What Happens to Time on a Clock at r=r_{s}/2 Inside a Schwarzschild Black Hole?

[snoopies622]

TL;DR

What happens when g_rr and g_tt are switched?

I notice that in a Schwarzschild black hole, at r=r_{s}/2, the c dt and dr terms are exactly the opposite of what they are in external, normal flat space (Minkowski metric). That is, one gets them by multiplying both terms by negative one. I'm having trouble grasping what this means. An observer far away from a Schwarzschild black hole — where spacetime is flat — cannot see a clock at r=r_{s}/2, but if he could, what would he see the clock doing? I understand that this question might be meaningless, but thought I'd give it a try. Thanks!

 

[Ibix]

The interior Schwarzschild metric has the same functional form as the exterior one in Schwarzschild coordinates. However, if you keep the coordinate labels then the Schwarzschild $r$ coordinate is timelike inside the black hole. This doesn't mean anything except that $r$ is a bad choice of coordinate label - often $t$ and $z$ are used instead of $r$ and $t$.

The fact that the metric becomes the Minkowski metric at a particular surface just means that the coordinate basis happens to be orthonormal (it's everywhere orthogonal, except on the event horizon where they're not defined, but only orthonormal at the surface you've defined).

You asked what an external observer would see if it weren't impossible to see a clock there. That question makes no sense. He can't see it. Apart from anything else, the surface where the coordinates are orthonormal is spacelike, so nothing stays there - it's an instant in time.

 

[Dale]

Summary:: What happens when g_rr and g_tt are switched?

I'm having trouble grasping what this means.

A spherical spacetime can be foliated as a bunch of nested spheres. The r coordinate gives the area ($4\pi r^2$) of the specified sphere. Outside the horizon two nearby spheres are separated by a certain number of meters. Inside the horizon two nearby spheres are separated by a certain number of seconds.

 

[PeterDonis]

in a Schwarzschild black hole, at the c dt and dr terms are exactly the opposite of what they are in external, normal flat space (Minkowski metric)

Do you mean just opposite signs? The magnitudes are not $1$.

If you mean opposite signs, this is only true in Schwarzschild coordinates. There are other coordinate charts that do not have this property.

An observer far away from a Schwarzschild black hole — where spacetime is flat — cannot see a clock at but if he could, what would he see the clock doing?

A clock cannot be stationary at any value of $r$ less than or equal to $r_s$. It must be falling inward. So it's meaningless to ask what a clock "at" such a value of $r$ would be doing.

And on top of that there is the fact, already pointed out, that it's meaningless to ask what an observer far away would see the clock doing when it's physically impossible for him to see the clock.

 

[snoopies622]

Thanks all, these answers are very helpful. I guess black holes are where the differences between GR and the Newtonian universe become quite stark!

 

[snoopies622]

Epilogue: I just noticed that if a clock could be stationary at r=r_{s}/2, then relative to the distant observer it would be moving in the +/- imaginary direction. Special relativity produces funny answers like that too if one assumes impossible things like clocks moving faster than light.

 

[PeterDonis]

if a clock could be stationary at r=r_{s}/2

It can't, because it would have to be moving faster than light, i.e., such a curve is spacelike, not timelike.

relative to the distant observer it would be moving in the +/- imaginary direction.

No, it wouldn't. There is no such thing as an "imaginary direction".

The correct statement is, as above, that a curve with constant $r$ inside the horizon is spacelike, not timelike.

Special relativity produces funny answers like that too if one assumes impossible things like clocks moving faster than light.

Yes, which is why we don't do that.

 

[snoopies622]

Couldn't sleep last night so played around with this a little more, and discovered something interesting: The time dilation of a stationary clock at radius r from the center of a Schwarzschild black hole (relative to another clock infinitely far away) is exactly the same as that of a clock moving in flat spacetime with speed v where v is the escape velocity of the first clock parked outside the black hole.

It's as if space itself is moving toward the center of the black hole at speed v and the clock has to "move through space" that fast in order to hover there.

I wonder if this is true with other gravitational fields like that of the Kerr metric.

 

[Ibix]

It's as if space itself is moving toward the center of the black hole at speed v and the clock has to "move through space" that fast in order to hover there.

That's called the "river" model. It doesn't generalise well. A better explanation is that gravitational time dilation is the same as frequency change (and hence energy change) of a light pulse climbing out of the gravitational field. Conservation of energy requires the same energy change for mass, and escape velocity comes from the energy change. So time dilation compared to clocks at infinity is intimately associated with escape velocity.

 

[snoopies622]

Thanks Ibix, very interesting!

 

[PeterDonis]

The time dilation of a stationary clock at radius r from the center of a Schwarzschild black hole (relative to another clock infinitely far away) is exactly the same as that of a clock moving in flat spacetime with speed v where v is the escape velocity of the first clock parked outside the black hole.

Yes.

It's as if space itself is moving toward the center of the black hole at speed v and the clock has to "move through space" that fast in order to hover there.

Yes, as @Ibix says, this is called the "river model" of black holes. A classic paper on this model is here:

https://arxiv.org/abs/gr-qc/0411060

wonder if this is true with other gravitational fields like that of the Kerr metric.

As the paper linked to above shows, you can construct a "river model" for the Kerr metric, but the "river" now has a "twist" in it as it flows inward.

I would imagine a similar construction could be done for the Kerr-Newman metric, which is the most general type of black hole metric (and also the most general type of metric in which a "gravitational field" can be defined), including both charge and spin, and containing Schwarzschild, Reissner-Nordstrom, and Kerr as special cases. But I have not seen one.

 

[snoopies622]

Thanks Peter, that's a great paper.