:

– All reference frames moving in uniform (non-All reference frames moving in uniform (non-accelerating), translational (non-rotating), motion; are accelerating), translational (non-rotating), motion; are perfectly valid for performing all types of physics perfectly valid for performing all types of physics experiments, including experiments with light (optics)experiments, including experiments with light (optics)

– The speed of light is constant in any reference frame no The speed of light is constant in any reference frame no matter what its speedmatter what its speed

Einstein’s solution in 1905Einstein’s solution in 1905(On The Electrodynamics of Moving (On The Electrodynamics of Moving

Bodies)Bodies)

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Einstein’s solution in 1905Einstein’s solution in 1905(On The Electrodynamics of Moving (On The Electrodynamics of Moving

Bodies)Bodies)

Einstein didn’t have a problem with the physical Einstein didn’t have a problem with the physical descriptions of matter and radiation (light)descriptions of matter and radiation (light)

He did have an issue with how it was measured; in He did have an issue with how it was measured; in particular he objected to the classical view of what particular he objected to the classical view of what were simultaneous events, or “Simultaneity”were simultaneous events, or “Simultaneity”

Einstein’s two postulates could be rewritten to say:Einstein’s two postulates could be rewritten to say:– All the laws of physics are the same in every inertial All the laws of physics are the same in every inertial

reference frame (positive statement)reference frame (positive statement)– No test of the laws of physics can distinguish one No test of the laws of physics can distinguish one

inertial reference frame from another (negative inertial reference frame from another (negative statement)statement)

(As a consequence)(As a consequence)– The measured value for the speed of light must be The measured value for the speed of light must be

the same for all of observersthe same for all of observers

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The Effects of Einstein’s The Effects of Einstein’s SolutionSolution

Clocks run slower in the reference Clocks run slower in the reference frame of a moving object relative to the frame of a moving object relative to the clocks of a reference frame at rest to clocks of a reference frame at rest to the firstthe first

Clocks slow to ‘zero time’ as its Clocks slow to ‘zero time’ as its reference frame, relative to one at rest, reference frame, relative to one at rest, approaches the the speed of lightapproaches the the speed of light

The dimensions of an object shrinks (or The dimensions of an object shrinks (or contracts) in its direction of travelcontracts) in its direction of travel

An object flattens to a plane as its An object flattens to a plane as its reference frame, relative to one at rest, reference frame, relative to one at rest, approaches the speed of lightapproaches the speed of light

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The Effects of Einstein’s The Effects of Einstein’s SolutionSolution

Time and space are now variable depending Time and space are now variable depending on one’s velocityon one’s velocity

Time and space are now connected in a new Time and space are now connected in a new metric called: Space-Timemetric called: Space-Time

Whereas space and time may vary, intervals Whereas space and time may vary, intervals of Space-Time are invariant (like light)of Space-Time are invariant (like light)

The speed of light has become a cosmic The speed of light has become a cosmic conversion factorconversion factor

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SimultaneitySimultaneity

To the track-side observer in the middle of the top To the track-side observer in the middle of the top picture, both lighting strikes occurred simultaneouslypicture, both lighting strikes occurred simultaneously

To the observer on the middle of the train, in the middle To the observer on the middle of the train, in the middle picture; the front lighting strike occurred firstpicture; the front lighting strike occurred first

http://astro.physics.sc.edu/selfpacedunits/Unit56.html

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In fact, between the on-board observers and the track-In fact, between the on-board observers and the track-side observers, there is a general disagreement as to side observers, there is a general disagreement as to what time the lighting strikes occurredwhat time the lighting strikes occurred

Their clocks are now desynchronized as wellTheir clocks are now desynchronized as well


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In order to properly measure something, one must do the In order to properly measure something, one must do the measurement at the same timemeasurement at the same time

Observers in the moving reference frame will not with agree Observers in the moving reference frame will not with agree the time, at which, the resting observers performed the the time, at which, the resting observers performed the measurementmeasurement

