# Tag Info

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 ...
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### 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 ...

### 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 ...
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### 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 ...

### 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 ...

### 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 ...

### 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 ...
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### 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 ...
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### 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 $U_p$ ...

### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...

### 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 ...
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### 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 ...

### 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-...

### 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 ...

### 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 ...

### 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 ...
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### 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 ...
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 ...