Acceleration
In Part II, we saw that uniform motion in a straight line was relative in the sense that it is not possible to tell if we are moving or standing still except relative to something else. Of course,
motion of objects is much more complicated than this. Objects speed up and slow down, change direction, and rotate. In other words, acceleration happens. Acceleration is not relative in the same sense, as we can tell if we
are in an accelerated frame of reference by the forces we feel.
Nevertheless, we still need to understand it in terms of relativity.
Kirk did such a good job before that we give him a longer, sleeker ship. We put in another seat for his evil twin Skippy. We also miniaturize the lightsticks (our clocks using light) and have them produce a flash of light every ticktock. We put one lightstick at the fore and one at the stern of the ship so that both Kirk and Skippy see a series of pulses from both lightsticks. The lightsticks are fast enough, and the ship is long enough, that there are always several pulses in transit from each lightstick toward Kirk and Skippy. The lightsticks are synchronized, so Kirk and Skippy see a flash from each lightstick at the same time.
++
 
 \ \ \ \ S / / / /  >
D / / / / K \ \ \ \ U >
 
++
The fore lightstick is labeled "U" for "up," and the aft lightstick is labeled "D" for "down." The reason for these labels will become clear in time. Ignoring the rest of the ship, we get this:
 \ \ \ \ S / / / / 
D / / / / K \ \ \ \ U
Skippy reasons that if he could arrange to get a little bit ahead, be a little bit more toward the direction that we call the right, he could see the next pulse from the U lightstick before Kirk does. So, he pulls the lever that separates Kirk from his chair, pushes him aside, and turns on the rocket. Kirk, who is not accelerating, has to see the same thing that he has been seeing. Skippy, who is still in his acceleration chair, is accelerating to the right and so shortens his distance to the next pulse of the U lightstick. He also increases his distance to the next pulse of the D lightstick:
 \ \ \ \ S/ / / / 
D / / / / K \ \ \ \ U
Skippy is closer to the next pulse from the U lightstick and farther from the next pulse from the D lightstick. He must therefore see the pulse from the U lightstick first. Looked at another way, the pulse from the U lightstick must reach Skippy before it reaches Kirk. The pulse from the D lightstick has already reached Kirk before it reaches Skippy.
Skippy can continue this indefinitely. What he will see is a series of pulses from the U and D lightsticks that look like this:
\ \ \ S/ / / /
/ / / \ \ \ \
The more frequent pulses from the U lightstick compared to the D lightstick will convince him that the U lightstick is running faster than
the D lightstick.
If we are in our reference ship that is not accelerating, how do we see the experiment? According to Special Relativity, we will have to agree with Kirk,
who is simply moving at constant velocity relative to us, that he sees the pulses at the same time. We will see Skippy a little bit to the right, so we will see him encounter a pulse from the U lightstick before Kirk does, and a pulse from
the D lightstick after Kirk does. We agree that Skippy must see the U lightstick as running faster.
There's another way to look at this in our frame of reference. Instead of having Skippy accelerate with the rockets, we tie a big tether to the prow of the ship and swing it around. Skippy still experiences acceleration, but it's always acceleration toward us. We see his lightstick running slower than ours, because he is moving around us, due to the time dilation of Special Relativity.
Skippy, however doesn't see us as moving in his accelerated frame of reference; he only sees us as
rotating. Since he's always accelerating toward us, he sees our lightstick just as he saw the U lightstick, running.
We saw in the comments of Part II how the "twin paradox" could be resolved using Special Relativity alone, knowing that acceleration happened, but avoiding talking about it. Now it is clear that we can do the same thing taking account of acceleration explicitly.
This does not require any new ideas beyond Special Relativity. The arguments about clocks in accelerated frames of reference are essentially similar to the arguments in inertial frames of reference. The answer to the "twin paradox" also comes out the same, which is what we'd expect.
What About Gravity?
Using the Lorentz Transformations instead of the Galilean transformations allowed Newton's theory of
motion to be brought up to date, but there was still one thing about
Newtonian physics that didn't quite fit: gravitation. According to
Newton, every mass instantanously attracted every other mass by an amount proportional
to the product of the masses and inversely proportional to the square
of the distance. The idea that gravitation is not instantaneous but travels at the speed of
