11
votes
Accepted
How does the covariant derivative satisfy the Leibniz rule?
Assuming that $T$ and $S$ are supposed to be $(1,0)$ tensor fields we can see that
Eq. (2) is wrong immediately, because the expression $ \nabla (S)^{ \nu \rho} $ has the wrong indices (it should have ...
- 619
9
votes
The limit of GR with infinite speed of light $c$
what would the universe be like if gravity was curvature but c was infinite?
The equivalence principle holds in Newtonian gravity. So you can geometrize standard Newtonian gravity.
That is called ...
- 92k
9
votes
Accepted
Is it possible to describe every possible spacetime in Cartesian coordinates?
As the choice of coordinates is arbitrary, can't I just "postulate" to use cartesian coordinates to describe any possible spacetime?
If by cartesian coordinates you mean a set of four ...
- 1,032
7
votes
How does the covariant derivative satisfy the Leibniz rule?
Carroll on p. 95 means that
$$\nabla_X \left(T \otimes S\right) ~=~ \left(\nabla_X T\right) \otimes S + T \otimes \left(\nabla_X S\right),\qquad X\in\Gamma(TM),$$
and hence
$$\nabla_{\mu} \left(T \...
- 193k
7
votes
Accepted
The limit of GR with infinite speed of light $c$
Well, you see there's a problem there. The actual kinematic symmetry group for non-relativistic physics is not the Galilei group, but the Bargmann group - its central extension. This is best seen by ...
- 1,252
7
votes
Is it possible to describe every possible spacetime in Cartesian coordinates?
If OP by Cartesian coordinates means a local coordinate system $(x^0,x^1,x^2,x^3)$ [say, in some local open neighborhood $U\subseteq M$ of spacetime] such that the components $g_{\mu\nu}$ of the ...
- 193k
6
votes
Is it possible to describe every possible spacetime in Cartesian coordinates?
Since the question makes no reference to the number of dimensions, you could ask it just as well for a universe that is 2-dimensions of space and 1 of time. If you can't do it even there, then the ...
- 1,252
5
votes
Accepted
Where does the negative signature case come from in the Pythagorean derivation of distances in spacetime?
(copied from my answer to Minkowski Metric Signature [with some modifications])
Here's an argument essentially due to Bondi.
It is physically motivated by radar measurements.
First, an introduction ...
- 11.4k
4
votes
Is it possible to describe every possible spacetime in Cartesian coordinates?
Why would you want to write it in Cartesian coordinates? Putting aside everything, the statement "... using Cartesian Coordinates we could easier think about the structure of spacetime itself.&...
- 392
3
votes
Is curvature localised in General Relativity?
The tensor of curvature is a function of the metric and its derivatives. The metric is a function of the point in space-time. So, as far as I understood the question, the curvature is localized, it is ...
- 15.3k
3
votes
Why doesn't Galilean relativity lead to a contradiction in SR?
The elapsed property time over a worldline is:
$$ \Delta\tau = \int_W\frac{dt}{\gamma(t)} $$
where gamma is computed in the initial rest frame.
Since the two $W$ are related by reflection, they will ...
- 30.7k
3
votes
Does Special Relativity require a "ruler postulate" analogous to the "clock postulate"?
Barring its mathematical statement, the clock postulate, in my view, has a crucial physical content: it assumes the existence of ideal clocks whose behaviour is not influenced by their accelerations. ...
- 67.9k
3
votes
Physical meaning of each component of the metric tensor in GR
This is like finding the meaning of individual coordinates of a classical position vector.
The vertical component has something of a special meaning because gravity is vertical and it matters to just ...
- 34.9k
3
votes
Where does the negative signature case come from in the Pythagorean derivation of distances in spacetime?
It comes from the most fundamental observation of physics: past is qualitatively different from future. A Euclidean model of spacetime cannot accommodate this: time would be just another dimension. ...
