Coordinate radius

In basic terms, we have shown that the Schwarzschild metric is essentially an update of the Minkowski metric of flat spacetime, which accounts for the curvature of spacetime due to the gravitational effects of mass, or possibly more accurately energy. However, before we use this metric, it is possibly useful to provide some further clarification of the coordinate radius [r] used in the metric.

1

In the diagram above, we are again making reference to the observers in the basic model. So, for all practical purposes, we can consider observer [A] to be so far removed from the gravitational effects of mass [M] that we may describe [A] as being positioned in flat spacetime.

But how do we define the position of observer [C] free-falling towards mass [M]?

Clearly, the implication of the Schwarzschild metric is that the position of [C] has to be affected by the curvature of space and time as it approaches mass [M]. As such, we need to establish how we might conceptually measure the radial position of [C] and then define a coordinate parameter to which this measure is to be linked. The first practical problem is that each [B] observer cannot simply extend a tape measure to the event horizon. So what the diagram is suggesting is that we might establish a series of radial observers, e.g. [B1, B2, B3], which maintain a stationary position with respect to mass [M]. Each of these radial observers would then calculate their radial position [rB1, rB2, rB3] by measuring the circumference of their orbital position and then dividing by [2π]. The practicality of this method is not questioned at this point, but even so:

How might these radial positions be affected by the curvature of spacetime?

In the top half of the diagram is a representation of the geometry of flat space, while in the lower half is a representation of curved spacetime. If spacetime surrounding mass [M] were flat, we could cross-check the radius [r] calculated by each [B] observer as follows:

[1]      1

As suggested by [1], [rB2] would be calculated from the circumference, while the radial distance to its neighbouring [B] observers could be directly measured, i.e. x1, x2. In flat space, we would expect the expression in [1] to hold true, although we might have to question this method in curved space. In the lower half of the diagram, we can see that if we project the flat measurement of [x1, x2] onto the curvature of space, the actual measurements would be [x1’, x2’], so:

[2]      2

Of course, this discrepancy is exactly what the Schwarzschild metric predicts and, as such, we may segregate out the relativistic effects on time [dt] and space [dr] as follows:

[3]      3

We might realise that the implications that follow from [3] is that the observed time in [B] is dilated with respect to an observer [A] sitting in flat spacetime, while the observed distance is expanded. However, we still need to answer the question:

Who determines the coordinate radius [r] used in [3]?

What [3] tells us is that the radius calculated by each [B] observer can be normalised to flat spacetime and used as the basis of a coordinate system associated with the Schwarzschild metric.

[4]      4

At this point, we might also highlight that the form [4] aligns to the assumption of the 'basic model' previously introduced. However, while the methodology suggested by the diagram above is conceptually possible, in practice, it is observer [A] who establishes the coordinate framework against which the Schwarzschild metric then uses to determine the effects of relativity on observers [B] and [C]. Therefore, the following discussions will often use the form [r/Rs] to denote the relative position with respect to mass [M], where:

[5]      5

If we go with the assumption that [M] corresponds to the rest mass, then the Schwarzschild radius [Rs] will be an invariant definition in all frames of reference. As such, we might summarise the positions of [B1, B2, B3] in the diagram above in the following way:

Bn rA=r/Rs 1-Rs/r rB
B1 6 0.8333 6.57
B2 4 0.7500 4.62
B3 2 0.5000 2.83
Bn 1.0001 0.0001 100.02

In the table above, we have aligned the coordinate radius [r] to the distant observer [A] in flat spacetime and normalised the radial position of [B1, B2, B3] in terms of the integer ratio [r/Rs]. The curvature of space, implied by the Schwarzschild metric, means that the shell observers at [B1, B2, B3] would calculate a different radius based on [Cn/2π] that aligns to the values of [rB] shown in the table above, which are also expressed as a ratio of [r/Rs]. The last value in the table is simply to show the extreme curvature of space as [r] approaches the event horizon defined by [Rs].