Theory of Relativity
Relativity can be described using space-time diagrams.
Contrary to popular opinion, Einstein did not
invent relativity. Galileo preceded him.
Aristotle
had proposed that moving objects (on the Earth) had a natural tendency to slow
down and stop. This is shown in the space-time diagram below.
Note the
curved worldline above. This shows a variable velocity, or an
acceleration. Galileo objected to Aristotle's hypothesis, and asked
what happened to an object moving on a moving ship.
Now it is
still moving in its final state. Galileo proposed that it is only relative
velocities that matter. Thus a space-time diagram can be transformed by painting
it on the side of a deck of cards, and then skewing the deck to one side -- but
keeping the edges along a straight line:
Straight
worldlines (unaccelerated particles) remain straight in this process. Thus Newton's First Law is
preserved, and non-accelerated worldlines are special. This Galilean
transformation does not affect the time. Thus two observers moving with respect
to each other can still agree on the time, and thus the distance between two
objects, which is the difference in their positions measured at equal times, can
be defined. This allowed Newton to describe an inverse square law for gravity.
But Galilean transformations do not preserve velocity. Thus the statement "The speed limit is 70 mph" does not make sense -- but don't try this in court. According to relativity, this must be re-expressed as "The magnitude of the relative velocity between your car and the pavement must be less than 70 mph". Relative velocities are OK.
But 200 years after Newton the theory of electromagnetism was developed into
Maxwell's
equations. These equations describe waves with a speed of
1/sqrt(epsilono*muo), where epsilono is the
constant describing the strength of the electrostatic force in a vacuum, and
muo is the constant describing the strength of the magnetic
interaction in a vacuum. This is an absolute velocity -- it is not relative to
anything. The value of the velocity was very close to the measured speed of
light, and when Hertz generated electromagnetic waves (microwaves) in his
laboratory and showed that they could be reflected and refracted just like
light, it became clear that light was just an example of electromagnetic
radiation. Einstein tried to fit the idea of an absolute speed of light into
Newtonian mechanics. He found that the transformation from one reference frame
to another had to affect the time -- the idea of sliding a deck of cards had to
be abandoned. This led to the theory of special relativity. In special
relativity, the velocity of light is special. Anything moving at the speed of
light in one reference frame will move at the speed of light in all
unaccelerated reference frames. Other velocities are not preserved, so you can
still try to get lucky on speeding tickets.
Because the
speed of light is special, space-time diagrams are often drawn in units of
seconds and light-seconds, or years and light-years, so a unit slope [45 degree
angle] corresponds to the speed of light. The set of all light speed world lines
going through an event defines the light cones of that event: the past
light cone and the future light cone. An example of light cones is shown above.
The fancy light picture on the left shows both the past and future light cones
of the event where the two worldlines cross, while the schematic version on the
right is easy to use in more complicated diagrams.
Thus in the situation shown in 3 space-time diagrams below, the central
section shows the worldline of one stationary observer, one observer moving to
the right, and two events on the future light cone on the event where the two
observers' worldlines cross.
The
left-hand section of the figure shows the Galilean transformation into the frame
of reference of the moving observer. The events on the future light cone have
shifted to the left, but they are still at the same time. Since the coordinates
x and t just provide a way of describing space-time, and are not the space-time
themselves, the two events are still on the future light cone. But now slopes of
the light rays have changed, so the speed of light has changed. The Lorentz
transformation appropriate for special relativity is shown on the right hand of
the figure. The events on the future light cone have shifted to the left as
before, but now their times have changed, so the slopes of the light rays do not
change. The speed of light is invariant in Einstein's special relativity.
What is the evidence for the invariance of the speed of light? The hypothesis
that the speed of light is c relative to its source can easily be disproved by
the one-way transmission of light from distant supernovae. When a star explodes
as a supernova, we see light coming from material with a large range of
velocities dv, at least 10,000 km/sec. Because of this range of velocities, the
spectral lines of a supernova are very broad due to the Doppler shift. After
traveling a distance D in time D/c, the arrival time of the light would be
spread out by dt = (dv/c)(D/c).
However,
this DOES NOT happen. For the Crab supernova, with D/c = 6000 years, dv = 10,000
km/sec would give a range of arrival times of 200 years. But the Crab was only
bright for 1 year. For very distant supernovae with D/c = 5 billion years,
modern observations with spectrographs show that the redshifted and blueshifted
light arrives at the same time: within 10 days. This limit on the spread is 5
billion times smaller than the prediction of the "bullet" model of light.