This is because:This is because:– Synchronization of clocks is frame dependent. Different Synchronization of clocks is frame dependent. Different

inertial frame observers will disagree about proper inertial frame observers will disagree about proper synchronizationsynchronization

– Simultaneity is a frame dependent concept. Different Simultaneity is a frame dependent concept. Different inertial frame observers will disagree about the simultaneity inertial frame observers will disagree about the simultaneity of events separated in spaceof events separated in space



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The importance of the The importance of the relativistic factor (Gamma)relativistic factor (Gamma) Gamma appears as a velocity based variable Gamma appears as a velocity based variable throughout Special Relativity (recall Lorentz)throughout Special Relativity (recall Lorentz) It is the key mathematical solution for telling It is the key mathematical solution for telling

us “by how much” does time slow down us “by how much” does time slow down (dilate) and space shrinks (contracts) (dilate) and space shrinks (contracts)

==

Gamma grows to infinity as the v approaches Gamma grows to infinity as the v approaches the speed of light, and shrinks to unity when the speed of light, and shrinks to unity when one approaches rest (see next slide)one approaches rest (see next slide)

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The importance of the The importance of the relativistic factor (Gamma)relativistic factor (Gamma)

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The Lorentz TransformationThe Lorentz Transformation

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When we are rest, we are actually When we are rest, we are actually traveling in the time dimension at the traveling in the time dimension at the speed of lightspeed of light

When we divert that some of that speed When we divert that some of that speed over the three dimensions of space, i.e. over the three dimensions of space, i.e. we go into motion; then we travel through we go into motion; then we travel through less timeless time

The amount that time slows is a factor of The amount that time slows is a factor of one’s velocity relative to a reference one’s velocity relative to a reference frame at restframe at rest

How does the speed of light How does the speed of light affect our experience with affect our experience with

time?time?

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If t’ is the time in the moving If t’ is the time in the moving reference frame, then the amount reference frame, then the amount by which time appears to dilate is by which time appears to dilate is t, shown by the following formula:t, shown by the following formula:

t=t’/ t=t’/


time?time?

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When the two reference frames are rest When the two reference frames are rest relative to each other, their time dimensions relative to each other, their time dimensions are parallel to each other and perpendicular are parallel to each other and perpendicular their respective space dimensions (orthogonal)their respective space dimensions (orthogonal)

When one of the reference frames goes into When one of the reference frames goes into motion, it begins to rotate with respect the motion, it begins to rotate with respect the reference frame at rest while its time reference frame at rest while its time dimension must stay orthogonal to its space dimension must stay orthogonal to its space dimensionsdimensions

This causes the measuring rod’s ends to This causes the measuring rod’s ends to desynchronize with the measuring rod at rest desynchronize with the measuring rod at rest causing a visible foreshortening causing a visible foreshortening


space?space?

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If x’ is the length of a measuring If x’ is the length of a measuring rod in the moving reference frame, rod in the moving reference frame, then the amount by which length then the amount by which length appears to contract is x, shown by appears to contract is x, shown by the following formula:the following formula:

x=x’/x=x’/


space?space?

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The Lorentz Contraction on Time and The Lorentz Contraction on Time and SpaceSpace

Space-Time Diagrams are a graphical tool to show the Space-Time Diagrams are a graphical tool to show the effects of the Lorentz Contraction on space and on time. effects of the Lorentz Contraction on space and on time. These diagrams represent a frame of reference at rest, These diagrams represent a frame of reference at rest, there is no motion yet.there is no motion yet.

