- "

- The laws of physics are the same for all non-accelerating observers.
- The speed of light in vacuum is independent of the motion of all observers and sources, and is observed to have the same value.

If you conduct an experiment
in a moving vehicle (provided it is moving at a constant velocity relative
to the ground) the experiment will give exactly the same result as one
conducted in a laboratory at rest relative to the ground. This is why we
can drink a can of soda just as well in a vehicle moving at a constant
velocity as in one that is at rest relative to the ground. The first postulate
says that there is *no* experiment we can do that can determine whether
it is we who are moving, or the ground, or both. The most that any observer
can do is to determine their speed *relative *to something. The earth
goes around the sun at a relative speed of 30 km/s. But this value
is just the speed *relative* to the sun. The earth has also a speed
*relative* to the galactic center. Einstein proposed that there is
no absolute meaning to the phrase: the earth's speed through space.

The second postulate
says that the speed of light is *always* observed to be the same however
we, or the source, might be moving. It is a universal invariant.

The consequence of
Einstein's two postulates are radical: time and space become intertwined
in surprising ways. Events that may be simultaneous for one observer can
occur at different times for another. This leads to length contraction
and time dilation, the slowing down of time in a moving frame. Every observer
has her own personal time, caller **proper time**.
That is the time measured by a clock at the observer's location. Two observers,
initially the same age as given by their proper times, could have different
ages when they met again after traveling along different spacetime paths.

A

Figure 1.

Figure 2.

The angle a light ray (and therefore the cone) makes with the time axis is determined by the speed of light. If we choose our units so that the speed of light, denoted by c, is equal to 1 unit, this angle will be 45 degrees. Because nothing can travel faster than light the wordline of any material object must make an angle with respect to the time axis that, necessarily, is smaller than that of a light ray.

All points that lie
within the forward cone are accessible, in principle, from O by a messenger
from O traveling at a speed less than that of light. These points define
the *future* of O. Similarly, the points within the backward pointing
light cone define the *past* of O. They are all the points from which
a messenger could have reached O without exceeding the speed of light.
*All other points are neither in the future nor the past of event O*!
This is so because it is not possible to send a messenger from O to any
of these points, for the messenger would have to travel at a speed greater
than that of light, which by assumption is impossible.

- ds

If
**ds ^{2} > 0**

the two nearby events are said to separated by a

If
**ds ^{2} <
0**

the two nearby events are said to be separated by a

Finally, **if****
ds ^{2} = 0**

the two events are said to be separated by a

- ds

Now consider setting dt = 0. In this case

- ds

Proper time and proper distance are very important concepts because they give us a way of mapping out spacetime so that all observers throughout spacetime would agree on the mapping. This is particularly important in cosmology, where we have to think about the entire universe in some sensible way.

- ds

- ds

- ds

- ds/c = dt[1

By
continuity, we conclude that, in fact, dt =
ds/c* is the proper time not only for observers at rest in the coordinate
system but also for moving observers,* provided that they move only
along timelike worldlines. This then gives us the rule for computing the
proper time of any observer, whether at rest, or in motion, relative to
our (arbitrarily chosen) coordinate system: the elapsed proper time between
any two nearby events, along the worldline of an observer (regardless of
their state of motion) is just the spacetime interval between the nearby
points divided by the speed of light:

- dt

Observers that are fixed in space (dl = 0) have particularly simple worldlines; this makes it easy to compute their elapsed proper times. For more complicated worldlines the calculation is not quite as easy because we then have to worry about displacements both in coordinate time dt as well as in coordinate space dl.

In characteristic fashion, Einstein turned this observation on its head. He hypothesized that these two kinds of mass are, in fact, one and the same and he sought to deduce the remarkable consequences of this hypothesis. Einstein was very good at using simple, but profound, physical reasoning to get to the heart of things. His discussion of free-falling elevators is a classic example, that we shall now repeat.

You
are in an elevator that is at rest relative to the earth's gravitational
field. The gravitational force on your body, called your **weight**,
pushes you down onto the floor of the elevator. However, because you are
neither going through the floor nor being thrown into the air it follows
that the floor must be pushing up on you with exactly the same force. You
experience this reaction force as your weight.

Suddenly,
disaster strikes; the elevator cables snap. Undaunted by the imminent termination
of your worldline, as a brave seeker of truth, you decide to take stock
of what's happening in the elevator. The most striking thing is that you
have become weightless. *Because of the equivalence of inertial and gravitational
mass all objects fall freely with the same acceleration*. The floor
of the elevator is accelerating towards the center of the earth as fast
as you are. This has two consequences: it cannot impart any force on your
body and you remain at rest relative to the elevator. You take a pen from
your pocket and let go of it. The pen appears to remain suspended in thin
air. Again, this is because the pen is in free fall and is accelerating
towards the earth at the same rate as the elevator.