light had been around since the late 19^{th} century, and Poincaré published a paper concurring with this idea just a few days before Einstein's 1905 paper on Special Relativity. Einstein and others tried to figure out
an interpretation of gravity consistent with Relativity and the
Lorentz transformations, and for many years they failed.
Weight and Mass
In our everyday experience, massive objects such as cars and bowling balls have two properties:
 They resist pushing
 They resist lifting
The former is usually called "mass," and the latter is usually called
"weight." On the Earth,
at least, the weight and the mass always seem to be the same. They
are so similar that we confuse the two in our units. People in the
U.S. talk about mass in pounds, a unit of force, whereas to keep the
units right, it should
really be in slugs. Similarly, people in the rest of the world describe weight
in terms of grams or kilograms, whereas it should really be Newtons.
The fact that the weight effects of mass and the inertial effects
of mass seem the same suggests the possibility that they are always
the same. This is known as the Weak Equivalence Principle. Galileo,
in his Dialogues, came up with an elegant philosophical justification of why
they should always be the same, no matter what. It has also been tested
empirically, notably by Henry Cavendish, who designed an experiment
that is still done today.
Einstein's Equivalence Principle
Einstein had an epiphany when he was still a patent examiner. What if someone fell from a roof? During the time of the fall, the person would feel no gravity, and so in that frame of reference, gravity would not exist. He described this as the happiest thought of his life.
Einstein began to think of gravity as not really a force but rather the
absence of a
force.
When we are sitting in a chair, it's not that gravity pushes
us downward; it's that the Earth by way of the chair pushes us upward.
A frame of reference that is falling freely, like a satellite in orbit,
is the logical analogue of a frame of reference that is just moving
along at a constant velocity in space.
Gravity is, therefore, a fictitious force. It is like the centrifugal force that we feel outward on a carousel. The real force is the centripetal force of the floor pushing our feet inward. The idea of the centrifugal force comes from ignoring the fact that the
carousel is rotating, but it's really just inertia. It's just like
Skippy in the ship flying around us as if on a tether. Skippy feels
a force away from us, but that is because his ship is being accelerated toward us.
Einstein extended the Weak Equivalence Principle, making the bold
guess that as the gravitational and inertial effects of a falling mass
should always be the same, it should therefore, even in principle,
be impossible to tell the difference between acceleration and gravity.
Of course, on the Earth, if we move around or just wait a while, the
gravity will be in a different direction. Also, if we move far from
the Earth, the gravity will be less. Einstein's Equivalence Principle
only applies over a small local region.
Einstein looked at Newton's second law of motion, which relates force to rate of change of momentum or, in the classical view, mass and acceleration. He also looked at Newton's law of gravitation, which also specifies a force, this time from gravitation. These are two seemingly
unrelated equations. He reasoned that if the Equivalence Principle
were always true, the forces had to be equal. This made the force drop
out of the equationone fewer concept to worry about. The mass of
the falling body appeared on both sides of the equation, so it, too, dropped
out. The resulting equation related gravitational acceleration on one side
to a mass and a distance on the other side.
Knowing that under Special Relativity, strange things happened to masses
and distances, he was careful only to use this exact equation at low speeds
and small distances. Nevertheless, he made the bold guess that the spirit
of the equation should always hold.
If Einstein was correct, and gravitation is really indistinguishable
from acceleration, then we must expect that all of the things that happen
with acceleration also happen with gravitation. If Skippy saw his U clock
going faster than his D clock under acceleration, then he must also see
the same thing when his ship is upended and resting on the Earth. His U
clock would then really be up, and his D clock would then really be down, and he would have to see his U clock running faster than his D clock, due to gravity.
Curved Spacetime
From Part II, we have the notion of spacetime as space and time in
four dimensions. From Newton, there is
the idea that objects will move in straight lines in the absence of an
external force. However,
this is not true in an accelerated frame of reference. When Skippy
released Kirk from his chair,
he saw Kirk falling in a curved path. However, Kirk thought it was
