New answers tagged curvature
3
votes
Einstein field equations from covariant derivative of a general linear gauge transformation
None. Nothing of this has anything to do with gravity. Your equation for "R" defines the gauge field curvature $F_{\mu\nu}$, not the Riemann curvature $R_{\alpha \beta\mu\nu}$.
- 41.9k
8
votes
Is there a version of the Einstein field equations that uses the Riemann curvature tensor instead of the Ricci curvature tensor?
Here is the Lichnerowicz' form of Einstein's equations:
\begin{eqnarray}
R^μ{}_{νρσ;μ}=J_{νρσ},\\
R_{μνρσ;α}+R_{μναρ;σ}+R_{μνσα;ρ}=0,\\
\end{eqnarray}
where the “current” tensor is
$$J_{νρσ}=\left(T_{...
- 12.6k
10
votes
Is there a version of the Einstein field equations that uses the Riemann curvature tensor instead of the Ricci curvature tensor?
If you mean an equation such that the stress energy tensor at a point of the spacetime determines the Riemann tensor at that point, then the answer is negative for physical reasons. Such an equation ...
- 57.9k
6
votes
Is there a version of the Einstein field equations that uses the Riemann curvature tensor instead of the Ricci curvature tensor?
You can start with Ricci decomposition in a $n$ dimensional Riemannian manifold:
$$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_{...
- 887
14
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_{\...
- 52.1k
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 ...
- 943
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 ...
- 8,054
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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7
votes
Accepted
Confusion about Einstein's field equations
$R = g^{\mu\nu} R_{\mu\nu}$ does NOT mean $R_{\mu\nu} = R g_{\mu\nu}$.
- 21.1k
0
votes
How would someone discover the Einstein-Hilbert Action?
A physicist is liable to build a theory as the simplest option that does what they want. The truth is we can't "deduce" a unique theory; @Andrew's answer gives a great overview of more ...
- 22k
5
votes
Accepted
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 ...
- 9,260
7
votes
Accepted
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 ...
- 34.4k
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}) &...
- 819
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 ...
- 943
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 \...
- 12.3k
0
votes
Is space-time made of something?
General relativity can be viewed another way than space-time curvature: e=mc2 therefore gravitation energy has mass that follows Newton. Of course, this is a feedback situation: when gravitational ...
- 101
3
votes
Does solving Einstein's field equation depend on Newtonian equations?
I recently heard, that there is a derivation of the field equations not using the Newtonian limit. The one I know of uses it though. We get $g_{00}=1-\frac{2\Phi}{c^2}$ out of the Newtonian limit of ...
12
votes
Accepted
Does solving Einstein's field equation depend on Newtonian equations?
When Einstein derived his equations the Newtonian limit (and it may be taken also in the present of sources) was an important check for the theory. It is also useful to understand the limits of the ...
- 7,318
2
votes
Does solving Einstein's field equation depend on Newtonian equations?
If, in some limit, GR didn't reduce down to $\kappa \nabla ^{2}\phi = \rho$ for some constant kappa, (where $\phi$ is a gravitational potential such that ${\vec g} = {\vec \nabla}\phi$), then it ...
- 38.8k
5
votes
Is space-time made of something?
In theoretical physics there is always a 'choose your battles' judgement call to make.
My favorite example in the history of physics is Newton proposing the inverse square law of gravity. Many of his ...
- 15.4k
0
votes
Is space-time made of something?
In a nutshell, General Relativity is an interpretation of gravity where instead of mass simply being attracted to mass, mass exists within a conglomeration of space and time and causes it to curve. ...
- 56
3
votes
Is space-time made of something?
What exactly is the "spatial points" that can be curved?
As far as we know, spacetime is not made of “things”. When we talk about curvature we are not talking about a material “thing” that ...
- 66.3k
0
votes
Can 1D beings in 1+1D determine if they are in a curved universe?
Are you assuming that the 1D beings can only learn by sensory stimulus? Assuming they have sufficient mathematics, why couldn't a 1D being determine that it was living in a single dimension of a ...
3
votes
Can 1D beings in 1+1D determine if they are in a curved universe?
There is no intrinsic curvature in 1D. If you work out the math you find that there are $n^2(n^2−1)/12$ independent components of the curvature tensor in $n$ dimensions. For 1D that is zero, and for ...
- 66.3k
0
votes
What should be the shape of spacetime to repel opposite masses from one another?
If you have a positive and a negative mass the negative mass will repel the positive one and the positive one will attract the negative one, so together they will form a runaway pair.
The ...
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1
vote
Could the curvature of spacetime, as in general relativity, result from the interaction of quantum fields?
Is it possible that the curvature of spacetime, as in general relativity, results from the interaction of those quantum fields with the fields of quantum matter?
I find this phrasing a little weird, ...
- 8,054
-1
votes
Einstein field equations in empty space, question about non-zero curvature
It is a good question.
Suppose there is only empty space, and nothing changes, no field no nothing, then what? ... well, as nothing changes there is no time (because time ... is change). For there to ...
1
vote
Could the curvature of spacetime, as in general relativity, result from the interaction of quantum fields?
I'm gonna write a pretty quick and not much detailed answer, but to give you a quick idea.
The first taught thing to quantize gravity is actually, that its quantum field behaves as a spin 2 particle ...
- 554
4
votes
Why are distances to event horizons linear with mass when gravitational effects fall off as $1/r^2$?
Forget black holes, just think about Newtonian physics.
The escape speed from some radius around a spherically symmetric mass is just where the sum of the kinetic and potential energy is zero. This ...
- 111k
-1
votes
Why are distances to event horizons linear with mass when gravitational effects fall off as $1/r^2$?
Let's say that a 2 kg dumbbell is hanging 100 m above event horizon. Let's say that when said dumbbell is dropped into the hole the mass of the hole increases by 100 grams. (This can happen ...
- 1,576
2
votes
Is space — as opposed to space-time — curved by a gravitating mass?
"Can we relatively freely rotate our 4 dimensional coordinate system for the universe's spacetime such that what was space before (time fixed, say at zero) is rotated "into" the time ...
6
votes
Accepted
Is space — as opposed to space-time — curved by a gravitating mass?
A few facts:
In a 4-dimensional manifold such as spacetime you can pick any timelike direction and call it time in the vicinity of any given event. Directions orthogonal to this will then make up '...
- 47.1k
12
votes
Is space — as opposed to space-time — curved by a gravitating mass?
Let's suppose you are an observer and you have a clock to measure time and rulers to measure distance. You construct a coordinate system by placing yourself at the origin and using your clock and ...
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