However, light could travel at speed c relative to a medium -- the ether. If
it did, then the rate of a "bouncing photon clock" moving with respect to the
ether
would
depend on the angle between its photon bouncing axis and its velocity. A
stationary bouncing photon clock has a period of P = 2L/c. If it is moving
parallel to its axis at velocity v, and light moves at speed c with respect to
the ether, then the speed relative to the clock when the photon is moving
"upstream" is c-v and the one-way time is L/(c-v). When the photon is moving
"downstream" the speed relative to the clock is c+v so the one-way time is
L/(c+v). The period is the sum of these times:
P(par) = L/(c-v) + L/(c+v) = [2L/c]/(1-v2/c2).
If the clock is moving perpendicular to its axis, the light has to move a distance L sideways and a distance vt "upstream" to keep up with the clock, where t is the one-way time. The total distance traveled is ct, which is the hypotenuse of a right triangle with sides L and vt. Thus the period is given by:
(ct)2 = L2 + (vt)2 so t = L/sqrt(c2-v2) P(perp) = 2t = [2L/c]/sqrt(1-v2/c2).
Thus the ether model predicts that
dP/P = [P(par)-P(perp)]/P = 0.5*v2/c2.
Brillet and Hall (1979, PRL, 42, 549) actually built a bouncing photon
clock (a laser stabilized to a Fabry-Perot etalon) on a rotating table and
compared its rate to an atomic clock (a laser stabilized to a methane line).
The observed
dP/P was (1.5 +/- 2.5)*10-15. For the minimum possible velocity of 30
km/sec, due to the orbit of the Earth around the Sun, this is at least a million
times smaller than the ether model prediction. The 370 km/sec velocity of the
Solar System with respect to the cosmic background radiation gives an ether
model prediction 100 million times larger than the Brillet-Hall limit. For this
velocity even fourth order effects (v4/c4) can be strongly
ruled out.
Michelson and Morley used two bouncing photon clocks at right angles to each other, but without the lasers and counters which didn't exist. This left an L-shaped interferometer. But they were able to show that dP/P was essentially zero instead of the ether model prediction.
The constancy of the speed of light allows the use of radar (RAdio
Detection And Ranging) to measure the position and time of events not on an
observer's worldline. All that we need are a clock and the ability to emit and
detect radar pulses.
If we send
out a radar pulse at time ts, which is reflected at the event E, and the echo
arrives back at time tr, then we know that the light was traveling at c for the
entire round-trip journey out to E and back, so the distance of E is D(E) =
c*(tr-ts)/2. In our frame of reference we are stationary, and light travels at
the same speed coming back from E as it did going out, so the time of the event
E is obtained by averaging the send and receive times, t(E) = (tr+ts)/2.
Armed with radar, we can determine the time of two events on the worldline of
an observer moving with respect to us. We can then compare the time interval we
measure to the time interval measured by the moving observer. Consider the two
observers A and B below.
They both
set their clocks to zero at the event Z where their worldlines cross. A sends
out a radar pulse at the event S at time tA(S) = 1. This is received by observer
B at event R with time tB(R) = k. The factor k depends on the relative velocity
of A and B, but since the light takes some time to travel between A and B, we
know that k will be larger than 1. The notation tA means times determined by A,
while tB are times determined by B. If A sends out a pulse at some other time,
tA = x, it will be received by B at time tB = x*k by the principle of similar
triangles. In particular, if A sends out a pulse at time tA = k it will be
received by B at time tB = k*k. Now consider the radar pulse reflected by B at
event R. It starts at time tB = k. When will A receive it? Since the speed of A
with respect to B is the same as the speed of B with respect to A, this time has
to be tA(T) = k*k. We can now compute the distance of event R from A's worldline
and time of event R according to A, tA(R). These are DA(R) = c(k*k-1)/2 and
tA(R) = (k*k+1)/2. Thus the speed of B according to A is
v = D/t = DA(R)/tA(R) = c(k*k-1)/(k*k+1).
We can solve for k giving
k = sqrt((1+v/c)/(1-v/c))
which is the relativistic Doppler shift formula. But we also find that tA(R) > tB(R), so A says that B's clock is running slow. The amount of this time dilation is
(1+k*k)/(2*k) = 1/sqrt(1-v2/c2).
Thus moving clocks run slow. Note that B will also find that A's clock is running more slowly than his. There is a symmetric disagreement about clock rates.
