12
votes
Accepted
Do Killing vectors form a Lie-algebra?
Let us start from the property of the Lie derivative $$[L_X,L_Y] :=L_XL_Y -L_YL_X = L_{\{X,Y\}}.\tag{1}$$
The Killing condition for the vector field $X$ is $L_Xg=0$ where $g$ is the metric. That is ...
- 67.8k
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
10
votes
Is gravitational binding energy or gravitational self-energy a source of gravity?
Is gravitational binding energy or gravitational self-energy a source of gravity?
Yes. If gravitational theory satisfies the strong equivalence principle (SEP) then the gravitational binding energy ...
- 15k
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 ...
- 947
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 ...
- 91.9k
9
votes
Why does the Weyl tensor not show up in the Einstein field equations?
Why does the Weyl tensor not show up in the Einstein field equations?
I think that the “moral” reason for that is that the
Weyl tensor represents locally free, propagating part of the curvature and ...
- 15k
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
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 ...
- 1,242
7
votes
Accepted
Why is the Riemann curvature tensor not zero?
I cannot see the problem. Fix $p\in M$, then $$(R^d_{cab}V^c)_p=((\nabla_a\nabla_b-\nabla_b\nabla_a)V^d)_p$$
implies that $(R^d_{cab}V)_p=0$ if the vector field $V$, defined in a neighborhood $U_p$ ...
- 67.8k
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
$$\nabla_{\mu} \left(T \...
- 193k
7
votes
Is gravitational binding energy or gravitational self-energy a source of gravity?
To work out the gravitational mass of a star you would have to include the rest mass of its components, their internal energy and their gravitational potential energy. The sum of the latter two terms ...
- 126k
6
votes
Is it possible to describe every possible spacetime in Cartesian coordinates?
Since the question makes no reference to the number of dimensions, you could ask it just as well for a universe that is 2-dimensions of space and 1 of time. If you can't do it even there, then the ...
- 1,242
5
votes
Accepted
Why does rotation make black holes smaller?
First, let us note that “horizon radius” (at least when talking about black holes without spherical symmetry) is a coordinate dependent term, so it is better to use coordinate independent measure of ...
- 15k
4
votes
Is it possible to describe every possible spacetime in Cartesian coordinates?
Why would you want to write it in Cartesian coordinates? Putting aside everything, the statement "... using Cartesian Coordinates we could easier think about the structure of spacetime itself.&...
- 355
4
votes
Accepted
Allowed Topologies for General Relativity
The theory only deals with the local curvatures, not the global topology. Hence any manifold with an allowed metric is allowed. These can be infinitely many, especially for negative curvature space-...
- 30.4k
4
votes
Tensor Index Manipulation
Let $A$ be any matrix and $A^{-1}$ its inverse, so $AA^{-1}=I$. Then
$$
0 =dI = d(A^{-1})A+ A^{-1}dA
$$
Thus, on multiplying by $A^{-1}$ on the right we have
$$
d (A^{-1})= -A^{-1} (dA) A^{-1}
$$
Now ...
- 50.3k
4
votes
Why does rotation make black holes smaller?
isaacg asked: "Why does faster rotation shrink the outer event horizon?"
It doesn't. In terms of the irreducible mass $\rm \mathcal{M}$ and the cartesian radius $\rm R=\sqrt{x^2+y^2+z^2}$
$$...
- 10.4k
4
votes
Is it possible that if the universe collapses, it reaches the same state as in its beginning?
A collapsing universe would not return to its initial state. Whereas the initial state is very uniform, the final state would be highly inhomogeneous.
Our universe originally had density variations of ...
- 2,658
4
votes
Accepted
What size is the smallest black hole stable at Standard temperature & pressure?
The rate of decrease of mass $m$ of a black hole due to Hawking radiation goes as
$$\frac{dm}{dt} = -\frac{\hbar c^4}{15360 \pi G^2} \frac{1}{m^2} \,, $$
where the symbols have their usual meaning. If ...
- 947
3
votes
Accepted
Why can't the answers to equations be infinity?
No. Anyone saying that an "infinity is a way of telling you have made a mistake" is being too playful with words. Usually, infinities are a way of telling that you have done something ...
- 355
3
votes
Accepted
Schwarzschild line element in Eddington-Finkelstein coordinates
What your are encountering here is a core feature of general relativity: time is inherently, and fundamentally a local quantity. While each observer will have an unambiguously defined local proper ...
- 10k
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
Accepted
No hair theorem and Klein-Gordon equation
… what am I missing?
You are missing the meaning of (various) no-hair theorems. Those theorems are talking about equilibrium configurations corresponding to end points of gravitational collapse and ...
- 15k
3
votes
Carter Constant with a Cosmological Constant
Here is a paper that gives the Conformal Killing-Yano tensor for any member of the Plebanski–Demianski family of solutions. This is the most general family of type D vacuum solutions to the Einstein-...
- 10k
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 ...
- 34.8k
3
votes
Why is the Riemann curvature tensor not zero?
I think the OP is correct to call $V^d$ as a vector field rather than a vector. Say we define covariant derivative
$$ \nabla_X Y$$
Then, while evaluating the covariant derivative at a point $p$, it is ...
- 947
2
votes
Can two relativistic black holes' event horizons overlap and separate again?
The Question: Arpad's initial query revolved around a cosmic conundrum. Given the significant momentum of two speeding black holes and their distant centers of gravity, how could a mere touch bring ...
2
votes
Accepted
Communication issue between two orbiting bodies
Hint: There is no reason a slow-moving clock can't show the right time. Try setting your alarm clock foeward an hour, then let it run slow.
It will show the right time eventually.
Now: At any given ...
- 14.5k
2
votes
Is there a GR explanation for cosmological coupling causing mass increase as the universe expands?
Perhaps this is a late answer, but in addition to TimRias answer:
Our universe is not just de-Sitter but FLRW. The latter is more general in the sense that the scale factor $a(t)$ can take arbitrary ...
- 947
2
votes
Why does the Weyl tensor not show up in the Einstein field equations?
The Einstein field equations relate the curvature of spacetime to the distribution of matter and energy. While the Weyl tensor is important for understanding gravitational waves and certain aspects of ...
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