Questions tagged [curvature]
Use this for questions pertaining to curvature of manifolds. Does not need to be specific to general relativity, but also for curvature of e.g. a Calabi-Yau manifold.
1,633
questions
0
votes
0
answers
24
views
Extrinsic curvature, Gauss equation and Christoffel symbol contribution
This question is in the context of geometry of hypersurfaces and ADM formalism. In a $4$-dimensional manifold, we define a $3$-hypersurface with space-like tangent basis $e_a$, $a=1,2,3$, and a normal ...
- 133
-3
votes
0
answers
77
views
Meaning of Einstein's equation (Baez & Bunn) - derivation question
See this page, and this paper.
I am mostly happy with the steps from equation (7) onward.
Just before equation (6), the authors make a choice of sign that carries through the rest of the derivation. ...
- 1,891
4
votes
0
answers
83
views
GR Action and Ashtekar Connection
Palatini action $S_{Pal}$ is (assuming Cosmological constant $\Lambda=0)$:
$$ S_{Pal}[e,\omega]=\int e\wedge e\wedge R[\omega].$$
Motion equations (varying this action) gives us Einstein Equation and
...
- 373
4
votes
2
answers
138
views
Why isn't the curvature scale in Robertson-Walker metric dynamic?
$$ds^2=-c^2dt^2+a(t)^2 \left[ {dr^2\over1-k{r^2\over R_0^2}}+r^2d\Omega^2 \right]$$
This is the FRW metric, here k=0 for flat space, k=1 for spherical space, k=-1 for hyperbolic space. $R_0$ is the ...
- 133
-1
votes
0
answers
34
views
Problem on Christoffel symbols and Riemann tensor [closed]
this is the question related to Christoffel symbol and Riemann tensor and non vanishing Christoffel symbol.
- 1
-1
votes
1
answer
100
views
When doing general relativity in practice, how do we choose the appropriate manifold describing the scenario?
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-...
- 7,793
2
votes
6
answers
2k
views
Is it possible to describe every possible spacetime in Cartesian coordinates? [duplicate]
Curvature of space-time (in General Relativity) is described using the metric tensor. The metric tensor, however, relies on the choice of coordinates, which is totally arbitrary.
See for example ...
- 63
-1
votes
1
answer
117
views
According the theory of general relativity, what is the role of causality in the changes of the curvature of spacetime? [closed]
In Einstein's equations the curvature of spacetime and energy-momentum-pressure density are correlated. Is it clear when changes in matter energy density affect causally to curvature and when changes ...
- 11
0
votes
1
answer
60
views
How do changes in energy distribution update the curvature of spacetime? [closed]
Let's give the name "gravitational signaling" to the information that affects changes in curvature. For example, the Moon draws changes in the curvature controlled by the Sun as it orbits ...
0
votes
1
answer
77
views
How to prove This Equation between Riemann Tensor and Killing Vector?
How to prove This Equation between Riemann Tensor and Killing Vector?
$$
[\nabla_\mu, \nabla_\rho]\xi_\sigma = R_{\sigma\nu\mu\rho}\xi^\nu
$$
I know
$$
R(\vec{X},\vec{Y},\vec{Z})=[\nabla_{\vec{X}}, \...
- 127
0
votes
0
answers
22
views
How you would you detect weak negative spatial curvature that only existed in cosmic intergalactic voids?
If the large space voids between galaxies had uniformly-distributed "negative" gravitational lensing, would its presence be obvious from photos in the same way that the presence of Einstein ...
- 119
0
votes
1
answer
67
views
Is curvature localised in General Relativity?
Is the curvature of spacetime in General Relativy localised?
- 349
0
votes
1
answer
44
views
How is radius of curvature defined at the end of a circular path?
This is my first time asking a question here, apologies in advance.
How is the radius of curvature defined at the end of a circular path? For example, take a particle moving on a semi-circular path ...
1
vote
0
answers
44
views
How to derive and understand the covariance property of the curvature bivector in Gauge Theory Gravity?
In David Hestenes' Gauge Theory Gravity paper (RG), the curvature tensor is defined via
$$ [D_\mu, D_\nu] M = R(g_\mu \wedge g_\nu) \times M $$
therefore
$$ R(g_\mu \wedge g_\nu) = D_\mu \omega_\nu - ...
- 123
-2
votes
0
answers
29
views
My question learning gravity [duplicate]
so einstein told us two things: number one matter curves spacetime so if you put the earth on you assume so einstein said assume that there is a curved sheet or a curved rubber sheet and if you put ...
- 45
3
votes
2
answers
333
views
Why is the Riemann curvature tensor not zero?
The Riemann curvature tensor for a torsion-free connection is given by:
$$R^d_{cab}V^c=(\nabla_a\nabla_b-\nabla_b\nabla_a)V^d$$
Where $\nabla_a$ and $\nabla_b$ are the covariant derivatives in the $a$ ...