What
this shows is that gravity can be made to vanish merely by going to a frame
of reference that is in free fall. If gravity can be so easily banished,
Einstein reasoned that what we call the *force* of gravity may be
an illusion; perhaps, gravity is not a force at all, but is somehow related
to free motion in spacetime. Einstein went further: since gravity has been
transformed away within the elevator all experiments conducted therein
should give the same results as experiments carried out in a region far
away from gravitational influences. Einstein summarized the results of
his reasoning in his Principle of Equivalence, which can be stated thus:

(By the way, note the
use of the word *local*. Local means in this context a region of space
that is sufficiently small so that the gravitational field can be considered
uniform. If the region is too large we would notice the effects of the
non-uniform gravitational field. Then the equivalence principle would not
apply. Think about what would happen to two particles in a large falling
elevator: the particles are falling towards the center of the earth along
radial trajectories; therefore, we would see the particles come progressively
closer as the elevator approaches the center of the earth. This would tell
us that we were near a massive object, like the earth, rather than at a
place far away in space. The equivalence principle would not be valid.)

Consider the figure
below. In a rocket in free space (somewhere far away from stars and planets)
a laser beam is emitted from one side of the rocket to a light detector
on the opposite side. The astronaut sees the laser beam travel in a straight
line from one side to the other. Now consider the same rocket in free fall
near a planet. According to the equivalence principle the astronaut will
again see the laser beam travel in a straight line across the cabin. So
far, nothing strange! But now consider the same experiment viewed from
the vantage point of someone who is at rest relative to the planet. The
stationary observer also sees the event: *laser beam hitting the light
detector*. (Events can't be changed just by changing your point of view!)
But, by the time the light beam has crossed the cabin, the rocket and the
light detector will have fallen a small distance. So the observer who is
stationary with respect to the planet will see the laser beam follow a
*curved path*.

Since the stationary observer believes herself to be in a gravitational field (because she feels her weight) she will conclude that gravity bends light. Einstein assumed that light, nonetheless, travels in as straight a line as possible. The fact that light's natural motion is curved could be understood if the spacetime through which it traveled were itself curved. Similar reasoning led Einstein to the further conclusion that clocks in a gravitational field run slower than clocks far from gravitational influences. Einstein sought to explain these effects as consequences of natural (that is, unforced) motion in curved spacetime.

After ten years of arduous intellectual searching, in
1915, Einstein succeeded, finally, in translating his profound physical
intuition about Nature into a rigorous mathematical theory of free motion
in curved spacetimes. Thus was born the **general
theory of relativity. **Einstein's
equations

- G

The German scientist Karl Schwarzchild found the first
exact solution of Einstein's equations just months after the publication
of these equations which, given their complexity, was a pretty impressive
feat. The **metric** (as the expression
for spacetime intervals is called) found by Schwarzchild is

(1) ds^{2}
= c^{2}(1-2MG/c^{2}r)dt^{2}
- dr^{2}/(1-2MG/c^{2}r)
- r^{2}(dq^{2}+sin^{2}qdf^{2}).

It describes the spacetime geometry surrounding a spherically symmetric object of mass M, situated at the spatial coordinate r = 0. This metric gives an excellent description of the spacetime geometry around objects like the sun and the earth, that are, to a good approximation, spherical. This metric is the basis of the three classic tests of Einstein's theory of general relativity: 1) the perihelion advance of Mercury, 2) the bending of starlight by the sun and 3) the slowing down of clocks by gravity. A detailed understanding of the first two phenomena requires familiarity with differential equations, and shall therefore not be considered here. To understand the third requires nothing more than basic algebra; so let's take a look at it.

If we follow our rule and set dr = dq = df = 0, and then divide by c, in equation (1), we obtain

- (2) dt
= (1-2MG/c

- (3)
dt1/dt2
= (1-2MG/c

- (4) rc = 2MG/c

To finish off we shall apply equation (3) to the earth.
We'll set r1 = R, the earth's radius, and r2 = R+h, where the height h
is taken to be much smaller than R. We can get a simpler (approximate)
expression for the ratio of proper times by noting the that for the earth
2MG << c^{2}R. (<< means very
much less than.) With this approximation and using the formula (1+x)^{n}
= 1 + nx, which is valid for x << 1, we obtain, from equation (3),

- (5) dt1/dt2
= (1-MG/c

- (6) 1-dt1/dt2
= (MG/c

- (7) (n2-n1)/n2
= (MG/c

This effect has actually
been observed. In 1960 the scientists R.V. Pound and G.A. Rebka
(Phys. Rev. Letters **4**, 337 (1960)) shot the 14,400 electron-Volt
gamma rays from radioactive iron (Fe^{57})
up the 21.6 meter tower at Harvard University, and tried to absorb the
gamma rays in similar iron nuclei at the top of the tower. But since the
frequency of the gamma rays is predicted to be (slightly) lower than the
natural frequency of the iron the gamma rays were absorbed less efficiently
than normal. Then Pound and Rebka introduced an inspired trick (based on
an effect discovered shortly before by Mossbauer): they changed the natural
frequency of the iron absorber by moving the iron nuclei upwards, at just
the right speed, thus causing a lowering of the natural frequency due to
the Doppler effect. The gamma rays were then readily absorbed by the moving
nuclei. The scientists determined that the frequency of the rising gamma
rays was less than the natural frequency of the stationary iron nuclei,
at the top of the tower, by a fractional amount equal to 2.56 x 10^{-15},
in excellent agreement with the prediction from equation (7) of 2.46 x
10^{-15}. The Pound-Rebka
experiment is one of the most beautiful of 20th century science.