to General Relativity
Problems with Newtonian
Newton was fully aware
of the conceptual difficulties of his action-at-a-distance theory
of gravity. In a letter to Richard Bentley Newton wrote:
"It is inconceivable,
that inanimate brute matter should, without the mediation of something
else, which is not material, operate upon, and affect other matter without
mutual contact; as it must do, if gravitation, ...., be essential and inherent
in it. And this is one reason, why I desired you would not ascribe innate
gravity to me. That gravity should be innate, inherent, and essential to
matter, so that one body may act upon another, at a distance through vacuum,
without the mediation of anything else, by and through their action and
force may be conveyed from one to another, is to me so great an absurdity,
that I believe no man who has in philosophical matters a competent faculty
of thinking, can ever fall into it."
So, clearly, Newton believed
that something had to convey gravitational influence from one body to another.
When later it became clear that influences travel at finite speeds it was
reasonable to suppose this true of gravity also. But Newton's law of gravity
did not incorporate the finite travel time of gravitational influences.
If right now the sun were to be destroyed by a passing black hole
we would not feel the gravitational effects until about 8 minutes had elapsed.
Because Newton's law did not include such retardation effects, and permitted
violations of special relativity, it was clear that Newton's law
had to be an approximation to the correct law of gravity.
In 1905 Albert Einstein
introduced his theory of special relativity. With this theory Einstein
sought to make the laws of motion consistent with James Clerk Maxwell's
(1831-1879) laws of electromagnetism. Those laws predicted that light in
vacuum traveled at a speed c (about 300,000 km/s) that was independent
of the motion of the observer of the light and of the light source.
Newton's law of motion, however, predicted that the speed of light should
depend upon the motion of the observer. Einstein basically sided with Maxwell!
Special relativity makes two postulates:
A non-accelerating observer
is said to be in an inertial frame of reference.
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.
It is extremely useful
to think of space as filled with points, each labeled so that we can distinguish
one point from another. These labels are called coordinates.
We are free to use whatever labels we like. But, in practice, we use
numbers as the labels because numbers are easy to manipulate.
A coordinate system
is a set of rules that tell you how the labels are related to each other
and to the points they label. For example, in a Cartesian coordinate
system the symbols x, y and z represent three numbers that
label the points in 3-dimensional space. These numbers can be arbitrarily
assigned. We can, for example, choose any point in space to be the origin,
that is to have the label (0,0,0), of the coordinate system; and we are
free to use any coordinate system that's convenient. In astronomy it is
often convenient to use a coordinate system called spherical polar coordinates,
with each point labeled by the three numbers p
= (r, q,
as shown in the figure below.
The labeling of points
is arbitrary, and has no intrinsic significance. But, the distance dl
between two nearby points is an intrinsic geometrical property of the two
points. The distance is independent of how the points are labeled. It is
therefore independent of the coordinate system.
An event is a given place
at a given time. Einstein, and others, suggested that we should think of
space and time as a single entity called spacetime. An event
is a point p in spacetime. To keep track of events we label each by four
numbers: p = (t,x,y,z), where t represents the time coordinate
and x, y and z represent the space coordinates (assuming a Cartesian coordinate
A good way to visualize
ideas about spacetime is to use spacetime diagrams. Usually, we
draw the time axis vertically with a single spatial coordinate, x say,
horizontally. But sometimes we represent space as a two-dimensional plane.
(As far as we know space is really 3-dimensional, as shown in the figure
above, but it is impossible to fully visualize a 4-dimensional world.)
In the figure below we represent space as a two-dimensional plane in x
and y, with time increasing upwards. As a particle moves
in space and time (that is, in spacetime) it traces a path called a worldline.
The plane represents the present which,
from the viewpoint of the particle, is of course always changing.
Suppose at some event
(that is, a given place at a given time), labeled O in the figure, there
is a flash of light. From the viewpoint of an observer at O a spherical
shell of light will expand away from event O at the speed of light. However,
in spacetime this spherical shell looks like a hyper-cone. If we
restrict space to a 2-dimensional plane in x and y, then the
set of all paths followed by the light rays from O will form not a
hyper-cone but a cone.
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.
