Einstein’s Theory of relativity
The theory of relativity has two parts. The Special Theory of
Relativity relates the measurements of the same events made
by two different observers moving relative to each other with
constant velocity (inertial observers) . The General Theory of
Relativity includes the effects of gravity.
The Special Theory of Relativity
This is a very successful, well verified theory, discovered by
Einstein based upon what were called the two postulates.
• The Postulates of Einstein’s Special Relativity:
(1) The laws of physics are the same for every inertial (nonaccelerating) observer.
(2) The speed of light, c = 300, 000 km/sec, is absolute. That
is, the speed of a pulse of light measured by an observer is
independent of the speed of the source relative to the observer.
• Galilean Relativity:
Galileo would agree with the first postulate, but not with the
second. Say we have two observers, O is on the ground, and
0
O is riding on a car moving with a speed v relative to O. Ob0
server O would say that observer O is moving toward the left
with the same speed v. An object, maybe a ball, is moving
with a speed u relative to observer O as shown below.
u
u'
O'
O
v
0
The speed of the ball relative to O , which we call u0 , is found
1
in classical (Galilean) relativity from the formula
u0 = u − v
This is the Galilean (everyday, common sense) speed subtraction formula. So if the object is a pulse of light, u = c, classical
physics says c0 = c − v, which is not equal to c, and disagrees
with the second postulate. It turns out that this formula is
valid only if both v and u are much smaller than c.
• Einstein’s Speed Subtraction formula:
Einstein found a different formula,
u0 =
u−v
.
1 − uv
c2
If u and v are much smaller than c, then uv/c2 is much smaller
than 1, and the denominator becomes approximately equal to
1, and then this formula gives that u0 ≈ u − v, in agreement
with the classical result. However, if the object is a pulse of
light, so u = c, then u0 = c also, no matter what v is. So both
observers measure the same speed c. The speed of light is very
special! Every observer measures the same speed c, no matter
how they are moving, even if they are accelerating, and even
also if they are in gravitational fields. It is absolute. But how is
this possible? After all, measuring the speed of something just
involves measuring the distance traveled and the time elapsed,
and then speed = distance/time
• Time contraction and length contraction:
The answer is that the two observers do not agree about measurements of lengths and time intervals. Say there are two
events that happen at the same place in the moving frame of
2
0
reference. Say that the traveler O goes to bed in his vehicle
(event 1), and gets up (vent 2) a time ∆t0 later as measured
by his clocks.
O' measures a length L' for the trip, L'= vΔt'
O' measures a time interval Δt' from event 1 to 2
event 2
event 1
O' gets up here
O' goes to bed here
v
v
O
O measures a time interval Δt from event 1 to 2
lenght L of trip as measured by O, L= vΔt
Observer O would measure a different time ∆t with his clocks,
where
r
v2
0
∆t = ∆t 1 − 2 .
c
The square root is a number less than 1, so ∆t0 < ∆t. So, say
an astronaut goes on trip and returns to Earth. The Earth
time ∆t > ∆t0 . If v is much less than c, the square root is
almost equal to 1, and the two times are essentially the same.
But if v = 0.99c, then ∆t0 < 0.141∆t, or ∆t > 7.1∆t0 , and
if v = 0.999999c, then ∆t0 < 0.0014∆t, or ∆t > 707∆t0 . So
an astronaut goes on a trip that to him lasts for ∆t0 = 1 year.
When he returns to Earth, ∆t = 707, and 707 Earth years have
elapsed and all his friends have long since died! This amazing
result has not been verified with astronauts, but it has been
3
verified with radioactive particles, than seem to have a much
longer half life when they are moving than their sisters that
have remained at rest.
How does the astronaut explain that he measured a shorter
time? To him the distance traveled is shorter, because
r
v2
0
0
0
L = v∆t , while L = v∆t, so L = L 1 − 2 , and L0 < L.
c
So, to the astronaut the time was shorter because the length
of the trip was shorter.
Length and time measurements are relative.
• Simultaneity is relative:
Observers moving relative to each other disagree on whether or
not two events are simultaneous. Observer O is on the ground
0
and observer O is standing at the middle of a train moving
0
with speed v relative to O. As O passes in front of O two
0
0
lightning bolts hit at points A and B on the train, and also
make burns marks on points A and B on the track, as shown
in the figure to the left.
The flashes of light reach O at the same time. Observer O then
verifies that the burn marks at points A and B were at the same
distance from the point where he was standing. Therefore,
since the distances were the same, the two flashes must have
been simultaneous, as far as he is concerned.
4
0
But as the flashes traveled toward O, observer O was moving
toward point B, so the flash coming from the front of the train
would reach him before the other flash coming from the rear.
