H5     Epstein Diagrams - Flat or Rolled

For periodic processes such as the up and down oscillations of a spring pendulum in a gravitational field or the somewhat fictitious free fall through a tunnel which traverses the center of the earth, the moving object repeatedly returns to the same place. Such processes can be very beautifully depicted in a rolled version of an Epstein diagram. The first picture shows a stationary object in both a flat and in a rolled Epstein diagram. The time axis is not curved; there is no gravity at work; we are in the realm of STR:

What do we get when we roll the curved diagram of section H1? A lampshade, or more scholarly, a truncated cone:


In his diagrams [15-146ff] Epstein occasionally suggests by using glasses or parasols the ‘down’ direction of the gravitational field (all diagrams in this section are taken from Chapter 10 of his book). We once again nicely see how an object that is not held in place by force moves in a straight line through space-time toward the ‘bottom’. The angle φ between this 'fall-line' and the circles on the surface of the cone, which are at a fixed location and centered on the axis of symmetry of the cone, continually increases. sin(φ) = v/c is still correct. It just indicates that the relative speed between fixed points and the free-falling object is increasing - just as we have found in section H1.

Let’s consider the paths of various fast moving objects in gravity-free space-time. We are familiar with the representation in a plane: o lies at the origin, a is pretty fast, b is even faster, and c gives the behaviour of a photon:

All four speeds are represented in the following rolled version. We get four simple paths on a cylindrical surface:

We now could study how this is presented in a homogeneous gravitational field, with g and thus the curvature of space-time being constant. A lampshade is a good representation of the local situation at a given distance from the center of the field generating mass. But large spatial movements can, however, not be represented!

Which is the corresponding solid of revolution, if the curvature is increasing with proximity to the Earth's surface - or, in other words, when an stationary orbit continually claims more proper time? The answer: Something that looks like the bell of a trombone or an ear trumpet. Epstein calls this body a horn:

This diagram presents a large section of the x-axis on both sides of the earth together with the “rolled” time axis. The situation in the earth’s interior must be presented in the gap in-between. There, however, gravity (and hence the curvature) decreases linearly toward the earth’s center. Therefore, between the two horns a sphere with detached polar caps must be added:

The ‘barrel’ in the middle is actually an excised sphere, as we will presently prove. Imagine a cylindrical tunnel directly through the earth’s center from Switzerland to, let’s say, New Zealand. The tunnel axis is identical to our x-axis and the origin corresponds to the center of the earth. We can now, starting from any point on the x-axis, send an object, with or without an initial velocity, on a journey through the earth along the x-axis.

Objects falling near the earth swing like a spring pendulum back and forth around the center of the earth. In the interior of the earth the gravitational force obeys Hook’s Law: F = -k • Δx. The oscillation period - as with the spring pendulum – is independent of the initial degree of deflection from the resting position, and thus the paths rolled into our space-time diagram must be the same length, regardless of the initial velocity! This is exactly satisfied if the ‘barrel’ between the two horns is a section of a sphere! The ‘straight’ paths, which correspond to free-falling are only on a sphere’s surface always closed, and also have in all cases the same length per orbit:

But throw an object with a positive initial speed in this tunnel, it will go beyond the spherical region and will also take somewhat more proper time per orbit than the pendulum inside the earth (sketch below on the left). If the initial velocity is even larger than the escape velocity of about 11.2 km/s, the object will escape (sketch below on the right):

So much for the curvature of space-time as the cause of effects which we normally blame on forces in Newtonian physics. But that is only half the story: Even the metric of space itself is distorted. We will study these effects - again with Epstein - in the next section.

Adam Trepczynski has also produced a Shockwave animation of the rolled space-time diagrams of Epstein which he has kindly made available.