Just as we can define
the distance between two points in space, dl, we can also define the distance
between two nearby events in spacetime, ds. And like ordinary
spatial distance, ds (which is called the spacetime
is invariant; that is, it is measured to be the same by all observers
and its value does not depend on how we choose to label the events. The
spacetime interval is defined to be
where c is the speed
of light and dt is the difference between the time labels at the two nearby
events. The difference dt is called the coordinate
time difference to remind us of the fact that its value depends
on how we choose to label the events, in contrast to the interval ds which
does not. Notice something peculiar about the interval. Its square can
be positive, zero or negative! It turns out that this is the only way we
can get an invariant length in spacetime.
ds2 > 0
the two nearby events
are said to separated by a timelike interval.
This means that a messenger could travel from one event to the other at
less than the speed of light. In other words, a messenger could set
off from one spatial point, at a given time, and arrive at the nearby spatial
point at the time which together with the spatial point define the second
the two nearby events
are said to be separated by a spacelike interval.
This means that no messenger could travel from one event to the other because
to do so would require the messenger to travel at superluminal speeds (speeds
greater than that of light).
ds2 = 0
the two events are
said to be separated by a lightlike, or
null, interval. That is, a messenger could connect these two
events if the messenger were to travel at exactly the speed of light.
That is, you could start from one point, at a given time, and arrive at
the other just in time, if your speed were exactly that of light. It follows
that two null separated events can be connected by a light ray. Indeed,
worldlines that are null intervals are precisely the paths followed by
light rays in spacetime.
Consider a spacetime interval
that has dl = 0. In this case
What does this mean? This
means that the two nearby events are actually at the same spatial
point, but are separated in time by an amount dt. In this case the coordinate
time difference dt coincides with the proper time
between the two events; dt is the proper time that has elapsed between
the two events because it is the time measured by a clock at that point.
The elapsed proper time is just ds/c, that is, the spacetime
interval divided by the speed of light.
Time and Proper Distance
Now consider setting
dt = 0. In this case
This shows that when two
events have the same time, that is, when they are simultaneous the
spatial distance between them is invariant, that is, measured to be the
same by all observers. This distance is called the proper
distance between the two points.
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.
As noted above the proper time is the time measured by a
clock at an observer's location. So far we have considered the proper time
of a special class of observers, namely, those who are in a fixed spatial
location relative to a coordinate system; that is, those for whom dl =
0. But what about the proper time of observers who are moving relative
to the coordinate system. How do we compute their proper times? Let's start
with the formula for the spacetime interval
dividing throughout by c2 and pulling out
a factor of dt2 on the right-hand side
we can re-write the interval as
Time of Moving Observers
= dt2[1 - (dl/dt)2/c2].
But, by definition, v
= dl/dt is the observer's speed, relative to the coordinate system; so
we can write
= dt2[1 - (v/c)2].
assume observers can travel only along timelike intervals; therefore, their
speed v is always less than the speed of light, c. So in the above expression
(v/c)2 is always less than 1. We now take
the square root of that expression to obtain:
ds/c = dt[1 -
spacetime interval between two nearby points along the worldline of our
moving observer is, by definition, just ds. The latter has the dimensions
of distance; but when we divide it by the speed of light we convert ds
to a quantity dt =
ds/c that has the dimensions of time. Notice that dt
= ds/c is always less than or
equal to dt. It is only equal to dt when our observer is at rest in the
coordinate system. This is the case we considered earlier, when we identified
ds/c as the proper time of an observer fixed relative to the coordinate
system (that is, dl = 0).
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:
= dt2 - (dl/c)2
By adding up (that is,
integrating) all the small elapsed proper times dt
along the worldline we can compute
the total elapsed proper time measured by a clock attached to any observer.
The formula that relates dt to
dt shows that the former is always less than or equal to the latter: the
clocks of moving observers run slower relative to a clock that is stationary
in the coordinate system. This is the famous time
dilation effect of relativity theory.
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
principle of equivalence
Newton's theory two kinds of mass appear: inertial mass in his law of motion
and gravitational mass in his law of gravity. In Newton's theory there
is no physical reason why these masses should be related to each other.
