Radial Paths

In some ways, the subject of relativity is based on perspective, i.e. one or more observers that occupy different frames of reference may have a different perspective of spacetime. To provide some cross-reference to 3 distinct perspectives, the following model has been used by way of an introduction:

1

Given that this basic model has already been outlined, we will only mention the salient points in this thread before expanding on some of the implications of the Schwarzschild metric on each of the following perspectives:

  • The distant observer [A] infinitely far from any gravitational mass [M].
  • A shell observer [B] at a finite distance from the gravitational mass [M].
  • A free-falling observer [C] accelerating towards the gravitational mass [M].

In essence, the distant observer represents a conceptual frame of reference, which the Schwarzschild metric can use to transpose the radial distant [dr] and time [dt] into some measure of spacetime in another frame of reference. There is an argument that the use of the distant observer, as a preferred frame of reference, leads to interpretative anomalies that are an effect of the coordinate system being used rather than a real effect of spacetime curvature. This criticism is discussed in a later section entitled `Gullstrand-Painleve Coordinates` that also outlines several solutions of the Schwarzschild metric for a free-falling observer constrained to a radial path. So, having previously introduced the Schwarzschild metric, we might now ask ourselves what it tells us about the nature of spacetime around uncharged, non-rotating, spherically symmetric body of mass [M]. Initially, we shall examine solutions that only consider a radial path so that we might get some understanding of the perspectives from each of the frames of reference previously outlined in connection with the basic model. However, let us start with the full metric:

[1]      1

A radial path inline with the equator of a gravitational body allows us to simplify the metric as follows:

[2]      2

In the case of a free-falling observer [C], the change in the time [dt] and radius [dr] are being associated with [C], but measured by [A]. As such, we might clarify this specific example by subscripting the relevant variables to identify the corresponding frame of refrence as shown in [3]:

[3]      3

You can test the orientation implied in [3] by considering the case where the velocity of [C] is very high, such that the distant travelled in 1 second would be very small. In this example, [drA] might be thought to approach zero, such that [3] reduces to the form:

[4]      4

We might cross-check the form of [4] with the equations in the diagram above and realise that it represents the observed time dilation in [C] by observer [A]. While the subscript notation is a bit clumsy, it will be repeated on occasions to clarify the frames of reference associated with any specific example being discussed, i.e.