Calculating distance between two points in potential

[JimLad]

Calculating distance between two points in spacetime

Hi,

I have spacetime surrounding a black hole described by Scharzchild metric and I want to find the distance a stationary observer would measure between two stationary points separated by dr radially.

I thought I could use the expression for the line element (stationary), ds = (1-2GM/rc^2) c dt, and integrate from r->r+dr. However it is in terms of dt...so I must have gone wrong somewhere.

Thanks

 

[pervect]

Here's a few tips:

1) Finding the distance will be finding the length along some path. Draw out the path.

2) Is the path parameterized by dt or by dr? Is the path time-like or space-like?

3) Beware the sign conventions. A -+++ sign convention is good for finding the length of space-like paths, a +--- sign convention is good for finding the length of time-like paths. It might be useful to see how to work the problem using the 'wrong" sign convention (it can be done, it's just less convenient), but only after one sees how to get the answer at all.

 

[smallphi]

Distance in SR and GR is defined as the the integrated spacetime interval between two events that define the two ends of the distance and happen 'at the same time'. Different observers in SR and GR disagree about what 'at the same time' means. Also, since there are gazillion of paths connecting two events, the distance will depend on the choice of path.

For the case at hand, we assume 'the same time' means the same coordinate time in Schwarzschild metric. The two events happen at (t, r, theta, phi) and (t, r+dr, theta, phi). You will have to set all differentials in the metric to zero, except dr, and then integrate if you need the result for finite delta_r. The path connecting the two events for finite delta_r is assumed to be radial (no change in the coordinate time or angles).

Of course the question is why would a static observer care about the coordinate time. After all, coordinate time is just a label and observers use the time measured by the clocks they carry i.e. their proper time. One can derive from the metric how the increment of proper time of the static observers depends on the increment of coordinate time. That dependence is different for static observers at different radii. All the static observers in the small neighbourhood of the one at r, will have proper time that runs approximately the same with respect to the coordinate time. For that small neighbourhood of observers, 'the same time' of the two events defining the measured length means the same local proper time (measured by their local clocks) which means approximately the same coordinate time if dr is small.

 

[MeJennifer]

Proper distance or coordinate distance?

JimLad are you asking for the physical proper distance between two events if we were to travel between them at a given speed or are you asking for the coordinate distance in various coordinate charts?

The former is a question about physics the latter is an exercise in differential geometry.