The vertical axis which is time, is labeled ‘ct’ so that the The vertical axis which is time, is labeled ‘ct’ so that the speed of light can be shown as a 45-degree angle (slope=1)speed of light can be shown as a 45-degree angle (slope=1)

Only the x-axis is shown for simplicity; y and z are Only the x-axis is shown for simplicity; y and z are suppressed, so that all motion continues down the x-axissuppressed, so that all motion continues down the x-axis

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Diagram A shows the original reference frame at rest Diagram A shows the original reference frame at rest (un-primed), and a new one in motion (primed)(un-primed), and a new one in motion (primed)– Try not to think of ct’-axis and x’-axis as contracting in Try not to think of ct’-axis and x’-axis as contracting in

toward the c-line, but rather rotating about it.toward the c-line, but rather rotating about it.– Say the that ct’-axis is lifting off the slide towards you as the Say the that ct’-axis is lifting off the slide towards you as the

x’-axis is rotating away from you beneath the plane of the x’-axis is rotating away from you beneath the plane of the slideslide

Diagram B shows a faster moving frame of referenceDiagram B shows a faster moving frame of reference– Rotated more about the c-line Rotated more about the c-line


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This is the Lorentz Transformation at workThis is the Lorentz Transformation at work Say an event (A) like a pulse of light was heading away from the Say an event (A) like a pulse of light was heading away from the

origin of both reference framesorigin of both reference frames Diagram A shows how the un-primed frame would measure itDiagram A shows how the un-primed frame would measure it Diagram B shows how the frame in motion would measure itDiagram B shows how the frame in motion would measure it Important to note: The ct’ and x’-axis’ are still at right-angles to Important to note: The ct’ and x’-axis’ are still at right-angles to

each other; so are the measurement lines out to Event Aeach other; so are the measurement lines out to Event A


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In both reference frames is one measuring rod at In both reference frames is one measuring rod at different times and at rest with respect to its different times and at rest with respect to its frame (it only travels in the time dimension)frame (it only travels in the time dimension)

Even though in B, the reference frame is in motionEven though in B, the reference frame is in motion Note how the rod must always stay parallel to the Note how the rod must always stay parallel to the

x or x’-axisx or x’-axis


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We wish to compare the length of the moving rod with the one We wish to compare the length of the moving rod with the one at rest at time ct1at rest at time ct1

During this time both the right and left ends of the moving rod During this time both the right and left ends of the moving rod will be ‘seen’ at different times in the resting reference framewill be ‘seen’ at different times in the resting reference frame

In B, we catch the moving rod at ct1 when its left end is In B, we catch the moving rod at ct1 when its left end is aligned with the left end of the rod at restaligned with the left end of the rod at rest


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Because the observer at rest can only measure parallel Because the observer at rest can only measure parallel to his x-axis at time ct, the extent of his measurement to his x-axis at time ct, the extent of his measurement can only go to the right end’s trajectory path (Diagram A)can only go to the right end’s trajectory path (Diagram A)

He then measures from there straight down (or parallel He then measures from there straight down (or parallel to his time axis) to his x-axis (Diagram B)to his time axis) to his x-axis (Diagram B)

We now see the rod in motion as foreshortenedWe now see the rod in motion as foreshortened

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At ct2, a moment later, the moving rod’s right At ct2, a moment later, the moving rod’s right end aligns with the resting rod’s right endend aligns with the resting rod’s right end

But the moving rod is still foreshortenedBut the moving rod is still foreshortened


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The same measurement of time shows the aspects of The same measurement of time shows the aspects of Time DilationTime Dilation

Even though the clocks were synchronized at the start Even though the clocks were synchronized at the start they continue to see each other as running slower they continue to see each other as running slower because of the requirement to measure parallel to their because of the requirement to measure parallel to their own x-axisesown x-axises

Ct3’ sees ct2 as running slower and ct2 sees ct2’ as Ct3’ sees ct2 as running slower and ct2 sees ct2’ as running slowerrunning slower


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On board an all-glass bus moving at .75c, a On board an all-glass bus moving at .75c, a (strong) person throws a ball from the back of (strong) person throws a ball from the back of the bus towards the front at a velocity of .75c the bus towards the front at a velocity of .75c relative to the busrelative to the bus

How fast would this ball appear to go relative How fast would this ball appear to go relative to an observer at the bus stop (at rest)?to an observer at the bus stop (at rest)?