a straight path. Skippy sees spacetime as curved while he is
accelerating. The spatial aspect of the curvature is obvious: Kirk follows
a parabolic path, at least until he hits the stern bulkhead. We also have
to incorporate time into our ideas.
Imagine living on a curved surface. This shouldn't be too hard,
as we all live on one. It's called the Earth. We have all seen pictures of the Earth from space. We all have the idea that it's a ball embedded in three dimensions. We go up in buildings and fly in airplanes, but mostly we move along the surface of the Earth.
How can we tell that the Earth is a ball and not a flat plane? We could,
of course, launch a satellite and take pictures, or watch shadows from
the sun on a stick in the ground, or watch a Lunar eclipse, but all of these ways involve having
something off the surface (the camera or the Sun or the Moon). Let's say that we were really limited to
just the surface. Perhaps we couldn't look or move up and down, or
we just don't think about it. How can we tell?
We could sail in one direction, and eventually we'd come back from the opposite direction. This already shows that something odd is going on. We could also try to make figures using Euclidean geometry. Let's say we're trying to make a triangle with three straight lines. We expect the angles to
add up to 180 degrees. For small figures, they come out very close to the Euclidean ideal. However, as we try to make the triangles bigger on the Earth, no matter how hard we try to keep the lines straight, the angles add up to more than 180 degrees.
To see this, imagine cutting an orange. Make one cut, cutting it into halves. Make another cut, crosswise to the first one, and another, crosswise to both of the others. We'll end up with eight orange sections, all the same. The edge of each section will be a triangle over the surface of the orange. Three sides, as straight as we can make them. Three angles, each 90 degrees. They add up to 270 degrees, which is greater than 180. There's something funny going on.
We would see similar problems no matter what we tried to construct.
We could construct a circle. We know that the ratio of the circumference
to the diameter is supposed to be π (pi). As we make the circle larger and
larger, we notice that the circumference is smaller than we'd expect from
the diameter.
If we were living on a saddle instead of a ball, we would see the opposite happen.
The angles of the triangle would add up to less than 180 degrees, and the
circumference would be greater than we would expect. This is called negative curvature, as compared to the positive curvature
of a ball.
We can come up with all sorts of different ideas for the surface. Maybe there's a mountain
valley on the Earth with a ridge crossing it, which makes that part more like a saddle than a ball.
Maybe there's a mound in the middle of the valley, which makes that part
more like a ball again.
With our 3D insight, this kind of curvature is easy to see with a 2D surface. We can
imagine a ball. We can see how the circumference of a circle would be smaller
that expected, because the diameter is humped over the curved surface of the ball and has to be longer. However, we don't need that insight into embedded surfaces to detect curvature or even measure how much there is. Even if we didn't know from up and down, we could
still know that we were on a curved surface. Thinking of a 2D surface like
a ball helps us visualize it, but it isn't necessary to detect curvature.
Acceleration curves spacetime, so we should see similar effects, even if
we cannot see any obvious ball or saddle. Earlier
we swung a ship on a tether to produce acceleration. Instead of just swinging one
ship around, we'll use a carousel in the same way. The carousel is a circle. When it is stopped, we can compare the circumference at the rim with the distance to the center,
and it comes out as we would expect. Now we set the carousel spinning
and sit at the center. We see the rim moving rapidly. From Part II,
we know that rapidly moving objects contract in the direction of travel;
this is called Lorentz contraction. If the entire rim contracts along
the direction of travel around us, then the circumference of our
circle must be shorter than what we would expect. This is similar
to what happens on a ball.
By the equivalence principle, what happens under acceleration
should also happen in gravity. Let's try to construct a rectangle in spacetime in the presence of gravity. We'll use one dimension
of space and another dimension of time. The units and size of the rectangle don't really matter much, but to make things simple, we'll use one
of the ideas suggested in Part II, where the unit of time is a
nanosecond, and the unit of distance is the time that light travels in
a nanosecond, about a foot. It's slightly larger than a foot, so we'll
call it a bigfoot.
We do the experiment on Earth by setting up two clocks, the U clock several
bigfeet above the D clock:
U