This slow down factor is exactly the slow down calculated above in the ether
model for a bouncing photon clock moving perpendicular to its bounce axis. The
clock moving parallel to the axis slows down by the same amount under special
relativity because of the Lorentz-Fitzgerald contraction of moving
objects in the direction of motion.
The
space-time diagrams above both show a rod moving past an observer. On the left
the rod is moving, while on the right the same situation is shown in the rod's
frame of reference. The observer moving with respect to the rod makes a radar
determination of its length, as does an observer moving along with the rod. The
observer on the rod sees a length of 5 light-ticks because it takes 10 ticks for
light to make the round trip to the end of the rod and back. The observer moving
with respect to the rod at v = 0.6*c measures only 8 ticks for the round trip
and thus gives the length of the rod as 4 light-ticks. Thus the length of a
moving rod appears to be reduced by a factor of
sqrt(1-v2/c2). Thus length contraction changes P(par) for
the bouncing photon clock to
P(par) = [2L*sqrt(1-v2/c2)/c]/(1-v2/c2)
= [2L/c]/sqrt(1-v2/c2) = P(perp)
so the rate of a bouncing photon clock does not depend on the angle between its velocity and its bouncing axis.
Because the clocks of different observers run at different rates, depending
on their velocities, the time for a given observer is a property of that
observer and his worldline. This time is called the proper time because
it is "owned" by a given particle, not because it is the "correct" time. Proper
time is invariant when changing reference frames because it is the property of a
particle, not of the reference frame or coordinate system. In general, given any
two events A and B with B inside the future light cone of A, there is one
unaccelerated worldline connecting A and B, just as there is one straight line
connecting two points in space. In the frame of reference of the observer
following this unaccelerated worldline, his clock is always stationary, while
clocks following any other worldline from A to B will be moving at least some of
the time. Because moving clocks run slow, these observers will measure a smaller
proper time between events A and B than the unaccelerated observer. Thus the straight worldline between two events has the largest proper
time, and all other curved worldlines connecting the two events have smaller
proper times. This is exactly analogous to the fact that the straight line
between two points has the smallest length of all possible curves between the
points. Thus the "twin paradox" is no more paradoxical than the statement that a
man who drives straight from LA to Las Vegas will cover fewer miles than a man
who drives from LA to Las Vegas via Reno. Now we come to a matter of gravity: how can gravity be an inverse square law
force, when the distance between two objects can not even be defined in
Einstein's special relativity? Special relativity was constructed to satisfy
Maxwell's equations, which replaced the inverse square law electrostatic force
by a set of equations describing the electromagnetic field. So gravity was the
only remaining action-at-a-distance inverse square law force. And gravity has a
unique property; the acceleration due to gravity at a given place and time is
independent of the nature of the body.
The pair of
space-time diagrams above show quintuplets separated at birth. The middle
worldline shows the quint who stays home. The space-time diagram on the left is
done from the point of view of the middle quint. Each dot on a worldline is a
birthday party, so the middle quint is 10 years old when they all rejoin each
other, while the other quints are 6 and 8 years old. The space-time diagram on
the right shows the same events from the point of view of an observer initially
moving with one of the moving quints. When the quints come together their ages
are still 6, 8, 10, 8, and 6 years. Thus the straight worldline between two
events can be found by maximizing the proper time, just as the straight line
between two points can be found by minimizing the length. General Relativity
The
space-time diagram above shows a proton and an antiproton moving under the
influence of an electric field on the left and a gravitational field on the
right. Gravity accelerates all objects equally. This fact was known to Newton,
and tested to an accuracy of 1 part in 100 million by Eotvos. Later work by
Dicke and by Braginsky has improved the accuracy of the test to 1 part in a
trillion. A space mission called STEP
Thus through any event in space-time, in any given direction, there is only
one worldline corresponding to motion solely influenced by gravity. Compare this
to the geometric fact that through any point, in any given direction, there is
only one straight line. We are led to propose that worldlines influenced only by
gravity are really straight worldlines. But how can an accelerating body have a
straight worldline? It all depends on how you measure it. Suppose we plot a
straight line on polar graph paper, and then make a plot of radius vs angle as
shown below?
In the
radius vs theta plot the straight line is curved. I have shown adjacent two
lines dashed and labeled just like the proton and antiproton shown earlier to
emphasize that while there are an infinite number of curved lines in the radius
vs theta plot, there is only one straight line through the initial point with
the initial direction.