- 128
8
votes
2
answers
640
views
How does the covariant derivative satisfy the Leibniz rule?
In Carroll's "Spacetime and Geometry", he states on page 95 (section 3.2) that the covariant derivative, $\nabla$, is a map from $\left(k, l\right)$ tensor fields to $\left(k, l+1\right)$ ...
- 83
5
votes
2
answers
494
views
Why does the Weyl tensor not show up in the Einstein field equations?
In the Einstein field equations, the only tensor that shows up is the Ricci tensor and the metric tensor, together with the Ricci scalar. The Weyl tensor though is a tensor that is a part of the ...
- 165
0
votes
0
answers
102
views
How many independent degrees of freedom does the metric tensor have in vacuum (at every point)?
A field of metric tensors fully characterises the curvature of a vacuum space-time. (For example, the spacetime between some single point masses which are themself not part of the manifold)
The metric ...
- 63
-1
votes
1
answer
101
views
Variation of the Ricci Tensor
In my own research into the use of Clifford Algebras with the Standard Model, gravity appears, but in calculating the Lagrangian of the theory I get:
$$-\frac{3}{4}\sqrt{-g}R^{\alpha\beta}R_{\alpha\...
- 11
0
votes
1
answer
98
views
What happens to the "curvature term" in the equations of motion for a rotating fluid near Earth's poles?
For a rotating fluid in spherical geometry, one of the terms of the equations of motion is the "curvature term". For example, for the zonal component of velocity (corresponding to eastward ...
- 1,535
8
votes
3
answers
4k
views
Why is Spacetime described as flat even though we live in 3 dimensions of space?
I’ve always heard and seen diagrams that show spacetime as being “flat” or in 2 dimensions with curvature. How does this correspond to the 3 spacial dimensions that we perceive to exist in?
- 125
2
votes
1
answer
56
views
Can we use vielbeins on curved space?
My question is: Can we use vielbeins on (let's say ) an anti de-sitter space?
This is why I am confused:
To couple fermions with gravity in curved space, using vielbeins is a well-known approach. So ...
- 291
0
votes
0
answers
82
views
Deriving the Ricci tensor on the flat FLRW metric
I am currently with a difficulty in deriving the space-space components of the Ricci tensor in the flat FLRW metric
$$ds^2 = -c^2dt^2 + a^2(t)[dx^2 + dy^2 + dz^2],$$ to find:
$$R_{ij} = \delta_{ij}[2\...
- 1
0
votes
0
answers
43
views
Question on gravity and spacetime curvature [duplicate]
In General Theory of Relativity, it is explained that the fabric of reality i.e. spacetime bends around objects with mass, and that curvature causes other objects to come close to/ fall towards the ...
2
votes
1
answer
166
views
Can we infer Hawking radiation assuming the Unruh effect?
An observer near the event horizon of a black hole will experience an extremely strong gravitational field.
Due to the principle of equivalence, this observer cannot locally distinguish between this ...
- 415
1
vote
1
answer
64
views
How does Riemann curvature tensor make an appearance in the geodesic deviation equation? ("A relativist’s toolkit")
How does the switching of indices in lead to the appearance of $R$ in the equation (specifically in the derivation given in "A relativist’s toolkit")?
- 11
2
votes
3
answers
93
views
How do we know that the speed of light is constant everywhere, not just here? [duplicate]
It might well be that universal constants, say the speed of light, are only constant as far as we can tell in our chunk of the universe - in the same way that the Earth looks flat in the area you live....
- 121
1
vote
3
answers
842
views
Why does mass make curvature in spacetime? [duplicate]
According to Einstein's general relativity theory, matter with mass makes curvature in spacetime. The greater the mass, the curvature in spacetime will be greater.
My questions:
Why will mass make ...
- 19
2
votes
1
answer
73
views
How do you relate $\Omega_{k}$, the curvature term in the FLRW metric, to the radius of curvature?
I have assumed, for reasons a bit too detailed to go into here, that if $\Omega_{k}$, the curvature term in the FLRW metric, is equal to 1, then the radius of curvature is equal to 13.8 billion light ...
- 325
1
vote
1
answer
67
views
Indices of $(\text{Riem})^3$?
This question relates to writing higher curvature terms in momentum space with respect to GR as an effective field theory.
I know that $R_{\alpha\beta\mu\nu} \sim \partial_\beta\partial_\mu h_{\alpha\...
- 433
4
votes
6
answers
881
views
How does general relativity theory explain gravitational pull? [duplicate]
I watched some videos on YouTube that explain why gravity is not a force,
according to general relativity theory.
I can wrap my head around the idea that spacetime can be curved due to a massive mass,
...