D
This gives us an edge at the left of the page. The edge is through
distance in space, so we'll call it a "space edge." We need another
space edge and two time edges, one at the
top and the bottom. How do we get these "time edges"? Easy. We
just wait. Then we use the same space edge, after the wait, as
our right space edge. We don't physically move the space edge;
we just let it flow through time as things do anyway.
We also know that, on a rectangle, all of the angles should be
right angles. What can that mean when one of the axes is time?
It means that the time should be independent of space. From Special
Relativity, we know that all that strange Minkowski rotation and
skewing, where time becomes space and space becomes time, happens
when there is relative motion. So, we station ourselves as an
observer right in the middle:
U



O



D
We count the first two corners as the position and the time
of the two clocks. The second two corners will be the position and the time of the two clocks a few nanoseconds later. After a few
nanoseconds, we might imagine this:
UU'
 
 
 
O O'
 
 
 
DD'
We and the clocks haven't moved, but we draw time to the right.
If this is really
a rectangle, then the time of clocks U' and D' must be the same.
If we did this out in the middle of nowhere, in what we can call
flat spacetime, that would indeed be the case.
However, on Earth, the situation is different. The U clock is above
us, so we see it as running faster. We also see the D clock running
more slowly.
So, we will not see U' and D' as valid corners of
a rectangle. The U' clock will read later than the D' clock,
because it has been running faster.
We expected the times to be the same, if this were a rectangle, so it
cannot be a rectangle.
If we try to keep the elapsed time on the clocks the same, it is even more obvious that it is not a rectangle:
UU'
 \
 \
 \
O O'
 \
 \
 \
DD'
We cannot construct a proper Euclidean rectangle in spacetime in a
gravitational field if one clock is above the other. Like
the person constructing figures on a surface, we notice that the spacetime in which we live is curved. In this case, it has negative curvature, like
a saddle. We see that the presence of gravity curves spacetime.
Geodesics in Spacetime
Special Relativity, building on Newton's First Law of Motion, deals
with objects moving at uniform velocity along a straight line in flat
spacetime. We need to develop similar concepts for the curved spacetime
of General Relativity.
The concept of a straight line in space is intuitive. Given two points
A and B at different locations, the line is the shortest distance
between the two points. Our person trying to draw a triangle on a ball
made lines that were the shortest distance between two points on the
ball. Looking from outside the ball, the lines are obviously
curved. This is called a geodesic. Airplane magazines often
show routes between airports as geodesics, which appear curved on
flattened maps.
So, first we modify Newton's First Law of Motion to say that objects move along
geodesics rather than straight lines. Since, in flat spacetime,
a geodesic is straight, this does not violate any earlier ideas.
We also need to understand how geodesics work not just in space, but
in spacetime. We can set up points A and B at different locations and
different times, say with B at ten seconds later than A. We can make
a spacetime pen by putting a clock on the pen. From Newton's First
Law of Motion, we should expect a geodesic to be uniform motion
in a straight line from A to B. However, there are many other ways
of making the line look straight in space. We could wait until
the last second and then move the pen rapidly to B. We could zigzag
back and forth and eventually stop at B.
Fortunately, there's a simple way of telling the difference between
all these paths to determine which one is the geodesic, at least if
the pen is moving slower than light. We look at the
clock on the pen, which gives the proper time of the pen. When the
pen gets to B, because it has been moving, its proper time will read
a little bit less than ours. The more wildly we move the pen, the lower
its proper time will read. The geodesic is the path that will make the proper time be the greatest. In the case of flat spacetime, this is
motion at a constant velocity.
We can replace the idea of the shortest distance with the longest
proper time. In the case of flat spacetime this automatically minimizes distance. Any path other than the straight line in flat spacetime
means more distance, which means more speed, which means a lower elapsed
proper time of the pen.
An observer in an accelerating frame sees objects moving in curves.
From the equivalence principle, we should expect the same thing to
happen in gravitation, and it does. A thrown object, so long as it is
not thrown so far that the direction or strength of gravity matters, should
trace the
same curved parabolic path relative to us that Kirk did relative to Skippy.
How does this relate to maximizing the proper time? Let's say that we want to get
a ball to a friend at a distance, one second later. We could built a straight track and
send the baseball along the track. However, this would not maximize the
proper time of the ball. We know that clocks above us in a
gravitational field run faster. So, if we could just give the ball
some height, then its clock would run faster, and the proper time would
be greater. We can't make it go too high, because that would require
making it go too fast, and that would make the clock run too slow. So we
need some sort of compromise, making to ball go a little bit upward, but
not so high that the advantage is undone by the extra speed. The optimal
path is the curved parabolic path that we would see if we just
threw the ball.
The paths of bodies under the influence of gravitation, in the absence
of other forces, follow geodesics in spacetime. From a thrown rock to an orbiting satellite to the curving of light around a star, all paths in the absence of other forces are geodesics.
The Gravitational Field
General Relativity holds that spacetime is curved, but what is it about a really big rock that causes spacetime to
curve in its vicinity?