Would they see it travel at 1.5c?Would they see it travel at 1.5c? No, actually they would see it move at 24/25c No, actually they would see it move at 24/25c

(or .96c)(or .96c) In fact, no matter how fast the bus or the ball In fact, no matter how fast the bus or the ball

was traveling, you will never see an object hit was traveling, you will never see an object hit or exceed the speed of lightor exceed the speed of light

The Addition of VelocitiesThe Addition of Velocities

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Because of the addition of relativistic Because of the addition of relativistic velocities, you can only approach the velocities, you can only approach the speed of lightspeed of light

Einstein used the following formula to Einstein used the following formula to describe this effect; if v1 was the velocity describe this effect; if v1 was the velocity of the bus and v2 was the velocity of the of the bus and v2 was the velocity of the ball on board, then V would be the ball on board, then V would be the observed velocity:observed velocity:

V=V=


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The reason for what the resting observer The reason for what the resting observer

saw:saw:

– The observer would see a foreshortened bus The observer would see a foreshortened bus

– The clocks at the back and front of the bus would The clocks at the back and front of the bus would

be observed as very much out of synch with each be observed as very much out of synch with each

other, and more importantly, out synch with the other, and more importantly, out synch with the

observer’sobserver’s

– The observer would never agree, given the above The observer would never agree, given the above

conditions, that the ball was traveling as fast as conditions, that the ball was traveling as fast as

the person that threw it believed it was going the person that threw it believed it was going


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Here’s the space-time diagram Here’s the space-time diagram representation of the addition of representation of the addition of velocitiesvelocities


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Say two cars of identical mass, each Say two cars of identical mass, each traveling at .75c, hit each other head ontraveling at .75c, hit each other head on

According to the classical laws of the According to the classical laws of the conservation of momentum and energy, conservation of momentum and energy, the wreckage would come to a complete the wreckage would come to a complete halt in front of an Observer Ahalt in front of an Observer A

A

Relative MassRelative Mass(Einstein runs into trouble)(Einstein runs into trouble)

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Now say an Observer B was traveling along with Now say an Observer B was traveling along with the left-vehicle (in its inertial rest frame) the left-vehicle (in its inertial rest frame)

He would see the right-vehicle coming at him at a He would see the right-vehicle coming at him at a speed of .96c (Addition of Velocities)speed of .96c (Addition of Velocities)

At the moment of impact one would assume that At the moment of impact one would assume that Observer B would see the wreckage go by at half Observer B would see the wreckage go by at half the closing speed of the two vehicles, or at .48cthe closing speed of the two vehicles, or at .48c

AB


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How could Observer B pass the wreckage at .48c How could Observer B pass the wreckage at .48c

and yet pass Observer A at .75c when Observer A and yet pass Observer A at .75c when Observer A

was at rest to the wreckage?was at rest to the wreckage?

Was Einstein’s addition of velocities wrong, or Was Einstein’s addition of velocities wrong, or

was classical physics off (again) at relativistic was classical physics off (again) at relativistic

speeds?speeds?

A B


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Einstein posited that because the right-vehicle Einstein posited that because the right-vehicle

was the one in relative motion, what if it had was the one in relative motion, what if it had

gained more mass to push the wreckage passed gained more mass to push the wreckage passed

Observer B, not at .48c, but at .75c?Observer B, not at .48c, but at .75c?

But how much more mass would be needed?But how much more mass would be needed?

AB


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How about using Gamma again?How about using Gamma again?

Einstein use the equation: Einstein use the equation: mm== m’m’

(m = relativistic mass, m’ = resting mass)(m = relativistic mass, m’ = resting mass)

And the right-vehicle then had enough mass to And the right-vehicle then had enough mass to

push the wreckage passed Observer B at .75cpush the wreckage passed Observer B at .75c

Although this appears to only be an observational Although this appears to only be an observational

phenomena, it is actually a measurable fact in phenomena, it is actually a measurable fact in

particle-colliders with high speed electronsparticle-colliders with high speed electrons

AB

λλ

Relative MassRelative Mass(Gamma to the rescue!)(Gamma to the rescue!)