Therefore, the fact that, for all objects for which this has been checked,
the inertial and gravitational masses are equal in value, to a very high
precision, is an astonishing mystery in Newton's theory, that begs to be
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.
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.
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.
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:
experiments will give the same results in a local frame of reference in
free fall and in a local frame of reference far removed from gravitational
That is, there is no experiment
we can perform that will tell us whether we are in a free falling reference
frame (like the elevator above) or in a reference frame far away in space.
The consequences of this profound hypothesis are quite remarkable.
(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
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
connect matter and energy (the right-hand
side) with the geometry of spacetime (the left-hand side). Each superscript
stands for one of the 4 coordinates of spacetime; so what looks like one
equation is actually 4 x 4 = 16 equations. But since some are repeated
there are really 10 equations. Contrast this with the single gravitational
law of Newton! That alone gives a hint of the complexity of these equations.
Indeed, they are amongst the most difficult equations in science. Happily,
however, some exact solutions have been found. Below we discuss one such
exact solution, the first, found in 1916 by Karl Schwarzchild.
The Schwarzchild Solution
A solution of Einstein's equations is a formula that allows
us to compute the spacetime interval between any two nearby points, given
an assumed distribution of matter and energy. Given this formula we can
calculate the free motion of objects in the spacetime geometry the formula
describes, using the fact that freely moving particles travel on worldlines
that are as straight as possible, that is, on geodesics.
A well-known example of a curved geometry is the surface of the earth;
it is a 2-dimensional space whose geodesics are great
circles. These are circles whose centers coincide with the center
of the earth. Aircraft fly along great circles because the latter are the
shortest lines between any two points on the earth's surface.
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
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.
Clocks run more slowly in the presence
of gravity. To see this imagine two clocks (these could be two atoms of
hydrogen) fixed in space near a large mass M; one clock placed at the radial
coordinate r = r1 and the other at r = r2. What we want to do is to compare
the elapsed proper times of the two clocks. How do we compute their proper
times? Remember the rule: to get the proper time difference set the spatial
coordinate differences to zero in the metric, that is, in the formula for
the spacetime interval and then divide the interval by the speed of light.
Since each clock remains fixed in space the elapsed time measured by each
clock will be a proper time interval. That's what we want to compute.
The Gravitational Redshift
If we follow our rule and set dr
= dq = df = 0, and
then divide by c, in equation (1), we obtain
the elapsed proper time dt on
clock fixed at location r. We
see that dt is not the proper elapsed time. Now apply equation (2)
to our two clocks at r = r1 and r = r2 and compare the two proper time
intervals, for the same coordinate time difference dt. We find
It should be clear from equation (3) that if r1 < r2 the
numerator will be smaller than the denominator; that is, the clock that
is closer to the mass M measures a shorter elapsed proper time than the
clock that is further out. The closer clock runs slower. Notice, also,
a very strange thing: as the closer clock (clock number 1) is moved ever
closer its elapsed proper time gets progressively shorter. At some point
r1 will be equal to the critical radius
At that radius the elapsed proper time is zero; the clock
appears to have stopped! The critical radius is called the Schwarzchild
radius. It divides the spacetime around the mass M into two regions, an
outer region r > rc and an inner realm r < rc. Later we shall learn
that rc is the coordinate radius of the event horizon of a black hole.
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 << c2R. (<< 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),
= 1 - (MG/c2)[1/R - 1/(R+h)],
in which we have kept only terms linear in 1/R, since the
rest are much smaller and therefore can be neglected. The expression in
the square brackets is approximately h/R2
and with a minor rearrangement we arrive at
If we interpret the proper time interval dt2
as one complete period of oscillation of light waves at position
r = R+h, then we can compute by how much the frequency of a light wave
at the ground differs from one a height h above it, due to the slow-clock
effect. The light wave at ground level has only advanced by an amount dt1,
which is not yet a complete cycle. The proper duration dt1
of the complete cycle for the light wave at ground level is longer than
an amount dt2/dt1.
By definition, the frequencies of the light waves are n1
= 1/dt1 and n2 =
1/dt2, so our final answer is
So if a light ray leaves the ground and rises to a height
h it will be found to have a lower frequency n1 than
the frequency n2 of
a similar light ray that is already
at height h.
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 (Fe57)
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.