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Einstein’s Theory of Special Relativity

Lynn Umbarger 04/28/2005

Topics (46 slides)

- Einstein’s 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)

Einstein’s Thought Experiments

At the turn of the 20th century Einstein asked the questions:

- If I dropped a pebble from the window of a train carriage, I would see the stone accelerate toward the moving ground 4 ft. beneath my window in a straight line, then what would the person sitting on the embankment next to the tracks see? Would they not see it travel more than 4 ft. and in a parabolic trajectory? Whose right?
- If I ran at the speed of light and looked into a mirror at my face, would I see my reflection?

What is a Reference Frame?

- A place to perform physical measurements
- Could be thought of as a grid-work of meter-rods and clocks so that trajectories and timings can be performed
- Your reference frame always moves with you
- When someone or something is at rest relative to you, 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 be “inertial” because they are moving at constant velocity; no acceleration, no rotation.

What is a Reference Frame?

- The reference frame O is at rest to the reference frame O’ which is in motion at a velocity of v and in the direction of the x – axis of both reference frames
- Not shown (yet) are the dimensions of time t and t’

The state of physics up to the turn 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 tiny perfect 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

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

- 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 by a 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 everyone provided that they are in uniform motion
- “Absolute Rest” and “Absolute Motion” are meaningless unless they are relative to something (Galilean/Newtonian Relativity)
- He also implied with his rotating bucket experiment, that there existed a frame of reference at absolute rest

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

- Maxwell (1860)
- Unifies electricity and magnetism into “Electromagnetism” 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 be around 186,00 miles per sec (~300,000 km/sec)
- But to what was the speed of light relative?

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

- The “Æther” (ether) was then proposed
- A flexible substance enough to penetrate everything, yet rigid 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 the direction of travel through the ether by a factor of:

The Problem (at the turn of the century)

- There may exist a reference frame at absolute rest, relative to which, light is at a constant velocity of ‘c’
- If motion (mechanics) is relative to particular reference frames, then why isn’t light?

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 inertial reference frames, would be a change in constant velocity: Once you knew one inertial reference frame, then you knew them all
- Therefore, when one changes inertial reference frames, one should measure a different velocity in the speed of light

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

- Dispense with the concept of an ether
- There are no reference frames at absolute rest
- Einstein’s two 1905 postulates:
- All reference frames moving in uniform (non-accelerating), translational (non-rotating), motion; are perfectly valid for performing all types of physics experiments, including experiments with light (optics)
- The speed of light is constant in any reference frame no matter what its speed

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

- Einstein didn’t have a problem with the physical descriptions of matter and radiation (light)
- He did have an issue with how it was measured; in particular he objected to the classical view of what were simultaneous events, or “Simultaneity”
- Einstein’s 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 the same for all of observers

The Effects of Einstein’s Solution

- Clocks run slower in the reference frame of a moving object relative to the clocks of a reference frame at rest to the first
- Clocks slow to ‘zero time’ as its reference frame, relative to one at rest, approaches the the 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 the speed of light

The Effects of Einstein’s Solution

- Time and space are now variable depending on one’s velocity
- Time and space are now connected in a new metric called: Space-Time
- Whereas space and time may vary, intervals of Space-Time are invariant (like light)
- The speed of light has become a cosmic conversion factor

Simultaneity

- To the track-side observer in the middle of the top picture, both lighting strikes occurred simultaneously
- To the observer on the middle of the train, in the middle picture; the front lighting strike occurred first

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

Simultaneity

- In fact, between the on-board observers and the track-side observers, there is a general disagreement as to what time the lighting strikes occurred
- Their clocks are now desynchronized as well

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

Simultaneity

- In order to properly measure something, one must do the measurement at the same time
- Observers in the moving reference frame will not with agree the time, at which, the resting observers performed the measurement
- This is because:
- Synchronization of clocks is frame dependent. Different inertial frame observers will disagree about proper synchronization
- Simultaneity is a frame dependent concept. Different inertial frame observers will disagree about the simultaneity of events separated in space

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

The importance of the relativistic factor (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 space shrinks (contracts)
- =
- Gamma grows to infinity as the v approaches the speed of light, and shrinks to unity when one approaches rest (see next slide)