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Mass and EnergyMass and Energy

But where did the extra mass But where did the extra mass come from?come from?

Einstein assumed it came from the Einstein assumed it came from the kinetic energy (KE) that the right-kinetic energy (KE) that the right-vehicle had gainedvehicle had gained

Kinetic energy was related to the Kinetic energy was related to the relativistic mass minus the resting relativistic mass minus the resting mass, or: KE = m - m’mass, or: KE = m - m’

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KE = m - m’KE = m - m’ KE is measured in units of joules or KE is measured in units of joules or

kilograms times a meter per second kilograms times a meter per second squaredsquared

But seconds (time) and meters (length) But seconds (time) and meters (length) get varied at relativistic speedsget varied at relativistic speeds

Use the speed of light c, as a conversion Use the speed of light c, as a conversion factor to get rid of these unitsfactor to get rid of these units


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KE = (m - m’)cKE = (m - m’)c But when an object is at rest, it must also have a But when an object is at rest, it must also have a

resting energy E, and no relativistic mass m’, or:resting energy E, and no relativistic mass m’, or:

E = mcE = mc

2

2

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End of Special RelativityEnd of Special Relativity

Other effects of Special RelativityOther effects of Special Relativity– Relativistic EnergyRelativistic Energy

Energy gains at higher velocitiesEnergy gains at higher velocities– Relativistic MomentumRelativistic Momentum

Momentum gains at higher velocitiesMomentum gains at higher velocities– Relativistic AberrationRelativistic Aberration

How the surrounding star field would appear at higher velocitiesHow the surrounding star field would appear at higher velocities– CausalityCausality

Cause precedes effect as a function of the speed of lightCause precedes effect as a function of the speed of light– Light ConesLight Cones

Tool used to show causality and the limit of cTool used to show causality and the limit of c– Minkowski SpaceMinkowski Space

A mathematical “trick” to make space-time coordinate A mathematical “trick” to make space-time coordinate manipulation a little easiermanipulation a little easier

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General Relativity General Relativity The MotivationThe Motivation

Einstein sought to extend Special Relativity to phenomena Einstein sought to extend Special Relativity to phenomena including accelerationincluding acceleration

He wondered if he could modify Newtonian gravity to fit into SRHe wondered if he could modify Newtonian gravity to fit into SR But Newtonian gravity was (instantaneous) action-at-a-distance But Newtonian gravity was (instantaneous) action-at-a-distance

and it was a and it was a forceforce And Galileo (and before) understood gravity to accelerate all And Galileo (and before) understood gravity to accelerate all

different masses at the same rate (Universality of Free Fall (UFF) different masses at the same rate (Universality of Free Fall (UFF) 32 ft./sec sec)32 ft./sec sec)

Einstein thought if F=ma, and ‘a’ is a constant when ‘m’ varies, 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?then how can ‘F’ vary identically with ‘m’ in the case of gravity?– Is it really that smartIs it really that smart– Is it really that fast, exceeding the speed of light?Is it really that fast, exceeding the speed of light?

Newton said if the Sun were to disappear in an instant, the Earth Newton said if the Sun were to disappear in an instant, the Earth would immediately fly (tangent) out of its orbitwould immediately fly (tangent) out of its orbit

– Is gravity really a classical force?Is gravity really a classical force?

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General RelativityGeneral RelativityThe Equivalence PrincipleThe Equivalence Principle

In 1908 Einstein had another break through via one of his “thought In 1908 Einstein had another break through via one of his “thought experiments”:experiments”:– Gravitational mass, the property of an object that couples it with a Gravitational mass, the property of an object that couples it with a

gravitational field, and Inertial mass, the property of an object that gravitational field, and Inertial mass, the property of an object that hinders its acceleration, were identical to each otherhinders its acceleration, were identical to each other

– A reference frame in free fall was indistinguishable from a A reference frame in free fall was indistinguishable from a reference frame in the void of outer space (or in the absence of a reference frame in the void of outer space (or in the absence of a gravitational field)gravitational field)

