Moving Schwarzschild Black Hole

[Andrew Kim]

The Schwarzschild Metric (with $c=1$),
$$ds^2 = -\Big(1-\frac{2GM}{r}\Big)dt^2+\Big(1-\frac{2GM}{r}\Big)^{-1}dr^2+r^2d\Omega^2$$
can be adjusted to a form involving three rectangular coordinates $x$, $y$, and $z$:
$$ds^2 = -\Big(1-\frac{2GM}{R}\Big)dt^2+\Big(1-\frac{2GM}{R}\Big)^{-1}\big(dx^2+dy^2+dz^2\big)$$
where $R=\sqrt{x^2+y^2+z^2}$.

Is it possible for one to transform to coordinates that are boosted in the direction of one of the aforementioned coordinates, like one can do with the Minkowski spacetime? (In these coordinates, the black hole would be moving.)

 

[Dale]

Is it possible for one to transform to coordinates that are boosted in the direction of one of the aforementioned coordinates, like one can do with the Minkowski spacetime?

Yes, any coordinates can be used.

 

[pervect]

The Schwarzschild Metric (with $c=1$),
$$ds^2 = -\Big(1-\frac{2GM}{r}\Big)dt^2+\Big(1-\frac{2GM}{r}\Big)^{-1}dr^2+r^2d\Omega^2$$
can be adjusted to a form involving three rectangular coordinates $x$, $y$, and $z$:
$$ds^2 = -\Big(1-\frac{2GM}{R}\Big)dt^2+\Big(1-\frac{2GM}{R}\Big)^{-1}\big(dx^2+dy^2+dz^2\big)$$
where $R=\sqrt{x^2+y^2+z^2}$.

Is it possible for one to transform to coordinates that are boosted in the direction of one of the aforementioned coordinates, like one can do with the Minkowski spacetime? (In these coordinates, the black hole would be moving.)

There's good news and bad news. The good news is that there are some well-accepted coordinate system that's similar to what you wrote. The bad news is that they are not exactly the same as what you wrote.

If you look at https://en.wikipedia.org/wiki/Schwarzschild_metric#Alternative_coordinates, you'll see the isotropic coordinates listed as one of the "alternate coordinate systems", with a line element of:

$${\frac {(1-{\frac {r_{s}}{4R}})^{{2}}}{(1+{\frac {r_{s}}{4R}})^{{2}}}}{dt}^{2}-\left(1+{\frac {r_{s}}{4R}}\right)^{{4}}(dx^{2}+dy^{2}+dz^{2}$$

It might not be obvious why it's a very good idea to have the speed of light be isotropic, but it is. Basically, if your coordinates don't respect the isotropy of the speed of light, your physics when expressed in those coordinates won't respect isotropy either, the "laws of physics" in coordinate form willl be different in different directions, which is rather confusing.

It also won't be possible to globally transform the resulting coordinates via a Lorentz boost, though. It will be possible to locally transform a "frame field" via a Lorentz transform, though. Techniques for transforming coordinates do exist, but you can't do a global coordinate transform via a Lorentz boost. One way of doing a transform is using algebra and the chain rule, i.e noting that d (x*y) is x*dy + y*dx. But I don't have tome to get into the necessary details.

If you have the background, you might look up PPN coordinates. They are part of an approximation method that is widely used in weak field situations, they are based on the isotropic coordinates I mentioned but make further approximations.

Anyway, if you are willing to go to enough effort, you can eventually do something like what you think you want to do, but usually by the time you figure out all the little details of how to do it correctly, you realize that you don't really want to turn a nice, easy-to-work with stationary metric into a harder-to-work with one where the metric coefficients vary with time.

 

[Khashishi]

Maybe it's a stepping stone towards calculating the metric for two colliding black holes.