- 17.7k
3
votes
Time dilation query
I like this question because it contains several common misconceptions about relativity, which, if not dispelled, will hinder progress.
Misconception 1: Motion affects the internal mechanism of clock. ...
- 30.7k
2
votes
Time dilation query
The answer is that if you have two events that occur in the same place in one frame, then the time interval between them is always less in that frame than in any other frame in which they occur in ...
- 24.2k
2
votes
What exists in the world according to the special relativity?
Special relativity really doesn't change the rules in the way you might have thought. Locally, we all are moving along paths through spacetime, so you can still maintain your chess analogy in the ...
- 24.2k
2
votes
Accepted
Physical meaning of each component of the metric tensor in GR
For simplicity (and for easier drawing) let us
first consider a metric in 2-dimensional space
instead 4-dimensional space-time, i.e.
$$g_{\mu\nu}=\begin{bmatrix} g_{11} & g_{12} \\ g_{21} & g_{...
- 35k
2
votes
Physical meaning of each component of the metric tensor in GR
It is perhaps better to exercise with simpler examples before deal with GR to have an intuition on the metric tensor.
Plane coordinates $xy$ with non-orthogonal axis leads to a metric with non-zero ...
- 15.3k
2
votes
Is it true to say that there is a single electric field and it is inhabited by and affected by all charges in the universe?
It's a model. "All models are wrong, some are useful." So, the question is what the use of this idea is. There is some insight here: to achieve the isolated electric fields of textbooks ...
- 17.7k
2
votes
Is it possible to describe every possible spacetime in Cartesian coordinates?
I think you have a mixup between global and local properties here. Using differential geometry language, a space-time is a 4-dimensional manifold with a Lorentzian metric.
It is a theorem in ...
- 153
2
votes
How comes that we use space and time together in one manifold?
Start off with 3D vectors. If you have a vector in one set of coordinates, then rotate the coordinate axes, you end up with the same vector needing to be expressed in the new coordinate system. A ...
- 2,472
2
votes
How comes that we use space and time together in one manifold?
Let's suppose a 2D (1 + 1) non relativistic world, and a distance $d$ and time interval $t$ between 2 events in a given frame. Any space-time metric $f(t,d) \neq f(t)$ would not be invariant, because ...
- 15.3k
2
votes
How comes that we use space and time together in one manifold?
Michaelson and Morley demonstrated that light has always the same speed in all frames of reference. The simplest answer to your question, then, is that if you are in a model with absolute time and ...
- 288
2
votes
Questions about E. Minguzzi's article on Synchronization (arXiv:1009.3005)
Thank you for the interest in the paper. Let me mention that this work has not been published so far because soon after I posted it I worked on another version that expanded it while rearranging some ...
- 286
2
votes
Another dimensions
Dimensions denote the number of variables needed to describe exactly where something is located. But this happens inside some given space (like the Earth's surface which is 2 dimensional or the ...
- 30.4k
1
vote
Compactification of Minkowski spacetime
$r'$ is just the angular position on the cylinder's surface. In the Penrose diagram (Fig. 17.9), it's the horizontal coordinate.
From comments, I think you're confused because there is a second copy ...
- 24.2k
1
vote
Compactification of Minkowski spacetime
Although in the cylinderical diagram the time coordiante $t'$ has the range
$$ -\infty < t' < \infty$$
the relavent range is only from
$$ -\pi< t' < \pi $$
As such, the conformally ...
- 1,032
1
vote
According the theory of general relativity, what is the role of causality in the changes of the curvature of spacetime?
Yes. This has to do with energy conditions and how the causal structure of the spacetime are linked. While the causal structure is usually somewhat axiomatic, in that the causal principle and ...
- 392
1
vote
Accepted
How to get metric tensor components?
The metric tensor is a bilinear map that takes in vectors of the tangent space to the manifold. We can expand the metric tensor as $$g(X_i, X_j) = g_{ij}dx^idx^j$$ Now, say the metric is a function of ...
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