– A reference frame, in the void of outer space, being accelerated A reference frame, in the void of outer space, being accelerated ‘up’, was indistinguishable from a reference frame at rest on the ‘up’, was indistinguishable from a reference frame at rest on the surface of the Earthsurface of the Earth

We can no longer tell the difference between being at rest or being We can no longer tell the difference between being at rest or being acceleratedaccelerated

Einstein’s new reference frames were now ‘safe’ from effects of Einstein’s new reference frames were now ‘safe’ from effects of acceleration and/or gravity (but they were no longer inertial and they acceleration and/or gravity (but they were no longer inertial and they had to be small)had to be small)

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General RelativityGeneral RelativityIdentifying the Gravitational Identifying the Gravitational

FieldField

Next step was to identify the gravitational field through Next step was to identify the gravitational field through field equations (but not as a force)field equations (but not as a force)

Since acceleration was motion, and motion affects time Since acceleration was motion, and motion affects time and space, so must gravity affect time and spaceand space, so must gravity affect time and space

In 1912 Einstein realized the the Lorentz Transformation In 1912 Einstein realized the the Lorentz Transformation will not apply to this generalized settingwill not apply to this generalized setting

He also realized that the gravitational field equations He also realized that the gravitational field equations were bound to be non-linear and that the Equivalence were bound to be non-linear and that the Equivalence Principle would only hold locallyPrinciple would only hold locally

He said: “If all accelerated systems are equivalent, then He said: “If all accelerated systems are equivalent, then Euclidean geometry cannot hold up in all of them”Euclidean geometry cannot hold up in all of them”

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General RelativityGeneral RelativityEinstein Revisits GeometryEinstein Revisits Geometry

With the help of his good friend Grossman, With the help of his good friend Grossman, Einstein researches the works of:Einstein researches the works of:– Gauss – Theory of surface geometryGauss – Theory of surface geometry– Reimann - Manifold geometry Reimann - Manifold geometry – Ricci, Levi-Cevita – Tensor calculus and Ricci, Levi-Cevita – Tensor calculus and

differential geometrydifferential geometry– Christoffel – Covariant differentiation or Christoffel – Covariant differentiation or

coordinate-free differential calculuscoordinate-free differential calculus Einstein realized that the foundations (and Einstein realized that the foundations (and

newly developed aspects) of geometry have newly developed aspects) of geometry have a physical significance (in the theory of a physical significance (in the theory of gravity)gravity)

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General RelativityGeneral RelativitySpace-Time is CurvedSpace-Time is Curved

The paths of free-bodies define what we mean The paths of free-bodies define what we mean by straight in 4-dimensional space-timeby straight in 4-dimensional space-time

And if the observed free-bodies deviate from a And if the observed free-bodies deviate from a constant velocity, it must mean that space-time constant velocity, it must mean that space-time itself, in that locality, is non-linear or curveditself, in that locality, is non-linear or curved

In any and every locally Lorentz (inertial) frame, In any and every locally Lorentz (inertial) frame, the laws of SR must hold truethe laws of SR must hold true

The only things which can define the geometric The only things which can define the geometric structure of space-time are the paths of free-structure of space-time are the paths of free-bodies (the Earth or an apple)bodies (the Earth or an apple)

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General RelativityGeneral RelativityThe ConsequencesThe Consequences

Euclidean inertial reference frames are abandonedEuclidean inertial reference frames are abandoned Only a locally-inertial coordinate system for Only a locally-inertial coordinate system for

extremely small, tangent pieces of flat space-time extremely small, tangent pieces of flat space-time (Minkowski) can survive as a reference frame(Minkowski) can survive as a reference frame

Reference frames are now in a free-fallReference frames are now in a free-fall Objects in a free-fall follow straight lines in 4-d Objects in a free-fall follow straight lines in 4-d

space-time known as “Geodesics”space-time known as “Geodesics” In fact, the shortest distance between two events in In fact, the shortest distance between two events in

space-time is a geodesic, regardless of how curved space-time is a geodesic, regardless of how curved the space-time is in between these two eventsthe space-time is in between these two events

