Multiple competing operationally significant notions of disance
Way back in February 2003, franznietsche asked:
franznietzsche said:
i'm toying around with the schwarzschild metric using it to find distances from a star in sphereical coordinates... what is the significance of r? is it the radius of the star or the distance from the center of the star? ... how would i use this to calculate spacetime distances in the presence of a star?
The radial coordinate here keeps track of "distance" from the object which is the source of the field in this spacetime model, and it is partially but not fully analogous to the usual radial coordinate familiar from euclidean geometry.
One geometric interpretation of the Schwarzschild radial coordianate can be deduced by comparing the line element you wrote down:
ds^2 = -(1-2m/r) \, dt^2 + \frac{1}{1-2m/r}dr^2 + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right),
-\infty < t < \infty, \; 2m < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi
with the line element for Minkowski spacetime in polar spherical coordinates:
ds^2 = -dt^2 + dr^2 + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right), \; -\infty < t < \infty, \; 0 < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi
In both cases, fixing t=t_0, \, r=r_0 gives the metric of a round sphere of surface area A = 4 \pi \, r_0^2, namely
d\sigma^2 = r_0^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right), 0 < \theta < \pi, \; -\pi < \phi < \pi
A possibly more important interpretation is that rests upon a more sophisticated concept, the "optical expansion scalar". For those who have seen this term, the obvious outgoing radial null congruence has (in both cases) optical expansion scalar 1/r.
Next, consider two observers who hover at r = r_1, \, r_2 where 2m < r_1 < r_2 < \infty. Indeed let's put r_1 = R, \, r_2 = R+h and consider h to be some small quantity.
Then we might ask: what is the length of a radial curve in a hyperslice t=t_0 which connects the two observers? This quantity will will be the distance measured by a slowly moving walker who walks along a radial wire suspended between the two observers.
Let's compute this "pedometer distance" in the case where we have Putting dt = d\theta = d\phi = 0, we obtain ds = \frac{dr}{\sqrt{1-2m/r}}, which we integrate:
\int_{r_1}^{r_2} \frac{1}{\sqrt{1-2m/r}} \, dr
The answer is rather complicated! To understand it, we expand the multivariable Taylor expansion in the quantities m, h, 1/R, obtaining:
\rm{distance} = h + \frac{m \, h}{R} - \frac{m \, h^2}{2 R^2} + \frac{3 m^2 \, h}{2 R^2}
which is both simpler and more perspicuous than the exact result.
But there is another, even more nature notion of distance which is very often employed: light travel time distance. This is defined very simply: one observer sends a radar pip which strikes the second observer and returns to the first, who divides by twice the elapsed time as measured by an ideal clock which he holds.
Let's compute the light travel time measured by the closer observer. The radial null geodesics satisfy ds^2 = 0, \, d\theta = 0, \, d\phi = 0, so we can write:
\pm dt = \frac{dr}{1-2m/r}[/tex]<br />
With some care, you can see that we have<br />
\frac{\Delta t}{2} = \int_R^{R+h} \frac{1}{1-2m/r}<br />
This is half the elapsed coordinate time, so we need to multiply by the factor which converts an elapsed coordinate time to elapsed proper time (measured by a static observer), namely \sqrt{1-2m/r}. We find<br />
\frac{\Delta s}{2} = \sqrt{1-2m/R} \cdot \; 2 m \, \log \frac{R-2m+h}{R-2m} \approx h + \frac{m \, h}{R} - \frac{m \, h^2}{R^2} + \frac{3 m^2 \, h}{2 R^2}<br />
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Let's compare this with the light travel time measured by the farther observer. Proceeding similarly we find a result larger by a factor of<br />
\frac{\sqrt{1-2m/(R+h}}{\sqrt{1-2m/R}}<br />
which yields the multivariable Taylor expansion<br />
\frac{\Delta s}{2} \approx h + \frac{m \, h}{R} + \frac{3 m^2 \, h}{2 R^2}<br />
We can summarize our results as follows: to second order in m,h, 1/R, we have<br />
\rm{distance}_{\rm{LTT, near to far}} = \rm{distance}_{\rm{pedometer}} + \frac{m \, h^2}{2 R}<br />
\rm{distance}_{\rm{LTT, far to near}} = \rm{distance}_{\rm{pedometer}} - \frac{m \, h^2}{2 R}<br />
\rm{distance}_{\rm{pedometer}} = h + \frac{m \, h}{R} - \frac{m h^2}{2 R^2} + \frac{3 m^2 h}{2 R^2}With more thought and more effort, we could add further notions to this list, such as an "optical disk distance".<br />
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There are several points of interest here. First, the light travel time is not symmetric: in general, the distance from A to B is not the same as the distance from B to A. Second, there are multiple distinct but operationally signficant notions of distance which we can adopt in this spacetime model. These all give very similar results over small distances, but can differ appreciably over larger distances. In this case, all three distances which we computed are to first order \rm{distance} \approx h + h \, \frac{m}{R^2} \, R, where I wrote the result in a manner which suggests extracting the Newtonian gravitational acceleration m/R^2 and comparing with similar compuations in the well-known Rindler metric for Minkowski spacetime. I'll leave that as an exercise, but we find the same conclusion: even in flat spacetime, there are multiple reasonable but distinct operationally significant notions of distance which accelerating observers can employ, and these are in general not even symmetric.<br />
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Failure to recognize this key point is a source of many common misconceptions, even by some physicists.<br />
<br />
Chris Hillman
