I Why do strong fields complicate our understanding of singularities?

[iuvalclejan]

Spacetime always has Lorentzian signature, not Euclidean signature. I don't know what you're talking about here.

How do you know this for sure? Maybe there are regions that are Euclidean and small and haven't been discovered. Certainly Euclidean spacetime is used as a tool for path integration in QFT. The point of the question is to see why is the necessity for timelike Killing vectors in either the Komar mass, or the 4-vector gotten by integrating T over a 3-surface of constant time and radius less than some r. The Euclidean spacetime is a reductio ad absurdum to show that stationarity does not require a negative square norm Killing vector.

The squared norm of a Killing vector field does not have to be negative. I have never claimed that it did. Indeed, I gave you regions of Schwarzschild spacetime in which the squared norm of the Killing vector field in question is zero or positive.

But in order for a spacetime to be stationary, it has to have a Killing vector field that is timelike, i.e., negative squared norm. And a spacetime has to be stationary for the Komar mass to be well-defined. That is by definition. Again, if you want details on the rationale for the definition, you're going to need to consult GR textbooks where the Komar mass and its rationale is discussed in detail. (Or you could look up Komar's original paper, I suppose.)

OK, you say this quite dogmatically. I don't see it at all. Stationary seems like it just requires a Killing vector in the direction we call time. It doesn't have to have negative squared norm as far as I could tell. Or maybe it does for the Komar integral, but not for the other integral of T.

 

[PeterDonis]

How do you know this for sure?

Because spacetime includes spacelike, null, and timelike intervals. You need a Lorentzian signature for that.

Euclidean spacetime is used as a tool for path integration in QFT

As an intermediate mathematical tool, yes. But before extracting any physical observables, you always have to Wick rotate back to Lorentzian spacetime.

you say this quite dogmatically

No, I give the standard definition of a stationary spacetime that is in every GR textbook. If you don't like that standard definition, then you need to go convince all the textbook authors to change it. Good luck.

 

[PeterDonis]

Thread closed for moderation.

 

[Drakkith]

How do you know this for sure? Maybe there are regions that are Euclidean and small and haven't been discovered.

We don't know anything for sure. Everything Peter has said comes from a theory and various models that have proven extremely useful and accurate and have been verified with the best observations and experiments we can come up with. If there is some region of the universe that they don't apply, it isn't here and it isn't anywhere we've looked so far, barring perhaps some extreme conditions such as inside black holes and near the time of the big bang.

If you wish to go beyond standard GR and its associated models, then I wish you the best of luck.

Thread will remain closed since this conversation appears to be inherently about things which aren't supported by mainstream science.