Is 'charged black hole' an oxymoron?

stevendaryl

That doesn't have anything to do with what I said. What I claimed is that whether the metric components are time-varying (meaning: their derivative with respect to the time component is nonzero) is a coordinate-dependent fact. That's obviously true.

It's also true that if there is a timelike Killing vector field, then the metric is unchanged by translation along the Killing vector. But that doesn't mean that the metric is time-independent UNLESS the basis vector in the time direction happens to be the same as the Killing vector field; which is true of Schwarzschild coordinates and Rindler coordinates and inertial coordinates in flat spacetime.

stevendaryl

This relates to the question of time-varying metric components in the following way:
If there is a time-like Killing vector field, then there exists a coordinate system in which the metric components are independent of time. You seem to be interpreting this as: If there is a time-like Killing vector field, then in EVERY coordinate system, the metric components are independent of time. That's clearly not true.

TrickyDicky

You might as well take a look at any definition of KV fields and specifically the fact they are coordinate-independent, you don't need to take my word for it.

DaleSpam

Hi TrickyDicky, stevendaryl is correct. I think that you are confusing the coordinate independent concept of "static" and/or "stationary" with the coordinate dependent concept of "time varying".

stevendaryl

For a number of exchanges, I have made a statement X, and you've said, No, Y is true. But Y doesn't mean that X is not true.

Statement X: Whether the components of the metric tensor is time-varying depends on which coordinate system you are using.

Statement Y: Whether there is a timelike Killing vector field does not depend on which coordinate system you are using.

Statement X is true AND statement Y is true. They don't contradict each other.

Now, to be fair, it's possible that some people use "static metric" to mean "there exists a timelike Killing vector field". But I explicitly said that I was talking about whether the components of the metric tensor are independent of the time coordinate. Those are two different things, and you act as if you don't understand the distinction. The first is a coordinate-independent notion, and the second is a coordinate-dependent notion.

stevendaryl

Thanks, Dale.

TrickyDicky

Nope, I never used the concept "time varying". Stevendaryl did.
He is saying a static spacetime can have metric components not time-independent and I was merely reminding him that the timelike KV of static spacetimes preserves the metric. Are you confused about this?

TrickyDicky

Both notions are the same notion.
A metric is called stationary if its components are time-independent. And all static spacetimes are stationary.

TrickyDicky

And just to be clearer, it is of course possible to use coordinates that don't reflect the time-independence of the spacetime but here we are trying to ellucidate the physics of the problem, not coordinate artifacts. I guess I took that for granted.