Why Orbital Time is 6πGM in Schwarzschild Geometry

[Jonsson]

Hello there,

We know that for lightlike paths, there are circular geodesics at $r = 3GM$ in Schwarzschild geometry. Suppose an observer flashes his flashlight at $r=3GM$ and after some time the light reappears from the other side of the black hole. The time he measures is $6 \pi GM$. I accidentally obtained this correct result, but I don't know why my method worked. I doubt it's a coincidence, and I'd like to understand why this is right. Basically by assuming that the light travels round a circle of length $2\pi r$ at a speed $c = 1$. Write $\Delta \tau = \frac{2 \pi r}{c} = 6 \pi GM$

Why does this work? I expected this to only work using the coordinate time, i.e. i expected $\Delta t = 6 \pi GM$, not $\Delta \tau= 6 \pi GM$.

Thanks.

 

[PeterDonis]

I accidentally obtained this correct result

How did you obtain it?

 

[Jonsson]

How did you obtain it?

Distance over the speed:
$\Delta \tau = \frac{2 \pi r}{c} = 6 \pi GM$

I exected it to work if you replace $\Delta t \to \Delta \tau$, but not this way around.

 

[PeterDonis]

Distance over the speed

So how do you know this is in fact the correct result for $\Delta \tau$ and not $\Delta t$?

 

[pervect]

I'd like to see if the OP can answer the following questions, or their best attempt at it if they can't.

Start with the fundamental definition of the problem: if the Schwarzschild coordinates are denoted by $t, r, \theta, \phi$, what is the line element for the Schwarzschild metric? There are different sign convenitons and notations possible, this step is mainly to ensure we are all using the same ones. Rather than dictate the conventions as a textbook might, the OP can specify ones they are comfortable with.

Next, consider a single static observer at r=3M. Two events happen to this static observer at different times. What is the relationship between $\Delta t$ and $\Delta \tau$ for the time interval between the pair of events, where t is the Schwarzschild coordinate time (as above), and $\tau$ is the proper time of a static observer?

Consider the worldlines of two static observers, one observer s1 at $r, \theta, \phi$ and one observer s2 at $r, \theta, \phi + \Delta \phi$. The observer at s1 sends a light pulse towards the observer at s2. What coordinate time t does the light pulse arrive at s2, if the coordinate time of the emission was $t_0$?

What happens if we interchange the role of the two observers, so that s2 sends a light pulse towards s1?

Hopefully having expressions for light going from s1 to s2, and having considered the reverse case for light being transmitted from s2 to s1, we are in a position to write down the coordinate time $\delta t$ for the round trip, and using our first result for the relationship between proper time and coordinate time, the proper time $\delta \tau$ for the round trip as well. Hopefully the motivation for understanding why we are analyzing a round-trip is understood.

Knowing the proper time for the round-trip exchange of light signals, how do we compute the distance between the worldline of the static observer s1 and the worldline of the static observer s2?

Note that we've been rather formal about how we've defined distance here, hopefully the motivation is understood that we are aiming to find the distance between two positions in space, a distance that does not change with time. The wrinkle here is that positions in space are worldlines in space-time, so that talking about the distance between worldlines is the same as talking about the difference in "positions" where we suppress the time dimension.

We wish to relate this notion of distance to the SI definition of the meter, http://physics.nist.gov/cuu/Units/meter.html.

The meter is the length of the path traveled by light in vacuum during a time interval of 1/299 792 458 of a second.

This may be the key point. We need to know which time interval we mean when we apply the above. Do we mean the coordinate time interval, or the proper time interval .