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

- When we are rest, we are actually traveling in the time dimension at the speed of light
- When we divert that some of that speed over the three dimensions of space, i.e. we go into motion; then we travel through less time
- The amount that time slows is a factor of one’s velocity relative to a reference frame at rest

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

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

t=t’/

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

- When the two reference frames are rest relative to each other, their time dimensions are parallel to each other and perpendicular their respective space dimensions (orthogonal)
- When one of the reference frames goes into motion, it begins to rotate with respect the reference frame at rest while its time dimension must stay orthogonal to its space dimensions
- This causes the measuring rod’s ends to desynchronize with the measuring rod at rest causing a visible foreshortening

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

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

x=x’/

The Lorentz Contraction on Time and Space

- Space-Time Diagrams are a graphical tool to show the effects of the Lorentz Contraction on space and on time. These diagrams represent a frame of reference at rest, there is no motion yet.
- 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)
- Only the x-axis is shown for simplicity; y and z are suppressed, so that all motion continues down the x-axis

The Lorentz Contraction on Time and Space

- 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 toward the c-line, but rather rotating about it.
- 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 slide
- Diagram B shows a faster moving frame of reference
- Rotated more about the c-line

The Lorentz Contraction on Time and Space

- 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 and Space

- In both reference frames is one measuring rod at different times and at rest with respect to its frame (it only 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 and Space

- We wish to compare the length of the moving rod with the one at rest at time ct1
- During this time both the right and left ends of the moving rod will 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

- Because the observer at rest can only measure parallel to his x-axis at time ct, the extent of his measurement can only go to the right end’s trajectory path (Diagram A)
- He then measures from there straight down (or parallel to his time axis) to his x-axis (Diagram B)
- We now see the rod in motion as foreshortened

The Lorentz Contraction on Time and Space

- At ct2, a moment later, the moving rod’s right end aligns with the resting rod’s right end
- But the moving rod is still foreshortened

The Lorentz Contraction on Time and Space

- 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 the requirement to measure parallel to their own x-axises
- Ct3’ sees ct2 as running slower and ct2 sees ct2’ as running slower

The Addition of Velocities

- On board an all-glass bus moving at .75c, a (strong) person throws a ball from the back of the 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 ball was traveling, you will never see an object hit or exceed the speed of light

The Addition of Velocities

- Because of the addition of relativistic velocities, you can only approach the speed of light
- Einstein used the following formula to describe this effect; if v1 was the velocity of the bus and v2 was the velocity of the ball on board, then V would be the observed velocity:

V=

The Addition of Velocities

- 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 be observed as very much out of synch with each other, and more importantly, out synch with the observer’s
- The observer would never agree, given the above conditions, that the ball was traveling as fast as the person that threw it believed it was going

The Addition of Velocities

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

Relative Mass(Einstein runs into trouble)

A

- Say two cars of identical mass, each traveling at .75c, hit each other head on
- According to the classical laws of the conservation of momentum and energy, the wreckage would come to a complete halt in front of an Observer A

Relative Mass(Einstein runs into trouble)

B

A

- Now say an Observer B was traveling along with the left-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 the closing speed of the two vehicles, or at .48c

Relative Mass(Einstein runs into trouble)

B

A

- 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 Einstein’s addition of velocities wrong, or was classical physics off (again) at relativistic speeds?

Relative Mass(Einstein runs into trouble)

B

A

- Einstein posited that because the right-vehicle was the one in relative motion, what if it had gained more mass to push the wreckage passed Observer B, not at .48c, but at .75c?
- But how much more mass would be needed?

Relative Mass(Gamma to the rescue!)

B

A

- 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 high speed electrons

λ

Mass and Energy

- But where did the extra mass come from?
- Einstein assumed it came from the kinetic energy (KE) that the right-vehicle had gained
- Kinetic energy was related to the relativistic mass minus the resting mass, or: KE = m - m’

Mass and Energy

KE = m - m’

- KE is measured in units of joules or kilograms times a meter per second squared
- 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

KE = (m - m’)c

- But when an object is at rest, it must also have a resting energy E, and no relativistic mass m’, or:

E = mc

2

2

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

General Relativity The Motivation

- Einstein sought to extend Special Relativity to phenomena including acceleration
- He wondered if he could modify Newtonian gravity to fit into SR
- But Newtonian gravity was (instantaneous) action-at-a-distance and it was a force
- And Galileo (and before) understood gravity to accelerate all different 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?