Einstein proposed that the effects of gravity (in a small region of
spacetime) are equivalent to the effect of using an accelerated frame of
reference without gravity. As as example, consider the famous "Einstein
elevator" thought experiment. If an elevator far out in space accelerates upward
at 10 meters/second2, it will feel like a downward acceleration of
gravity at 1 g = 10 m/s2. If a clock on the ceiling of the elevator
emits flashes of light f times per second, an observer on the floor will see
them arriving faster than f times per second because of the Doppler shift due to
the acceleration of the elevator during the light transit time.
The
space-time diagram on the left above shows the clock on the elevator ceiling
emitting flashes of light. The light transit time is h/c where h is the height
of the ceiling, and the velocity change is a*h/c so the Doppler shift increases
the rate of flash arrival by a factor of (1+a*h/c2), so the flash
arrival rate is f' = (1+a*h/c2)*f. On the right is the same situation
with stationary clocks in a gravitational field. In order to have the flash
arrival rate faster by a factor of (1+g*h/c2), the clock on the
ceiling must run faster by this factor. In other words, clocks run faster when
they are high up in a gravitational field. This effect has been seen in the
laboratory by Pound and Rebka (1960, PRL, 4, 337) who used the Mossbauer effect
to measure a frequency shift (f'/f -1) = (2.57+/-0.20)*10-15 after
dropping photons a distance of 22.6 meters. The expected shift was
2.46*10-15.
The effect of gravity on clocks was tested to greater precision by Vessot etal (1980, PRL, 45, 2081) who launched a hydrogen maser straight up at 8.5 km/sec, and watched its frequency change as it coasted up to 10,000 km altitude and then fell back to Earth. The frequency shift due to gravity was (f'/f -1) = 4*10-10 at 10,000 km altitude, and the experimental result agreed to within 70 parts per million of this shift.
Because of the gravitational speedup for uphill clocks, an observer moving
between two events can achieve a larger proper time by shifting his worldline
upward in the middle. Going too far upward requires moving so fast that time
dilation due to motion reduces the proper time more than the gravitational
speedup, so there is an optimum curvature to the worldline that maximizes the
proper time.
The
space-time diagram on the left above shows 9 observers moving between two events
with different accelerations. The third from the right has the correct balance
between going uphill to get a faster clock rate and avoiding motion to avoid
time dilation. As a result, this observer has the largest proper time between
the two events. Note that the accelerations are negative for paths that have a
maximum height, so the third worldline from the right is plotted as the third
dot from the left on the chart. The optimum worldline curvature is the
acceleration of gravity, and it is negative because things fall down, not up.
Curved coordinates alone, such as the polar graph, do not provide a
satisfactory model for gravity. Two straight lines through the same point but
with different directions will never cross again, while two worldlines
influenced only by gravity which pass through the same event with different
velocities can cross again. Consider the Galileo spacecraft, which made two
Earth flybys. In between the flybys, Galileo was on an elliptical orbit with a 2
year period. In order to allow "straight" lines to cross multiple times, a
curved space-time is needed. As a familiar example of a curved space, consider
the surface of the Earth and the great circle arc connecting two cities. The
great circle is the shortest distance between two points on the surface of the
Earth, and it is the path followed by airliners.
The great
circle path from Los Angeles (34 N, 118 W) to Tel Aviv (32 N, 35 E) goes all the
way to 70 N latitude.
Plotting latitude vs longitude, as if longitude were time and latitude
position, gives the pseudo-spacetime diagram below.
The two
great circles through Los Angeles, one to Tel Aviv and one to Singapore, are
both "straight" lines, but they intersect in two places. This is impossible in
plane geometry but it does occur in non-Euclidean geometry. The pseudo-spacetime
diagram above is almost identical to a real spacetime diagram for objects moving
in a tunnel drilled through the center of a massive sphere. Gravity produces
oscillatory motions so worldlines for different objects, each influenced only by
gravity, can cross at many events.
Thus the fundamentals of relativity that are important for cosmology are:
The speed of light is a constant independent of the velocity of the source or the observer.
Events that are simultaneous as seen by one observer are generally not simultaneous as seen by other observers, so there can be no absolute time.
Each observer can define his own proper time -- the time measured by a good clock moving along his worldline.
Observers can assign times and positions to events not on their worldlines using radar observations.
Every observer will see his clock running faster than other clocks which are moving with respect to him, and this is a mathematically consistent pattern required by the properties of radar observations.
As a result, the unaccelerated worldline between two events will have the longest proper time of all worldlines connecting these events.
In the presence of gravity, the worldlines of objects accelerated only by gravity have the longest proper times.
Gravity requires that spacetime have a non-Euclidean geometry, and this curvature of spacetime must be created by matter.
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