Introduction to the Metric

So, as previously outlined, the following equation is the basic form of the Schwarzschild metric:

[1]      1

For reasons that will become more apparent we shall drop the generalised definition of the spacetime interval [ds] and substitute the equivalent form of proper time [c.dτ]. Many texts also drop the velocity of light [c] by setting its value to unity, but we shall retain its presence so that the normal SI units can be directly resolved. In fact, let us initially expand the form of [1] so that we can more easily perceive the central effect of mass [M] in this solution plus specifically highlight the reciprocal effect on time [dt] and radius [dr]:

[2]      2

At this point, we should also try to introduce all the variables before expanding on the validity of the relativistic factor and what it infers:

Group Variable Units Comments
1 r metres Coordinate radius
seconds Tick of Proper time
2 dt seconds Change in time
dr metres Change in radius
radian Change in longitude
3 θ radians Latitude,i.e. North-South
Rs metres Schwarzschild radius = 2GM/c2
M kg Mass of object
3 G metres3/kg*sec2 Gravitational constant
c metres/sec Speed of Light

As indicated earlier, the Schwarzschild metric is a specific solution of Einstein’s field equations of general relativity for an uncharged, non-rotating, spherically symmetric body of mass [M]. One of the difficulties when first looking at the Schwarzschild metric is trying to associate the variables to some given frame of reference:

  1. The coordinate radius [r] relates to the radial proximity to the mass [M], which in-turn affects the spacetime separation [ds] between 2 events.

  2. The variables [dt, dr, dφ] are the differential changes in the spacetime position of some object at the coordinate radius [r] taken by some distant observer in flat spacetime. As such, these measurements are subject to the relativistic effects caused by gravity.

  3. The variables [θ, Rs, M] can be described as fixed variables for some given gravitational system. For example, we want to describe the gravitational effects around some specific mass [M], e.g. a star, for which we can calculate [Rs]. If we set [θ=π/2], we can restrict the description of the Schwarzschild metric to equatorial orbits.

  4. The final grouping identifies the gravitational constant [G] and the speed of light [c], which many texts set to unity, but will be retained so that the units associated with any equation can be resolved.

We might guess that the relativistic effects of gravity are characterised by just one term within the Schwarzschild metric:

[3]      3

On the basis of [3], we might therefore draw two immediate inferences:

  1. As the coordinate radius [r] increases to infinity, the relativistic factor in [3] must approach unity, which corresponds to flat spacetime, i.e. when nowhere near a gravitational mass [M], spacetime is flat.

  2. As the mass [M] decreases to zero, the relativistic factor in [3] must again approach unity, which also corresponds to flat spacetime, i.e. when there is no gravitational mass [M], spacetime is flat.

Therefore, the effects of the Schwarzschild metric in [2] on time [dt] and space [dr] is in addition to that previously define by special relativity in terms of the Minkowski metric, which does not address the effects of a gravitational mass [M] on spacetime, i.e.

[4]      4

So while we have not actually provided a derivation of the Schwarzschild metric, we might still realise that the introduction of the new relativistic factor in [3] now represents the curvature of spacetime by mass [M]. The requirement for the differential notation is due to the restriction on the size of the variables associated with the potential curvature of spacetime by the presence of a spherical gravitational field of mass [M].

Note: Determining the separation [ds] in spacetime does not implicitly resolve to a proper time [dτ] as it depends on the characteristics of this separation being time-like or space-like. We are initially using proper time [dτ] simply because it is easier to perceive as the time experienced by a local observer subject to a gravitational field or velocity.

We can reduce the complexity of [1] or [2] by only considering radial motion and equatorial orbits, such that θ=90° or π/2, so that the [sinθ] term goes to unity as does the [dθ] term.

[5]      5

Again, with reference to special relativity, proper time [dτ] is derived from variables determined by an observer, who sits at rest in flat spacetime. In the context of general relativity, we might initially wish to place our observer far from the gravitational effects of the central mass [M] in question.

How many frames of reference are being represented?

What the Schwarzschild metric appears to be telling us depends on the frame of reference being considered, i.e. distant observer [dt] or onboard observer [dτ]. In part, the difference between the perception of spacetime between the distant and onboard observer results from the spacetime curvature factor in [3], which in-turn is linked to the definition of [Rs].  For consistency within the present discussion, we shall try to put the scale of Schwarzschild radius [Rs] into some better perspective, starting with a derivation based on the classical idea that the kinetic energy associated with an escape velocity has to overcome the potential energy of gravitation

[6]      6

However, in a sense, there are two ways of interpreting the meaning being associated with the Schwarzschild radius [Rs] for any given mass [M]. One interpretation is this radius contains all of the mass, such that escape speed would equal the speed of light [c]. This radius effectively defines the event horizon of a black hole, but this does not stop us from calculating an equivalent Schwarzschild radius [Rs] for any given mass [M], even though its escape velocity [v] may be less than [c], as illustrated in the table below:

Black
Hole
Neutron
Star
Sun Earth Units
mass = 1.98E+30 1.98E+30 1.98E+30 5.98E+24 kg
radius = 2.95E+03 9.81E+03 6.95E+08 6.37E+06 m
volume = 1.08E+11 3.96E+12 1.41E+27 1.08E+21 m3
density = 1.83E+19 5.00E+17 1.41E+03 5.53E+03 kg/m3
v = 2.99E+08 1.64E+08 6.16E+05 1.12E+04 m/s
Rs = 2.95E+03 2.95E+03 2.95E+03 8.92E-03 m
g = 1.51E+13 1.37E+12 2.73E+02 9.84E+00 m/s2
Rs/r = 1.00E+00 3.01E-01 4.25E-06 1.40E-09
(1-Rs/r) = 0 0.698969996 0.999996 0.9999999986

What can be seen in the table above is that the definition of a black hole requires it to have an escape velocity [v] equal to the speed of light [c]. This requirement really depends on the density of the object, rather than its mass [M]. Therefore, while a black hole can have the same mass [M] as the sun, it requires a much higher density, e.g. smaller volume. It should also be clarified that the actual density of a black hole is assumed to collapse to a spacetime singularity and, as such, its density would be infinite, as its physical radius is zero. Therefore, the density shown for the black hole, in the table above, corresponds to the mass [M] contained within the event horizon that hides the singularity. For the purposes of the Schwarzschild metric, we can calculate a corresponding Schwarzschild radius [Rs], even though there is no associated event horizon, as the escape velocity is always below the [c] threshold. Only the black hole contains all its mass [M] within the definition of the Schwarzschild radius [Rs]. However, the general point of interest is that we now have a value for the ratio [Rs/r], which can be used to determine the relativistic effects due to the gravitational mass of any object [M], as indicated in the table above.

Clearly, a black hole is an extreme gravitational object that appears to have the ability to distort spacetime to the point where the normal laws of physics may breakdown entirely. However, it is highlighted that the acceleration due to gravity [g] is a function of the mass:

[7]        7

Therefore, while the value of [g] for the example black hole in the table above might appear extreme, a black hole might have the mass of billions of sun-like stars, which might suggest that it is physically possible to pass through an event horizon and survive, at least, for a while. It should be noted in connection with [7] that the radius [r] does not constitute a physical surface as in the case of all the other mass objects in the table above. However, we will defer any further discussion of black holes until later.

1

The following discussions will initially consider a free-falling radial path towards mass [M] from the perspective of an observer on-board and then from the perspective of a distant observer far from the effects of the gravitational mass [M]. Afterwards, we will review the implications of a orbital path around mass [M].