General RelativityThe Equivalence Principle

- In 1908 Einstein had another break through via one of his “thought experiments”:
- Gravitational mass, the property of an object that couples it with a gravitational field, and Inertial mass, the property of an object that hinders its acceleration, were identical to each other
- A reference frame in free fall was indistinguishable from a reference frame in the void of outer space (or in the absence of a gravitational field)
- A reference frame, in the void of outer space, being accelerated ‘up’, was indistinguishable from a reference frame at rest on the surface of the Earth
- We can no longer tell the difference between being at rest or being accelerated
- Einstein’s new reference frames were now ‘safe’ from effects of acceleration and/or gravity (but they were no longer inertial and they had to be small)

General RelativityIdentifying the Gravitational Field

- Next step was to identify the gravitational field through field equations (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 were bound to be non-linear and that the Equivalence Principle would only hold locally
- He said: “If all accelerated systems are equivalent, then Euclidean geometry cannot hold up in all of them”

General RelativityEinstein Revisits Geometry

- With the help of his good friend Grossman, Einstein researches the works of:
- Gauss – Theory of surface geometry
- Reimann - Manifold geometry
- Ricci, Levi-Cevita – Tensor calculus and differential geometry
- Christoffel – Covariant differentiation or coordinate-free differential calculus
- Einstein realized that the foundations (and newly developed aspects) of geometry have a physical significance (in the theory of gravity)

General RelativitySpace-Time is Curved

- The paths of free-bodies define what we mean by straight 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, the laws of SR must hold true
- The only things which can define the geometric structure of space-time are the paths of free-bodies (the Earth or an apple)

General RelativityThe Consequences

- Euclidean inertial reference frames are abandoned
- Only a locally-inertial coordinate system for extremely small, 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-d space-time known as “Geodesics”
- In fact, the shortest distance between two events in space-time is a geodesic, regardless of how curved the space-time is in between these two events
- All measurements are done from these lines, but only for small distances from them

General RelativityUnderstanding Geodesics

- A geodesic is the straightest line one can travel through space or across a surface
- However in one dimension lower, this “straight line” (or its shadow) can appear 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 northwest in a 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 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 airliner’s 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 in 4-d space-time it never completes a circuit because when it returns to the “same spot”, one year in the time dimension has expired
- All free bodies (unforced) in space travel in geodesics

General RelativityTensors

- Lorentz Transformations can no longer be used
- In order to perform measurements now, one needs to “parallel transport “ vectors from free falling reference frames to other reference 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) and put 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 must output this result (linearly) as: another vector, scalar, or even another tensor
- If one pokes a toy gyroscope in a linear fashion (torque); the gyro will eventually re-align itself in a different orientation than before. The new orientation is linearly related to the original one, but only a tensor can describe how it got there

General RelativityEinstein’s Tensors

- Einstein’s success in General Relativity was attributable to his use of various tensors to describe his gravitational field equations. In addition to his own, the Einstein Tensor, he used the following tensors:
- Riemann Curvature Tensor, which was made up of:
- Ricci 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 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 tensor 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
- 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 are performed. It also establishes boundary conditions which can be tricky.

General RelativityGravitational Field Equations

Einstein’s Gravitational Field Equation:

- The Ricci Tensor
- The Ricci Scalar (these two define curvature)
- The Metric Tensor
- Einstein’s Cosmological Constant
- The Coupling Constant containing Newton’s Gravitational Constant ‘G’
- The Stress-Energy Tensor (this defines matter)

General RelativityGravitational Field Equations

- The left side of equation tells us how space-time curves (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 to curve, and the curvature of space-time tells matter how to move

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 Maxwell’s “field equations” predicted electromagnetic waves

General RelativityOther Solutions and Proofs

- Mercury’s perihelion rotates 43” every century
- Light at every frequency can be bent by gravity
- Gravitational red shift can occur
- Clocks run slower in a strong gravitational field
- Gravitational Mass and Inertial Mass are identical
- Black Holes exist
- Gravity has it’s own form of radiation
- Spinning bodies can rotate the space-time near them “Frame-dragging”
- Spinning bodies can create an electrical like attraction “Gravito-magnetism”
- Space can stretch during the expansion of the universe

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