Okay, so this is how
it goes. To understand general relativity, here’s a brief run-through of what
Einstein accomplished with special relativity and how he intended to counter
its deficits with his ideas of a general theory.
Special relativity
is rooted, essentially, in one of the principles of Galilean relativity: to
whit, the idea that it is impossible to say whether or not you are moving. For
instance, if you were running with respect to a stationary bus, you could just
as easily say that you’re standing at rest and that the bus was moving away
from you. Similarly, if you were standing at rest and a train whooshes by, you
could just as easily say that you’re in motion and it’s actually the train
that’s stationary with respect to you: there is no way to determine which one
of you is really in motion. That is what the principle of Galilean relativity
states: there is no test with which you can determine whether or not you are in
motion. There is only relative motion; determining absolute motion is
impossible.
Einstein decided that
this principle was a fundamental physical law, and tried to hypothesize what
would happen if you always measured the speed of light to be constant, keeping
the laws of physics the same in every frame of reference. After all, it’s a
clear violation of absolute motion: light is always moving; its speed is fixed,
and it can never be at rest. Furthermore, if you didn’t measure it to be moving
at the same speed always, you could use it as a test to determine if you’re
moving: an observer at rest while you’re in motion calls out a different speed
than the one you measure, and suddenly you know you’re in motion, simply by
working out the math. Fair enough. So Einstein rolled up his sleeves, and came
up with thought experiment after thought experiment to see what would happen.
The results? Time dilation and Lorentz contraction (the relevant thought
experiments that showed this I won’t go into, as you’re already familiar with
them). In your own frame of reference, moving at a particular, invariant speed,
you would observe fundamental quantities to be quite different from another
observer moving at another speed. If you tried to see whether or not you were
moving with respect to a photon, time and space would change for you so that
you would always measure the speed of light to be the same, regardless of how
fast you moved. Galilean relativity, with one major modification, has been
preserved: light is permitted to be in a state of absolute motion, and your
measurement of time and space would change so that you could no longer really
tell if you were moving with respect to the photon – you would measure the
speed of light to be the same in all reference frames, making it impossible to
use as a test for absolute motion. It was thus still impossible to determine a
state of absolute motion, except unless you excluded light from consideration.
That, then, was the
edifice on which special relativity was based on: preserving the idea that
absolute motion is a no-no. Yet special relativity is called special for a
reason: it only holds if you’re constantly moving at the same velocity. Indeed,
all the laws of special relativity held for the special case of when you
weren’t accelerating at all. That was Einstein’s problem: how do you preserve
the Galilean principle if you’re accelerating?
You see what it
means. Acceleration means taking inertia into account: you ‘feel’ a certain
force operate on you whenever the car you’re in accelerates or brakes, and you
can instantly tell that you’re in motion. True, Newton’s third law states that
an equal and opposite force operates on the car; but what if you were
accelerating with respect to a house twenty metres away? You couldn’t honestly
say that the house felt a similar force: its twenty freaking metres away, for
goodness’ sake, you’re nowhere near in contact with it. How do you accommodate
the force?
This occupied
Einstein’s mind for years. And one day, he got it.
Imagine, for a
moment, you’re in an elevator that’s initially moving at a particular speed,
say, down. Suddenly, it accelerates: you feel a rushing force as this happens,
and must conclude that you are, sighing as you do so, in motion. But wait!
Little did you know that, in actual fact, the elevator hasn’t accelerated at
all: it’s merely that the mass of the Earth has spontaneously changed (yeah, I
know it sounds ridiculous, but bear with me for a moment). Thus, the force of
gravity changed – so what you’re actually feeling is simply the force of
gravity.
If you think I’m
going barmy saying all this, here’s another way to think of it. Would you, as
an observer in that elevator, be able to distinguish between the two
situations? You could say, on one hand, that you were at rest with respect to
the elevator (you’d be moving at the same speed as it is, remember) and that
the elevator accelerated. Or you could also say that the elevator was perfectly
stationary (at rest with respect to you) and that its (or the Earth’s) mass
changed spontaneously, so you felt a force that made you feel as if you were in
motion. There is no test to determine which of these situations is correct.
This was the germ of general relativity: a theory of relativity that could take into account
accelerating frames of reference and not just those at a constant velocity.
Einstein’s great insight was to realize that a body in acceleration with
respect to a stationary observer is virtually indistinguishable (to the
observer) from a body that is at rest in a changing gravitational field with an
observer that is in motion. Einstein could account for the force now: he linked
it to a changing gravitational field. Accelerating and being stationary in a
changing gravitational field are indistinguishable. And thus the Galilean
principle was saved once more.
Now here’s a thought experiment from special relativity. It’s important to GR, so I’m going
to explain it to you.
