Transcript Einstein's Theory of Special Realtivity

Einstein’s Theory of Special
Relativity
Lynn Umbarger
04/28/2005
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Topics (46 slides)
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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)
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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?
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What is a Reference Frame?
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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.
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What is a Reference Frame?
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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
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Not shown (yet) are the dimensions of time t and t’
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The state of physics up to the
turn of the 20th century
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Aristotle (349 BC)
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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
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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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The state of physics up to the
turn of the 20th century
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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 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
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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
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The state of physics up to the
turn of the 20th century
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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)
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But to what was the speed of light relative?
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The state of physics up to the
turn of the 20th century
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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
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How do we find the existence of the ether?
– In 1887, the Michaelson-Morley experiment had a nullresult
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An explanation
– Lorentz proposed that space shrinks (or contracts) in the
direction of travel through the ether by a factor of:
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The Problem
(at the turn of the century)
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There may exist a reference frame at
absolute rest, relative to which, light is at a
constant velocity of ‘c’
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If motion (mechanics) is relative to
particular reference frames, then why isn’t
light?
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The Problem
(at the turn of the century)
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Newton, who created the Inertial Reference Frame
(constant velocity), said it extended indefinitely,
across the universe
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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
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Therefore, when one changes inertial reference
frames, one should measure a different velocity in the
speed of light
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Einstein’s solution in 1905
(On The Electrodynamics of Moving Bodies)
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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
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Einstein’s solution in 1905
(On The Electrodynamics of Moving Bodies)
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Einstein didn’t have a problem with the physical
descriptions of matter and radiation (light)
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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”
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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
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The Effects of Einstein’s Solution
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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
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The Effects of Einstein’s Solution
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Time and space are now variable depending on
one’s velocity
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Time and space are now connected in a new
metric called: Space-Time
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Whereas space and time may vary, intervals of
Space-Time are invariant (like light)
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The speed of light has become a cosmic
conversion factor
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Simultaneity
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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
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Simultaneity
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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
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Simultaneity
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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
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http://astro.physics.sc.edu/selfpacedunits/Unit56.html
The importance of the relativistic
factor (Gamma)
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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)
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Gamma grows to infinity as the v approaches the
speed of light, and shrinks to unity when one
approaches rest (see next slide)
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The importance of the relativistic
factor (Gamma)
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The Lorentz Transformation
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How does the speed of light
affect our experience with time?
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When we are rest, we are actually traveling in
the time dimension at the speed of light
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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
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The amount that time slows is a factor of one’s
velocity relative to a reference frame at rest
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How does the speed of light
affect our experience with time?
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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’/
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How does the speed of light
affect our experience with space?
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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)
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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
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This causes the measuring rod’s ends to
desynchronize with the measuring rod at rest
causing a visible foreshortening
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How does the speed of light
affect our experience with space?
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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’/
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The Lorentz Contraction on Time and
Space
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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
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The Lorentz Contraction on Time and
Space
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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 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
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Diagram B shows a faster moving frame of reference
– Rotated more about the c-line
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The Lorentz Contraction on Time and
Space
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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 28
The Lorentz Contraction on Time and
Space
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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
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x’-axis
The Lorentz Contraction on Time and Space
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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
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aligned with the left end of the rod at rest
The Lorentz Contraction on Time and Space
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Because the observer at rest can only measure parallel to his xaxis 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
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The Lorentz Contraction on Time and Space
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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
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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 the
requirement to measure parallel to their own x-axises
Ct3’ sees ct2 as running slower and ct2 sees ct2’ as running
slower
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The Addition of Velocities
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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
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The Addition of Velocities
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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=
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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 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
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The Addition of Velocities
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Here’s the space-time diagram representation
of the addition of velocities
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Relative Mass
(Einstein runs into trouble)
A
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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
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Relative Mass
(Einstein runs into trouble)
B
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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
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Relative Mass
(Einstein runs into trouble)
A
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B
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?
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Was Einstein’s addition of velocities wrong, or was
classical physics off (again) at relativistic speeds?
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Relative Mass
(Einstein runs into trouble)
B
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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?
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But how much more mass would be needed?
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Relative Mass
(Gamma to the rescue!)
B
A
λ
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How about using Gamma again?
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Einstein use the equation:
m=
m’
(m = relativistic mass, m’ = resting mass)
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And the right-vehicle then had enough mass to push the
wreckage passed Observer B at .75c
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Although this appears to only be an observational
phenomena, it is actually a measurable fact in particle- 42
colliders with high speed electrons
Mass and Energy
But where did the extra mass come
from?
 Einstein assumed it came from the
kinetic energy (KE) that the rightvehicle had gained
 Kinetic energy was related to the
relativistic mass minus the resting
mass, or:
KE = m - m’
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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
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units
Mass and Energy
KE = (m - m’)c 2
 But when an object is at rest, it must
also have a resting energy E, and no
relativistic mass m’, or:
E = mc 2
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End of Special Relativity
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Other effects of Special Relativity
– Relativistic Energy
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Energy gains at higher velocities
– Relativistic Momentum
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Momentum gains at higher velocities
– Relativistic Aberration
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How the surrounding star field would appear at higher velocities
– Causality
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Cause precedes effect as a function of the speed of light
– Light Cones
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Tool used to show causality and the limit of c
– Minkowski Space
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A mathematical “trick” to make space-time coordinate manipulation a little
easier
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General Relativity
The Motivation
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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?
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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 Relativity
The Equivalence Principle
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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
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acceleration and/or gravity (but they were no longer inertial and they
had to be small)
General Relativity
Identifying the Gravitational Field
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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”
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General Relativity
Einstein Revisits Geometry
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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
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Einstein realized that the foundations (and
newly developed aspects) of geometry have a
physical significance (in the theory of gravity)
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General Relativity
Space-Time is Curved
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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)
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General Relativity
The Consequences
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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
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General Relativity
Understanding Geodesics
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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
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General Relativity
Tensors
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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, redirected 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 54
one, but only a tensor can describe how it got there
General Relativity
Einstein’s Tensors
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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)
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Stress-Energy (or Energy-Momentum) Tensor
– This tensor represents the source of gravity, the distribution and flow of
energy and its momentum
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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
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are performed. It also establishes boundary conditions which can be tricky.
General Relativity
Gravitational Field Equations
Einstein’s Gravitational Field Equation:
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The Ricci Tensor
The Ricci Scalar (these two define curvature)
The Metric Tensor
Einstein’s Cosmological Constant
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The Coupling Constant containing Newton’s Gravitational Constant ‘G’
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The Stress-Energy Tensor (this defines matter)
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General Relativity
Gravitational 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
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General Relativity
Solutions to the Field Equations
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The Schwarzschild Solution:
– For concentrated mass, give the radius of a
massive object as it becomes a black hole
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The Friedman Solution
– Gives the solution for a homogenous, isotropic
universe which has an origin as well as a fate
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Gravitational Waves
– Gravitational waves are a prediction just like
Maxwell’s “field equations” predicted
electromagnetic waves
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General Relativity
Other Solutions and Proofs
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
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 “Framedragging”
Spinning bodies can create an electrical like attraction
“Gravito-magnetism”
Space can stretch during the expansion of the universe
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Thank You
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Questions and Answers
For a copy of this presentation, email:
[email protected]
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