I have introduced the idea of Epstein Space-Propertime Diagrams (ESPD) in my previous Blog post. To refresh, here is the ESPD's structural relationship between two relatively moving inertial frames. In this post I will extend the principle to accelerated motion.

Fig. 1 The case with no acceleration copied on the right

Three important features of the EPSD are:

1. Relative movement at a constant speed is equivalent to an Euclidean rotation around the origin by an angle: Φ = asin (v/c) e.g. the red Euclidean coordinate system relative to the blue one (where v/c is the relative velocity as a fraction of the speed of light). When v=c, the angle is pi/2, in contrast to the Minkowski diagram, where the angle would have been pi/4 (using atan Φ instead of asin Φ).

2. In flat spacetime (i.e. no gravity, spatial contraction or expansion), accelerated wordlines extend by a unity arc length per time unit, loosely stated as 'at the speed of light' around a 'curvature center'. This is a tad misleading, because it clashes with the standard meaning of speed and velocity, which are measured through space alone. The correct statement is that the magnitude of the 4-velocity (a vector through 3 spatial plus one time dimensions) remains constant.

3. This implies that while the acceleration lasts, the whole structure of the accelerated frame continually rotates around a shifting 'origin' over time. This is best pictured by a radius of curvature of individual worldlines around a shifting center of curvature. Fig. 2 depicts only one worldline going through the original origin for a small Δt.

Fig. 2: Initial acceleration from rest.

Note that in space-propertime diagramming, the coordinate time (Δt) is given by the arc, not by the vertical axis, which is propertime (Δtau) of course.

The initial center of curvature lies along the x-axis and the acceleration can be viewed as a centripetal force experienced by a hypothetical object moving at the speed of light around the center of curvature at radius R. This would cause centripetal acceleration of c/R² in the +x direction.

Constant proper acceleration (as observed by in the accelerating frame) is equivalent to a continuous small rotation of the worldline around a shifting center of curvature, while the extension of the worldline is a unit four-vector per time unit . This gives the curved (parabolic) worldline segment Δt as indicated in Fig. 3.

Fig. 3 Continued constant proper acceleration

In most spacetime diagramming, geometric units like seconds and light-seconds, or year (yr) and lightyear (lyr) are used, so that c=1 lyr/yr and can be eliminated for simplicity.

The equation (lifted from the eBook) for the radius of curvature embodies the constant acceleration (x-dot-dot) and the relativistic factor cos Φ = √(1-v²). It shows how the relative speed can never exceed 1, because the radius of curvature will tend to infinity and the worldline to horizontal, i.e. the speed of light.

The constant proper acceleration is measured by an on-board accelerometer, so in the reference coordinate system, observed acceleration will tend to zero as Φ tends to pi/2.

This is in line with the standard relativity concept of momentum and energy tending to infinity as the relative speed tends to the speed of light. Not to distract from the natural flow of the idea, I will discuss energy and momentum in space-propertime at a later stage.

Here is a pseudo-code algorithm (lifted from the eBook) for calculating such an acceleration curve from the Fig. 3 point of view:

It is largely self-explanatory, but just a few comments may be in order, I hope. The first 2 lines of code after the 'output' line simply rotates the present center of curvature to temporarily sit at a more convenient spot for calculation. The simpler calculation is performed and then the values are rotated back so that the new (x, tau) coordinate sits in the the correct place. One can't use precisely zero acceleration, but I stick in a value of 1E-12, being close enough to zero for all practical purposes.

I will now turn to a more practical-sounding example involving Alice and Bob, two space fairing friends, obtained from the algorithm above.

On the right is a plot for Alice accelerating away from Bob at 1 lyr/y², which is (interestingly) very close to 1g of proper acceleration, as we know it.

After one year on Alice's clock, with her accelerometer showing a constant 1g, she reaches a speed of 0.76c relative to the inertial Bob. Her distance from Bob is x=0.54 lyr, as shown. Bob's clock reads 1.18 yr at the same time, according to his definition of simultaneity (a horizontal line). Note that the blue vector and the red arc are of equal length.

The initial radius of curvature R = 1 lyr, but increases to R~2.4 lyr after the one year of onboard acceleration. In Bob's frame the acceleration drops from the initial 1g to 1/2.4~0.42g, due to the squared relativistic factor.

The Epstein diagrams are good for gaining insight in the workings, but relativists have derived exact solutions for calculating the curves analytically. The relevant equation are (using a for proper acceleration, t for coordinate time and T for propertime):

Edit: I have noticed that these equations were not simplified all the way, so I followed Einstein's advice...

x = (COSH(at)-1)/a , T=SINH(at)/a and v = TANH(at)

The hyperbolic trig functions show that the underlying spacetime structure is still hyperbolic in nature.

In a next Blog post I will use the analytical equations to plot more scenarios for Alice and Bob (and friends), which can be hugely informative.

-J