Gravitational time dilation on a non-spherical object

[Jonathan Scott]

Can somebody provide an example with what we have established so far? Just use a brick so I can see how it is done. Thanks.

I don't know a quick answer for this integral; it's rather messy and best handled by mathematical software nowadays.

You could for example use the Wolfram Online integrator to get each of the indefinite integrals in the triple integral and substitute the result into the next one (switching the variable names round cyclically if using the online integrator, as it always assumes dx). For a brick, you could substitute the limits as you do each integration, as the x, y and z limits are independent.

 

[Jonathan Scott]

well this works for metrics like the Reissner Nordstrom metric, but only for static metrics, from what I understand:
in the case of the Kerr metric, there is frame dragging, and there is not only dt^2 in the metric but dt.dphi too

furthermore, at the event horizon of a Kerr black hole, g_rr -> infinite often without g_tt =0
this means that space is infinitely contracted but time is not infinitely dilated, if only g_tt provides the time dilation.
in such case the factor of the time dilation would not be the factor of the length contraction.

in other words, the event horizon is not always the ergosphere

I assumed that this thread was referring to static situations (and outside any event horizon). If the object is at rest relative to the source and the coordinates themselves are static, I think the dt^2 term is all that is needed.

 

[George Jones]

I have the posts discussing "time dilation" and "length contration" for rotating black holes to the separate thread

https://www.physicsforums.com/showthread.php?t=416147

that relativity fan started.

 

[Jonathan Scott]

I've spotted a paper "Gravitational potential and energy of homogeneous rectangular parallelepiped" on the ArXiv which calculates the Newtonian potential for various shapes including a cube and a rectangular shape.

The PDF is at http://arxiv.org/pdf/astro-ph/0002496v1".

 

[Jonathan Scott]

Here is an example of a comparison of a cube and a sphere based on that paper:

Consider a cube of side $2a$ with density $\rho$.

The total volume of the cube is $8 a^3$ so its mass is $m = 8 a^3 \rho$.

The potential at a vertex (corner) of the cube is given in the paper (equation 12) as a multiple of $a^2 G \rho$ (that is, $Gm/8a$) as follows:

$${12 \, \ln \left (\frac{\sqrt{3} + 1}{\sqrt{2}} \right ) - \pi } \approx 4.76$$

Now consider the potential due to a sphere of the same mass at the same distance from the center as the corner of the cube, that is $a \sqrt{3}$, in the same units:

$$\frac{8}{\sqrt{3}} \approx 4.62$$

This shows that the gravitational effect of a cube is not very different from the effect of a sphere of the same mass produced by smoothing off the corners and pushing the excess material onto the faces.

(The paper seems to quietly ignore the usual convention that the potential expression -Gm/r has a minus sign).