Solving Exercise 13.7 MTW Using Light Signals

[TerryW]

I have managed to work out parts a and b of Exercise 13.7 from MTW (attached), but can't see how part c works.

I can see how it could work in (say) the example of taking a radar measurement of the distance to Venus, where we have the Euclidian distance prediction and the result of the radar measurement which produces a slightly longer path as the radar beam passed through the sun's gravitational field. But if we are just taking radar measurements, the distances we establish are directly related to the time taken for the pulse to return by x = ct.

Can anyone shed any light on this?RegardsTerry W

 

[PAllen]

I interpret the exercise as saying you have a coordinate system in place, determined 'however'. You now want to to measure how the metric would be expressed in these coordinates. Thus, you may take coordinate time and coordinate positions as 'available'. If this is the intended interpretation, then think about what you can measure about light signals that gives you information about the metric, given the - stated - equation for a null geodesic. Looked at this way, this part should be no harder than (b).

 

[TerryW]

Thank you for your reply. Of course, it will only give me numeric values for the g_αβ/g₀₀. I can also get first derivatives of the g_αβs from (b) so I will be able to produce maps of g_αβ provided I cover the same region of space with my measurements for (b) and (c ).Regards

TerryW