Is the Schwarzchild Radius Relativistic or Newtonian?

[martinbn]

No, there couldn't, because the metric inside is still Schwarzschild (by Birkhoff's theorem), and we already know Schwarzschild has no other Killing fields.

Of course, may be I wasn't clear. My remark simply says that to say $\partial_t$ becomes spacelike int the interior is not enough, one needs to show that no other Killing field is timelike there. What you say is a perfectly good proof why there aren't any other.

 

[bbbl67]

Just a pedantic comment about Schwartschild being static. To show that the interior is not even stationary one needs to prove that there aren't any timelike Killing fields there. The fact that the Killing field that is timelike outside becomes spacelike inside, by itself is not enough, there still could be another timelike Killing field there. Of course that is not true, but it needs to be shown.

What exactly is a Killing Field?

 

[Nugatory]

What exactly is a Killing Field?

Google will be your friend - if you add the words "general relativity" to your search for "killing field".

 

[stevendaryl]

What exactly is a Killing Field?

It's a gruesome phrase, isn't it? It sounds like it should mean some valley where a mass murder took place.

 

[bbbl67]

It's a gruesome phrase, isn't it? It sounds like it should mean some valley where a mass murder took place.

Yeah, that's what I was thinking, that it was some place where mathematical parameters stop working or something, like an upper limit. But it's just named after a guy named Killing. So reading the Wikipedia article about Killing vector fields does me absolutely no good in understanding it. So how exactly does it apply to General Relativity?

 

[Ibix]

What exactly is a Killing Field?

A vector field that satisfies Killing's equation. A Killing vector defines a direction in which spacetime looks the same. For example, if a spacetime is rotationally symmetric, there's a Killing vector field whose vectors point in the tangential direction.

A timelike Killing field defines a direction in which you can travel so that space is the same. It means that one can have a notion of time such that space isn't changing with time. There is a timelike Killing vector field outside a black hole, and sure enough you can hover at constant altitude and nothing changes. There isn't one inside the horizon, which is because all paths lead to the singularity in finite time, so their notion of space must have a changing "shape".