*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.*