0
Then O would verify that the burn marks at the front and rear
of the train are at the same distance from him, since he was
0
standing in the middle of the train. As far as O is concerned,
the burn at the front of the train must have happened before
the burn at the rear because, as far as he was concerned, the
flashes each had to travel half the length of the train, but the
flash from the front arrived first, so it must have happened
earlier than the one at the rear. Both observers are right!
Simultaneity is relative.
• Energy and momentum:
In classical physics a particle of mass m moving with speed v
has
kinetic energy KE =
1
mv 2 , and momentum p = mv.
2
But theses formulas are only approximately correct if v c.
The correct formulas at all speeds are
m
KE = q
1−
v2
c2
c2 − mc2 , and p = q
m
1−
v.
v2
c2
A mass at rest, not moving, has energy E = mc2 , where
c = 3 × 108 m/s is the speed of light. This means that energy and mass are equivalent. In a process, we may start with
an initial mass mi , and end up with a smaller final mass mf ,
then the difference mi − mf has been converted to energy, and
the amount of energy release is E = (mi − mf )c2 . If we could
5
do this an annihilate 1 kg of mass the energy released would be
E = 9 × 1016 Joules, which is enough to run a 1,000 Megawatt
plant for 9 × 107 seconds, which is almost three years. Note
that a coal fired 1,000 Megawatt electric plant consumes around
10,000 tons of coal per day!! In chemical reactions the amounts
of mass that are annihilated are immeasurably small, the energy released is reasonable, and it was thought that mass was
conserved. Now we know that energy, not mas, is really conserved, but only with nuclear reactions did this become clear.
The General Theory of Relativity
Objects under gravity fall with the same acceleration independently of their mass. When a car accelerates forward with an
acceleration a, all objects in the car that are not attached to
anything, as measured from the frame of reference of the car,
accelerate backwards with an acceleration a, independently of
their mass. This fact led Einstein to his principle of equivalence, which he used to arrive at a theory of gravity called
general relativity. This is a very mathematical theory in which
space is not nothing, but has properties that determine how
objects will move.
One can say that space-time is curved. In a gravitationally
curved space-time objects move in ”a straight line, a geodesic”.
The generalization of a straight line in a flat space is a geodesic.
6
This is a curve that is generated by translating a segment parallel to itself. On a sphere this follows a great circle. In flat
geometry the angles inside a geodesic triangle (made of straight
lines) always add up to 180◦ . On a sphere, which is a space of
positive curvature, the angles in a geodesic triangle always add
up to more than 180◦ , and two geodesics starting from a point
eventually come together.
On a saddle shaped surface, which is a surface of negative curvature, the angles in a geodesic triangle always add up to less
than 180◦ , and two geodesics starting from a point eventually
move further apart. All this is expressed in the mathematically
complicated language of differential geometry.
Some of the consequences are that clocks placed in a stronger
gravitational field, such as at the ground floor of a building,
run slower than clocks placed in a weaker gravitational field,
such as at the top of the building.
A dramatic prediction of General Relativity is the possibility of
black holes. This is a mathematical solution of Einstein’s equations discovered by Karl Schwarzschild in 1916 for the gravitational field caused by a point mass M . It was not really
7
understood because the solution had a ”singularity” at a particular distance from the mass, now called the Schwarzschild
radius Rs
2GM
Rs =
,
c2
which for the mass of the Sun would be only 2.95 km. The
Schwarzschild radius was understood in 1958. David Finkelstein and Martin Kruskal explained that is is the distance of
the ”event horizon”. Any object that approaches the mass M
to a distance smaller than Rs will never be able to get out
again, and will ultimately be driven toward the mass M where
the object will be crushed by ”infinite” gravitational forces.
This represents a challenge that seems to require that gravity
and quantum physics be reconciled, which has not been done
up to now.
Outside the event horizon time slows down as one approaches
it. If an astronaut falls toward the event horizon, we as far
away observers would see his clocks and his motion slowing
down more and more, so as far we are concerned it would take
an infinite amount of time for the astronaut to reach the event
horizon. However, the astronaut wold measure a finite time in
his clock, and he would actually pass through the event horizon,
never to leave again. If the black hole is ”small” the astronaut
would be torn apart by tidal forces as he approaches the event
horizon. But if ithe black hole is very large he would not feel
much of anything, but would be disconnected from our world
forever.
Do black holes exist? There is evidence for very large concentrations of mass in small spaces that attract stars and stuff
into a very hot rapidly rotating accretion disk as in the artist
conception shown below.
8
An actual observation of something that fits the theory for the
accretion disk around a black hole is shown in the figure below.
9