Imagine you’re in a
circular chamber that’s spinning round and round at a constant angular
velocity. For some reason you want to measure the value of pi: this is weird,
but you’re a mathematician who wants to be a theoretical physicist, so that’s
okay. Now what’s pi? The ratio of the circumference to the diameter of this
oh-so-wonderful circular chamber you’re in. Ergo, you have to measure both the
diameter as well as the circumference to arrive at a value of pi. So you steady
up your nerves, ignore your dizziness and set to work.
First, you measure
the diameter. So far so good. Because you’re measuring something perpendicular to the direction of the chamber’s motion, Lorentz contraction doesn’t happen:
your rod stays exactly the same length, and you manage to arrive at a reading
that is exactly what you’d find if the chamber was at rest. Hopes high, you
begin to measure the circumference of the chamber. But now you’re in the direction
of motion: Lorentz contraction makes your ruler shrink, except you don’t
realize this because you’re also moving at the same speed. Naturally, when you
finally check your readings, you’re surprised to see that the circumference is
actually longer than what you measured it to be at rest. And when you put those
two numbers together – a longer circumference divided by the same diameter –
you get a value for pi that is no longer 3.14159etc.
The value of pi –
your measurement of it - has changed while you were moving at a constant
velocity. You can be tempted to ignore it, but this will always be true. What
can you conclude from this?
If you are as
well-versed in mathematics as I suspect you are, then you probably already know
where this is going. Different values of pi are characteristic of regions that
are not Euclidean: that are not perfectly flat, so to say, that are curved in
one way or another. One example is the curved surface of a (soccer) football,
where it is perfectly possible to draw a triangle with three right angles, and
other weird things; such spaces are not flat, and are thus not classified as
Euclidean. Thus one is forced to conclude that an observer moving at a constant
velocity measures events to no longer conform to a Euclidean background: that
the events a speeding observer notices is virtually indistinguishable from
those that occur on a curved surface. Space and time distort from him in a way
that make sense only and only on a curved surface: thus, even in special
relativity, one finds evidence that space and time are curved for the observer,
and that the observer will accordingly behave as if he’s on a curved surface.
A word of caution
here. When I say space and time 'curve' for the observer, I don’t mean they
literally curve. Time and space are not, as a friend once told me, ‘fly rods
that can be bent over physically’. It is merely that your measurements of time
and space are such that they are typical of a curved surface: distances become
longer or shorter, the time taken to cover them vary, and so on. Time and space
do not ‘curve’: only your measurements of them – distance, length, time – do,
so that you could very well conclude you’re moving on a curved surface.
And all this
happens when you’re moving at a constant speed: within even the bounds of
special relativity. When you’re accelerating – switching from velocity to
velocity – your measurements of space and time are going to ‘curve’ more and
more: you will measure successively changing values of pi, longer distances,
longer times to travel. And since acceleration is indistinguishable from a
changing gravitational field, this means that objects that are really in a
gravitational field will measure the same things: they will begin to behave
exactly as if their measurements of space and time were similarly twisted, so
that they too were on curved surfaces that gained more and more curvature as
the force of gravity increased.
Thus, an object in
a gravitational field will begin to behave as if it is on a curved surface that
is steadily growing curvier (I’m sorry, that word evokes images of bikini
babes. Nevertheless, it is all I have). This, I think, is Einstein’s most
profoundly insightful idea. Gravity isn’t a force that changes the trajectory of objects around it; what actually happens is that objects within a
gravitational field are merely trying to obey Newton’s first law (i.e continue
unimpeded with the same velocity in the absence of a force) while on a curved
background. Geodesics are straight lines on their surfaces too, remember? There
is no ‘force’ involved: merely an object trying to follow Newton’s Euclidean
laws in a non-Euclidean world. I always find this magical.
And that’s it,
really.
To summarise:
Being at rest in a
changing gravitational field is indistinguishable from accelerating. When
you’re moving at a constant velocity, you are forced to make observations that
only make sense if you accept that your space and time are those that are
appropriate to a curved surface. Thus, when you’re accelerating, your
observations correspond to steadily changing curved surfaces, and you will
behave exactly like you were on a curved metric. Since accelerating is
indistinguishable from being in a gravitational field, objects that are in a
gravitational field behave as if they are on steadily changing curved
backgrounds too. They carve out geodesics instead of straight lines because, on
a curved surface, geodesics are the only ‘straight’ lines possible. Hence,
planets form ellipses around the sun: space and time are so warped for them
that their closest conception to a straight line is an ellipse.
That’s all there is
to general relativity. It’s probably not the best kind of explanation I could
give – I was trying to convey the main ideas the quickest way I could – so do
tell me if there’s anything I didn’t clarify enough.