7
votes
Accepted
How are spatial coordinate systems in physics defined?
This question has been explored in the context of global positioning systems, which need to account for general relativity. The traditional Minkowski coordinates $(t,x,y,z)$ of flat space-time do not ...
- 188k
7
votes
Accepted
Is there a feature mapping for this kernel $k(x,y) = (\frac{\min(x,y)}{\max(x,y)})^2$?
The native Hilbert-space of $K^2$ is well known.
I assume the domain of $K$ is $\mathbb{R}^{>0}\times\mathbb{R}^{>0}.$ Note that:
$$ K(x,y)=\begin{cases}
\frac{x}{y} \text{ for } x\leq y\\
\frac{...
- 316
6
votes
Why are isometries of Minkowski space necessarily linear?
The following general result is described in the answer I posted on math.stackexchange here: if $V_1$ and $V_2$ are finite-dimensional vector spaces of equal dimension over an arbitrary field and they ...
- 50.6k
3
votes
The stabilizer of a point in the connected Lorentz group
One may prove by direct computation that in order to stabilize $p$, your matrix $f$ must be a block matrix of the form:
$$ f = \begin{bmatrix}
1 & 0 & \vec{v}^T \\
0 & 1 & -\vec{v}^T \\...
- 131
3
votes
Would a closed universe with special relativity violate causality? Does the universe have to be simply connected?
In addition to Igor Khavkine's comment. There are some more recent articles about twin paradox in compact spaces:
https://arxiv.org/abs/gr-qc/0101014 (The twin paradox in compact spaces, by John D. ...
- 16.5k
1
vote
Variational principle for relativistic gas dynamics
There is by now also quite some literature on action principles for relativistic dissipative fluids. A random reference is this paper. The short answer to your question is that the action for Euler ...
- 76
1
vote
Variational principle for relativistic gas dynamics
For now I'll just mention that there's a small literature on variational principles for perfect fluids in relativity, though I'm not an expert on it. Here is a reference that discusses some approaches,...
- 21.5k
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