All measurements are done from these lines, but All measurements are done from these lines, but only for small distances from themonly for small distances from them

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General RelativityGeneral RelativityUnderstanding GeodesicsUnderstanding Geodesics

A geodesic is the straightest line one can travel through space or across a A geodesic is the straightest line one can travel through space or across a surfacesurface

However in one dimension lower, this “straight line” (or its shadow) can appear However in one dimension lower, this “straight line” (or its shadow) can appear to be curvedto be curved

On curved or spherical surfaces, geodesics are part of a “Great Circle”On curved or spherical surfaces, geodesics are part of a “Great Circle”– An airliner that departs from San Francisco for Tokyo, heads northwest in a An airliner that departs from San Francisco for Tokyo, heads northwest in a

straight path to get there. When this path is traced-out on a 2-d map of the straight path to get there. When this path is traced-out on a 2-d map of the Pacific Ocean (or manifold), it appears as an arc or curvePacific 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 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 can see its 2-d shadow deflect north and south across ridges and valleys on the surface of the Earth; the airliner’s 3-d path is a geodesicthe surface of the Earth; the airliner’s 3-d path is a geodesic

So to, does the Earth travel in a geodesic through 4-d space-timeSo 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 in 4-d It appears to travel in a circle (or ellipse) in the lower 3-d space, but in 4-d

space-time it never completes a circuit because when it returns to the space-time it never completes a circuit because when it returns to the “same spot”, one year in the time dimension has expired“same spot”, one year in the time dimension has expired

All free bodies (unforced) in space travel in geodesicsAll free bodies (unforced) in space travel in geodesics

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General RelativityGeneral RelativityTensorsTensors

Lorentz Transformations can no longer be usedLorentz Transformations can no longer be used In order to perform measurements now, one needs to “parallel In order to perform measurements now, one needs to “parallel

transport “ vectors from free falling reference frames to other transport “ vectors from free falling reference frames to other reference frames, along geodesicsreference frames, along geodesics

Tensors are the tool of choice to perform these translationsTensors are the tool of choice to perform these translations– Tensors are mathematical “machines” that take in one or more Tensors are mathematical “machines” that take in one or more

vectors (say, tangent to an event in space-time) and put out vectors (say, tangent to an event in space-time) and put out one or more vectors at another event in space-timeone or more vectors at another event in space-time

– If during translation, the vector(s) gets stretched, re-directed If during translation, the vector(s) gets stretched, re-directed or torsion is applied (twisted); then the tensor must output this or torsion is applied (twisted); then the tensor must output this result (linearly) as: another vector, scalar, or even another result (linearly) as: another vector, scalar, or even another tensortensor

If one pokes a toy gyroscope in a linear fashion (torque); the gyro If one pokes a toy gyroscope in a linear fashion (torque); the gyro will eventually re-align itself in a different orientation than before. will eventually re-align itself in a different orientation than before. The new orientation is linearly related to the original one, but only The new orientation is linearly related to the original one, but only a tensor can describe how it got therea tensor can describe how it got there

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General RelativityGeneral RelativityEinstein’s TensorsEinstein’s Tensors

Einstein’s success in General Relativity was attributable to his use of Einstein’s success in General Relativity was attributable to his use of various tensors to describe his gravitational field equations. In addition to various tensors to describe his gravitational field equations. In addition to his own, the Einstein Tensor, he used the following tensors:his own, the Einstein Tensor, he used the following tensors:

Riemann Curvature Tensor, which was made up of:Riemann Curvature Tensor, which was made up of:– Ricci Tensor – which curls or curves up in the presence of energy/matterRicci Tensor – which curls or curves up in the presence of energy/matter– Weyl Tensor - which is similar to the the electromagnetic-field tensor and as a 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 propagate result, it can be used in the Maxwell equations as “medium” to propagate gravity as a wave (at the speed of light) across the voids of space. Also, this gravity as a wave (at the speed of light) across the voids of space. Also, this tensor only curls locally in the presence of a spinning mass (frame-dragging)tensor only curls locally in the presence of a spinning mass (frame-dragging)

