Special relativity, part 2

Warning: I'm so depressingly out of practice at mathematical things that I could well have stupid mistakes in the following. Sorry! If you spot anything wrong, please mail and correct me...

Some more notes on special relativity as I traverse Taylor and Wheeler's Spacetime physics.

Subjective travel times

So, I like hyperbolic rotation stuff, because it makes addition of change of velocity behave in a nice additive fashion. We have the relativistic velocity as tanh(zeta), where zeta is the rapidity. I like rapidity because it's additive and at low speeds is the same as velocity. From the point of view of a person accelerating to a relativistic speed, they can treat the acceleration as a large number of small changes of velocity, and the rapidity is the velocity they would expect to have in a Newtonian universe.

In a Newtonian universe, travelling distance d would take time d / zeta. Subjectively, in a relativistic universe, things would move past you at velocity v = tanh(zeta). On the other hand, Lorentz contraction would make the distance d' = d / cosh(zeta). Subjectively it would take time d' / v = d / (cosh(zeta) * tanh(zeta)) = d / sinh(zeta) < d / zeta.

In other words, if you're going somewhere far away, subjectively it takes less time than in a Newtonian universe!

Interpretation of interval

The interval, t^2 - x^2, is a nice invariant, but what's its physical interpretation? The "straight line path" is the route that maximises the interval, and (for time-like paths) you can find a frame in which the x movement is 0 zero, in which case the interval is just the square of the most time you can experience going between the points in space-time. Which is also known as proper time.

About this point, I realise this is all covered about 4 pages ahead of where I got to in Spacetime Physics.

Change of velocity as shear

In Newtonian mechanics, a change of space or time reference point is a translation, and a change of velocity is a shear. In special relativity in its normal representation, a change of reference point is a translation, but change of velocity is nothing like a shear. Can we find a way of reformulating things so that it is a shear, at the cost of making a change of reference point into something quite different?

Yes we can! First of all, we make the y-axis into sqrt(t^2 - x^2). As each line of constant y now represents a particular interval, the y-axis value is unmodified by a change of velocity, as required for a shear. By using a square root, the y-axis is still a time axis for x = 0.

Then we want the x-axis to behave like a shear during change of velocity, with it being affected by a translation, proportional to the distance up the y-axis. A thing that behaves additively under change of velocity is rapidity, and moreover each point along the x-axis for a fixed value of the y-axis (interval) represents a specific rapidity required to reach that point in spacetime. So, if we change the x-axis to sinh^-1(x), we get the behaviour we want!

Admittedly the resulting representation of spacetime doesn't behave nicely under translation or have other properties we'd like, but still!

Uniform acceleration doesn't lead to uniform time

One of the things I found fairly unexpected is that, given two identically accelerated objects, positioned at different points in the direction of acceleration, they will age differently. This carries through to general relativity, so that objects in a uniform gravitational field age differently. In other words, you can tell that you're undergoing uniform acceleration, and it's not equivalent to genuine free-float, which I find utterly unexpected.

Time in the rest frame as action points

As I have little intuition about the Lorentz metric, I've been playing around in order to get a feel for it. This motivated the "relativistic change of velocity as shear" thing above. It's very tempting to try to reframe the Lorentzian metric as a normal Euclidean one. So, one way to look at it is to track the motion of a set of particles in a particular reference frame. Each particle, in this reference frame, can move some distance x and accumulate some proper time sqrt(t^2 - x^2), for a given amount of time in the reference frame, t.

In other words, a particle, measured against a reference frame, can "choose" to spend that reference frame time on moving or experiencing the passage of time. In the parlance of turn-based strategy games, time in the reference frame is actions points that can be spent on movement (in the reference frame) or action (subjective experience of the passage of time), albeit with a Euclidean l2 norm, rather than the l1 norm of those games (sum up movements plus actions).

Put another way, reference frame time is path length of a path representing travel through a Euclidean metric spacetime of reference frame space and subjective time. This does not look particularly helpful, since you are unlikely to want to match subjective times together, but I thought it a nicely different way to look at things.

Posted 2015-12-20.