This page provides some mathematical details of the theoretical physics behind my gravity page. It's optional to the discussion.
The math behind the explanation
This page provides some mathematical details of the theoretical physics behind my gravity page. It's optional to the discussion.
The formula for such a helical function, for a momentum p and energy E, is
f of x, y, z, and t equals the exponential of i times the quantity p z minus E t over h bar
where "h bar" is Planck's constant h divided by 2 pi. Particle physicists like to use units where h bar is defined to be 1, so I can leave out that denominator:
f of x, y, z, and t equals the exponential of i times the quantity p z minus E t
i here is the square root of negative 1, the unit imaginary number. If you're not familiar with complex exponentials, the above can be rewritten in terms of trig functions
f of x, y, z, and t equals the cosine of p z minus E t plus i times the sine of p z minus E t
and the cosine and the sine are the two components of the little arrow that traces out the helix. The helix slides along as time t increases.
Now, the arrow for the above wave function has the same length anywhere in space, so the particle has an equal chance to be anywhere. In the real world there would be at least some slowly varying function of x, y, and z, and time multiplying the above:
f of x, y, z, and t equals b of x, y, z, and t times the exponential of i times the quantity p z minus E t
which actually tends to blur out the momentum a bit from the value p, but that is another story (the Heisenberg uncertainty principle). In what follows I assume that b is slowly varying enough in all dimensions that it does not affect the arguments.
In special relativity, the Pythagorean formula for distance along a straight line in three dimensions
d squared equals x squared plus y squared plus z squared
is generalized in a peculiar way to space-time:
the quantity c tau squared equals the quantity c t squared minus x squared minus y squared minus z squared
where tau is the proper time, or time experienced by a system (or person) traveling in a straight, constant-speed path along the vector (x, y, z) over coordinate time t. c here is the speed of light. The closer you go to the speed of light, the closer the proper time gets to zero. For everyday velocities, the c t term dwarfs the others and there is almost no variation in the elapsed time. Again, physicists like to use units where c is 1, so that they can write it
tau squared equals t squared minus x squared minus y squared minus z squared
Now this can be written in a very different way:
tau squared equals x superscript m times g subscript m n times x superscript n
where "x superscript m" represents not x raised to some power, but the component labeled by m of the four-dimensional vector (t, x, y, z), and we use an interesting notation made up by Albert Einstein, in which whenever a component index shows up twice on the same side of an equation, we assume that we are summing over all possible values of the index. Where did all the minus signs go? They went into that "g subscript m n" thing, which looks like a matrix:
g subscript m n is the four-by-four square matrix with diagonal elements positive 1, negative 1, negative 1, negative 1.
Sum over all possible values of m and n, and we get the ordinary-looking version back. g subscript m n is called the metric tensor.
What was the point of writing a simple formula in such a bizarre way? Well, it actually makes a number of things more convenient. For one thing, if we restrict ourselves to first-order changes in tau under small variations in x, y, and z,
d tau squared equals d x superscript m times g subscript m n times d x superscript n
this form of the equation is actually unchanged under changes in the coordinate system: you can define x, y, z, and t in some crazy wobbly way, and the coordinate change just manifests itself in different components when d x superscript m and g subscript m n are written out as a bunch of numbers.
For another thing, this form for small changes is then valid even in Einstein's General Theory of Relativity, in which g subscript m n can be affected not only by differences of coordinates, but also by differences in the geometry of space-time.
Now in general things can get pretty complicated for strong gravitational fields such as you find near a neutron star or black hole, or when you consider scales so large that tiny components in the metric can make a big difference (like billions of years or light-years). Closer to home, things are not so bad. In a weak gravitational field, using an appropriate set of coordinates, general relativity says that the metric tensor looks approximately like this:
g subscript m n is the four-by-four square matrix with diagonal elements 1 plus 2 V, negative 1 plus 2 V, negative 1 plus 2 V, negative 1 plus 2 V.
What is V? It turns out to be the same thing as the gravitational potential familiar from Newtonian physics. The formula 1 plus 2V looks a little weird because we're using c=1 units; put those factors of c back in, and it's 1 plus 2 V over c squared. We're measuring V here relative to the potential out at infinity, so it's negative when we're down inside Earth's gravity well.
If the variations in x, y, and z are small (that is, if the time dilation from high speeds is negligible), we end up with a really simple gravitational time-dilation formula:
Since d t and d tau are small, we can neglect higher-order terms when taking the square root:
d tau equals the quantity 1 plus V over c squared, times d t
and if tau equals t where t equals 0, then
tau equals the quantity 1 plus V over c squared, times t.
As the potential gets more and more negative, tau, the proper time of the physical system, advances more and more slowly.
(You might wonder what happens if V approaches minus c squared. Well, before it gets there, the approximations we're making break down and the metric doesn't look like that any more; the other components start doing interesting things, and this choice of coordinate system is not even necessarily the most convenient one. See the sci.physics FAQ on black holes for some inkling as to what does happen.)
If the energy E is approximately m c squared, then (neglecting the slowly varying function b) the helical wave becomes
f approximately equals the exponential of i times the quantity p z minus m c squared t.
(Imagine the "approximately equal" in what follows.)
But wait: That "t" is really tau, the proper time experienced by the physical system. There's gravitational time dilation to figure in!
f equals the exponential of i times p z minus m c squared times t times the quantity 1 plus V over c squared,
or: f equals the exponential of 1 times p z minus t times the quantity m c plus m V.
Now suppose V varies with height. It's negative and it gets less negative as you climb into the sky. Over a small region we can approximate it as V equals V sub zero plus a z, for some values V sub zero and a. Then
f equals the exponential of i times p z minus t times the quantity m c squared plus m V sub zero plus m a z.
Collect the terms in the exponent that vary in z:
f equals the exponential of i times... z times the quantity p minus m a t minus t times the quantity m c squared plus m V sub zero.
The factor that multiplies z is the thing that determines the variation of the wave function with height-- that is, the momentum. The momentum is now not p, but
p prime equals p minus m a t
and the time derivative of the momentum-- that is, the force-- is
the derivative of p prime with respect to t, or F, equals minus m a
(negative because it's pointing downward), and a is revealed to be the acceleration of gravity! (I would have called it g, but I used that for the metric tensor.) The geometry of space-time is causing a change in the momentum by the effect of gravitational time dilation on the quantum wave function.