In Newton's Law of Gravitation, it was the mass. Newton's Law
works very well at short distances and low speeds, such as we see
in the solar system. It cannot just be disregarded. Any formulation
of General Relativity has to produce the same results where
and when Newton's Law is known to work.
On the other hand, mass seems a tricky idea. We saw in Part II that
we could think in terms of a relativistic mass, called m_{r}
or m_{γ}. We also saw in the comments to Part II
that it was just as good, in fact overwhelmingly preferred by
physicists, not to
use such a concept at all.
Einstein's original goal was to fit gravitation into relativity,
using concepts that transformed according to the Lorentz
transformations. What is needed is something that at low speeds is the
same as the mass, to keep consistency with Newton's Law where it is
known to work very well, but at the same time is consistent with
the Lorentz
transformations.
There is such a thing. In Part II we saw that we could make
a 4vector with space (3 dimensions) and time (1 dimension) that would
transform according to the Lorentz transformations. We also saw that
we could make a 4vector with momentum (3 dimensions) and energy (1
dimension) that also would transform. This is sometimes called the
energymomentum.
At zero speed, nothing is moving, so there is no momentum, and the
momentum portion of energymomentum is zero. That leaves the energy.
The energy at rest is equivalent to the rest mass according to E=mc^{2} from Part II. So, at
rest, the energymomentum is the same as the rest mass.
General Relativity relates gravity and the curvature of spacetime
to the energymomentum like this: Consider a small spherical volume of massive
particles. As time passes, the sphere will shrink due to gravity. Work
in the local rest frame of the sphere at the center of the sphere.
Take the energy density of the particles and add it to the flow of
xmomentum in the x direction, the flow of ymomentum in the y direction,
and the flow of zmomentum in the zdirection. Multiply
by the volume of the sphere. This is proportional to the rate at which the rate of
shrinkage of the volume increases.
Combine this with some constants, such as the speed of light c
and the gravitational constant g, presume that this is true everywhere, and it is in principle
possible to derive the Einstein Field Equation. This does not mean that it is easy. While the mathematics of Special Relativity
are easily within the grasp of any moderately well educated highschool
student, the mathematics of General Relativity require some
tricky concepts.
The correct form of the field equation took many years to get right. Einstein had been a student of Georg Riemann, who had a way of characterizing curved spaces. He also collaborated
with Marcel Grossman,
who let him know about the work of Gregorio RicciCurbastro. and helped him with a lot of the other mathematics. In 1914 Einstein published a version of the field equation, but it was wrong. It was enough to interest David Hilbert, who corresponded
with Einstein. They came up with the same correct field equation
within a few days of each other. In 1915, Einstein submitted his
paper on the field equation.
Consider an equation in basic mathematics, such as E=mc^{2}. We can look at
it and see the meaning intuitively, what we might call the poetic meaning of
the equation. Energy and mass are in some sense
the same thing, although possibly in different forms and looked at differently, related by the square of the speed of light. It is also good for doing
calculations.
If we have a gram of matter and a gram of antimatter, we can easily calculate how much energy will come out in the form of gamma rays if we put them together. In basic mathematics, the poetic and calculation forms of an equation are usually the same or at least pretty close.
In advanced mathematics, the poetic and calculation forms often look so different that it is hard to see them as the same thing. General relativity is no exception. In poetic form, the field equation can be written like this:
G_{ab}=8πg/c^{4}T_{ab}
If we use the appropriate units and some conventions, it can simply be written like this:
G=T
This equation uses tensor algebra to fit a lot of information into a small space. (A tensor is a kind of matrix in the same sense that a vector is a kind
of array. The transformation matrices used in computer graphics are usually tensors.) G is the Einstein tensor and describes the shrinkage of a ball
of particles mentioned earlier. T is the energy stress tensor and describes the energymomentum mentioned earlier.
Of course, this poetic form is useless for doing calculations. It must be expanded into a form where we can use simple addition, multiplication, etc. When expanded, it results in 16 nonlinear differential equations.
Nonlinear coupled differential equations are tough to solve anyway, let alone 16. By being clever about the symmetries in the tensors, we can reduce them down to 10 or so, which is not much help. Writing them all out results in something like
100,000 terms.
This is hideously unwieldy. Nobody would sit down with a calculator
and use them, let alone understand them completely. Fortunately, in some simple cases, nearly all of the coefficients go
to zero, and it becomes possible to solve analytically. Three analytical solutions of
particular interest:
 The Schwarzschild Solution assumes that all of the energymomentum is concentrated at a single point, similar to how Newton used point masses. It can be used for planets and the like and has been useful in describing neutron stars and black holes.
 The Friedmann Solution assumes that the energymomentum is evenly distributed over a volume and is useful for describing the structure of the universe as a whole.
 The Gravity Wave Solution looks remarkably like a generalization of Maxwell's equations and predicts gravity waves, much as Maxwell's equations predicted radio.
Given the sheer complexity of the equations, we might expect that General
Relativity would be far more controversial than Special Relativity. It has been, and the controversy continues to this day. That will be one of the subjects of the next installment.