Stress-Energy (or Energy-Momentum) TensorStress-Energy (or Energy-Momentum) Tensor– This tensor represents the source of gravity, the distribution and flow of This tensor represents the source of gravity, the distribution and flow of

energy and its momentumenergy and its momentum Metric TensorMetric Tensor

– Einstein’s “canvas” on which these other tensors will interact. It is with this Einstein’s “canvas” on which these other tensors will interact. It is with this tensor that the measurements of distance (space-time intervals) and angles tensor that the measurements of distance (space-time intervals) and angles are performed. It also establishes boundary conditions which can be tricky.are performed. It also establishes boundary conditions which can be tricky.

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General RelativityGeneral RelativityGravitational Field EquationsGravitational Field Equations

Einstein’s Gravitational Field Equation:Einstein’s Gravitational Field Equation:

The Ricci TensorThe Ricci Tensor The Ricci Scalar (these two define curvature)The Ricci Scalar (these two define curvature) The Metric TensorThe Metric Tensor Einstein’s Cosmological ConstantEinstein’s Cosmological Constant The Coupling Constant containing Newton’s Gravitational Constant ‘G’The Coupling Constant containing Newton’s Gravitational Constant ‘G’

The Stress-Energy Tensor (this defines matter)The Stress-Energy Tensor (this defines matter)

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General RelativityGeneral RelativityGravitational Field EquationsGravitational Field Equations

The left side of equation tells us how space-The left side of equation tells us how space-time curves (is also the same as the Einstein time curves (is also the same as the Einstein Tensor)Tensor)

The right side tells us about the matter presentThe right side tells us about the matter present(in other words)(in other words) Matter (energy) tells space-time how much to Matter (energy) tells space-time how much to

curve, and the curvature of space-time tells curve, and the curvature of space-time tells matter how to movematter how to move

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General RelativityGeneral RelativitySolutions to the Field Solutions to the Field

EquationsEquations

The Schwarzschild Solution:The Schwarzschild Solution:– For concentrated mass, give the radius of a For concentrated mass, give the radius of a

massive object as it becomes a black holemassive object as it becomes a black hole The Friedman SolutionThe Friedman Solution

– Gives the solution for a homogenous, Gives the solution for a homogenous, isotropic universe which has an origin as well isotropic universe which has an origin as well as a fateas a fate

Gravitational WavesGravitational Waves– Gravitational waves are a prediction just like Gravitational waves are a prediction just like

Maxwell’s “field equations” predicted Maxwell’s “field equations” predicted electromagnetic waveselectromagnetic waves

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General RelativityGeneral RelativityOther Solutions and ProofsOther Solutions and Proofs

1.1. Mercury’s perihelion rotates 43” every centuryMercury’s perihelion rotates 43” every century2.2. Light at every frequency can be bent by gravityLight at every frequency can be bent by gravity3.3. Gravitational red shift can occurGravitational red shift can occur4.4. Clocks run slower in a strong gravitational fieldClocks run slower in a strong gravitational field5.5. Gravitational Mass and Inertial Mass are identicalGravitational Mass and Inertial Mass are identical6.6. Black Holes existBlack Holes exist7.7. Gravity has it’s own form of radiationGravity has it’s own form of radiation8.8. Spinning bodies can rotate the space-time near them Spinning bodies can rotate the space-time near them

“Frame-dragging”“Frame-dragging”9.9. Spinning bodies can create an electrical like attraction Spinning bodies can create an electrical like attraction

“Gravito-magnetism”“Gravito-magnetism”10.10. Space can stretch during the expansion of the Space can stretch during the expansion of the

universeuniverse

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Thank YouThank You

Questions and AnswersQuestions and Answers

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Einstein’s Theory of Special Relativity

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