10 votes

Is it possible to derive the Weyl tensor from the Ricci tensor or Ricci scalar? If so, how?

No. The Weyl tensor and the Ricci tensor correspond to different "degrees of freedom" of the Riemann tensor. As an example, notice that there are many Ricci-flat solutions to the Einstein ...
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8 votes
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Number of Independent Components of Levi-Civita Christoffel Symbol

First, let's talk about the metric tensor; it has $\frac{N(N+1)}{2}$ independent components, because it is a symmetric tensor: $g_{ab}=g_{ba}$. Writing this out in a matrix format: \begin{align} [g]&...
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  • 2,930
8 votes

Is there are relationship between the Ricci scalar and the determinant?

The determinant of the metric tensor $\det g $ is a scalar density of weight $+2$, and thus, when expressed in terms of local coordinates, transforms with the square of the Jacobian of the mapping. It ...
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  • 943
7 votes

Where were the first galaxies formed in our observable Universe, at the center or at its outskirts?

Our universe does not have a “center.” The distinguishing feature of our location is that we live here. We have excellent evidence that the universe is “isotropic,” which means the time from the Big ...
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  • 71.9k
7 votes
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Confusion about Einstein's field equations

$R = g^{\mu\nu} R_{\mu\nu}$ does NOT mean $R_{\mu\nu} = R g_{\mu\nu}$.
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  • 21.1k
7 votes
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How would someone discover the Einstein-Hilbert Action?

I don't know the history of how the Einstein-Hilbert action was discovered originally, but from a modern point of view it can be justified in several ways. First, if you want to represent gravity as ...
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  • 34.4k
5 votes

Is there a version of the Einstein field equations that uses the Riemann curvature tensor instead of the Ricci curvature tensor?

If you mean an equation such that the stress energy tensor at a point of the spacetime determines the Riemann tensor at that point, then the answer is negative for physical reasons. Such an equation ...
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5 votes

Is there a version of the Einstein field equations that uses the Riemann curvature tensor instead of the Ricci curvature tensor?

Well, as you know the Ricci tensor/scalar are built from the Riemann tensor, so it's not entirely clear to me what you're asking. You can of course write $$G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}R g_{\...
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  • 52k
5 votes

Is it possible to derive the Weyl tensor from the Ricci tensor or Ricci scalar? If so, how?

To add up to the above answer, even though the Weyl tensor contribution to the Riemann curvature tensor is naturally an independent degree of freedom from the Ricci tensor, it cannot be fully ...
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  • 943
5 votes

Where were the first galaxies formed in our observable Universe, at the center or at its outskirts?

The following is a good analogy for the current big bang cosmological model as far as the answer to "where the center of the universe is". Take an ideal balloon,before inflating it, its ...
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  • 222k
5 votes
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Delta function singularity in curvature

A good analysis of this subject is contained in the paper R. Geroch, J. Traschen, Strings and other distributional sources in general relativity. They conclude that thin shells (delta singularity on a ...
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4 votes

Delta function singularity in curvature

The simplest example I can think of is to have a "bumpy" metric (also here): \begin{equation} g_{\mu \nu} = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 + \theta (x - x_{0}) &...
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  • 819
2 votes

Frequencies (ticking rates) of the two ${}^{87}\text{Sr}$ clocks in the experiments at the Tokyo Skytree broadcasting tower

Is it correct to conclude that … Is it instead correct to conclude The distinction you are drawing is a distinction without a difference. It is only a matter of your arbitrary choice of reference ...
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  • 66k
2 votes

Is there a version of the Einstein field equations that uses the Riemann curvature tensor instead of the Ricci curvature tensor?

You can start with Ricci decomposition in a $n$ dimensional Riemannian manifold: $$R_{abcd}=C_{abcd}+\frac{1}{n-2}(S_{ad}g_{bc}+S_{bc}g_{ad}-S_{ab}g_{cd}-S_{ac}g_{bd})+\frac{R}{n(n-1)}(g_{ad}g_{bc}-g_{...
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  • 846
2 votes

Doubt about General relativity

You can have curvature without having a non-zero energy momentum tensor, indeed. In fact, these are special solutions to Einstein Equations, called the Vacuum solutions, where you set $T_{\mu \nu}=0$.
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  • 267
2 votes
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Where were the first galaxies formed in our observable Universe, at the center or at its outskirts?

If I understand your question correctly (based also on your disagreement with the interpretation of the other answers), what you are asking is whether the first galaxies were formed at the particle ...
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  • 819
2 votes
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Doubt regarding 4-velocity of a particle in Kerr space-time

From what I read, the author wants to show that there are no static observers under ergosurface. So he is not interested in observers moving in the direction of $\partial_r$ and $\partial_\theta$ - he ...
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  • 5,588
2 votes

Can a (conservative) four-force be derived from a scalar potential?

Your calculation is consistent, a bit misleading. You should view it as a constraint on the possibilities of $\Phi$ since the force $f$ must be spacelike. This is the case here since $\Phi$ has only ...
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  • 1,702
2 votes

Wave equation in curved spacetime

We have $\square=g^{\mu\nu}\nabla_\mu\nabla_\nu$ (or with a minus sign depending on convention) and because of the covariant derivative, the expansion depends on if it is applied to a scalar or tensor ...
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2 votes

Is space-time made of something?

Supposing $M >> m$, we can describe the movement of $m$ using Newton's law of gravity, and its second law:$$\mathbf F = m\frac{ \mathbf {d^2r}}{dt^2}= -GMm\frac{\mathbf {\hat r}}{r^2} \implies \...
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1 vote

Are dual bases and the Hodge dual "entirely distinct" uses of the word "dual", as per MTW?

MTW (Gravitation by Misner, Thorne, and Wheeler) sec 2.7 makes no use of the volume element, while MTW sec 3.5 (Ex 3.14) does make use of the volume element. The duality in MTW 2.7 is akin to the ...
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  • 8,687
1 vote

Spacelike geodesics of FLRW radiation universe

There's nothing especially odd about solutions of this form given the shape of FLRW spacetimes and the nature of FLRW (or conformal) coordinates. FLRW coordinates are essentially polar; you can think ...
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  • 17.5k
1 vote

Palatini Action

Your question revolves around proving that you can always use the vielbein isomorphism to convert coordinate tensor indices to frame tensor indices, and vice versa with the use of the inverse. In ...
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1 vote

Can change in mass of charge, change value of charge (according to theory of relativity)?

We all know that for presence of charge it is necessary of presence of mass The word "mass" in this statement refers to the invariant mass. That is the usual meaning of "mass" as ...
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  • 66k
1 vote

Inside the Photon sphere of a black hole

Just as one constructs effective potentials for orbiting bodies in Newtonian gravity, we can do this in the Schwarzschild geometry (the spacetime of a black hole) for both massive objects and light ...
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1 vote

Inside the Photon sphere of a black hole

Imagine pointing a laser pointer in the direction of a black hole. Two things can happen: If you point close enough to the black hole the laser beam will disappear into the black hole. If you point ...
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  • 8,205
1 vote

What are the meaning of geodesics?

Newton's first and second laws of motion respectively state momentum is unchanged without a net force and the rate of momentum is equal to net force. Every student naturally wonders what the point is ...
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