- 141
2
votes
0
answers
51
views
Functional derivative of Ricci tensor respect to metric [closed]
I am working on following functional derivative
$$
\dfrac{\delta R_{\mu\nu}}{\delta g_{\alpha\beta}}=C^{\alpha\beta}_{\mu\nu}
$$
Intuitively, it should be nonzero but I can not work it out.
- 21
0
votes
0
answers
16
views
Space expansion and change of electromagnetic radiation wavelenght
The expansion of the universe acts to a photon ray equally as an expanded baloon on points drawn on it just increasing the distances between this differential photons so the integral photon looks like ...
- 2,223
0
votes
1
answer
46
views
Need clarification/input on a curvature dilemma
Even though I said I'd never waste this much energy arguing with a flat earther, I have a dilemma and need input.
I'm in the Vancouver, Canada area. I've been shown a picture that the person claims is ...
-1
votes
1
answer
82
views
Is the Einstein-Hilbert Lagrangian a constant of motion?
In QMechanic's Answer, it is stated that
$L$ is a constant of motion (COM), and the constant of motion ≠0
is not zero.
Is the Einstein-Hilbert Lagrangian a constant of motion? Does the square of its ...
2
votes
2
answers
120
views
Calculating the Ricci tensor
I am currently working through an exercise to calculate the component $R_{22}$ of the Ricci tensor for the line element $ds^2=a^2dt^2 -a^2dx^2 - \frac{a^2e^{2x}}{2}dy^2 +2a^2e^xdydt -a^2dz^2$. The ...
- 99
5
votes
2
answers
184
views
Meaning of zeros in the metric tensor
I'm trying to find the $g_{0i}$ components of the metric I mentioned here, but it has turned extremely difficult. My current strategy is to equate Ricci tensor components gotten from the Christoffel ...
- 503
2
votes
2
answers
328
views
Is there a relation between spacetime curvature and radiation?
To my understanding, the curvature of spacetime is determined by the stress-energy tensor.
I was wondering if we could calculate some of those components using radiation.
Is it possible that objects ...
0
votes
2
answers
87
views
Proof of the second Bianchi identity
Suppose $\omega$ is a connection one-form, and the curvature tensor is defined as
$$R^a_{~~b} = d\omega^a_{~~b} + \omega^a_{~~c}\wedge \omega^c_{~~b}~,$$
where the latin indices refer to the fact that ...
- 571
1
vote
3
answers
83
views
Where is the normal force that pushes us up comes from if gravity is not a force according to general relativity?
https://youtu.be/XRr1kaXKBsU?t=530
I was watching this video and at this point he said that since gravity is not a force as per GR, we are left with only these normal forces pushing you up that ...
- 21
0
votes
0
answers
52
views
Proposition 4.4.5 in The Large Scale Structure of Space-Time
In the text by Hawking and Ellis they set the following proposition (pg. 101):
Proposition 4.4.5
If $R_{ab} K^a K^b \geq 0$ everywhere and if at $p = \gamma(v_1)$, $K^c K^d
K_{[a}R_{b]cd[e}K_{f]}$ ...
- 61
3
votes
0
answers
66
views
Why are departures from flat spacetime geometry small on scales smaller than the Hubble radius?
In Chapter 5 of Baumann's cosmology book where he discusses structure formation starting from Newtonian perturbation theory, Baumann mentions at the beginning that
Newtonian gravity is a good ...
- 179
0
votes
0
answers
29
views
The radius of curvature of the lens and how to measure it
what is the curvature of the lens flare? Is it different from the normal lens radius? How can it be measured with measuring devices?
4
votes
2
answers
570
views
Characterising Minkowski spacetime as a flat manifold with some other property?
It is known that flat manifolds can be characterized as follows
If a pseudo-Riemannian manifold $M$ of signature $(s,t)$ has zero Riemann
curvature tensor everywhere on $M$, then the manifold is ...
- 1,556
4
votes
3
answers
155
views
How do we know if spacetime is bent?
For example I'm at certain location in outer space.
How do I know if the spacetime in front of me is bent, e.g. by some dark matter?
- 141
7
votes
4
answers
303
views
Can you get $g_{\mu\nu}$ from $R^\lambda_{\alpha\beta\gamma}$?
From the Christoffel symbols it is easy (although cumbersome) to get the Riemann tensor $R^\lambda_{\alpha\beta\gamma}$ from the metric tensor $g_{\mu\nu}$. But is it possible to reverse the procedure?...
- 503
13
votes
3
answers
2k
views
In general relativity, why is Earth able to accelerate?
I was told and convinced that gravity is not a force, and in free fall you're an inertial frame and experience no force, and when on the surface of Earth you would be accelerating upwards.
What I ...
- 320
4
votes
1
answer
118
views
Is a conformally coupled scalar always massive?
Maybe this is trivial, but the action of a conformally coupled scalar is
$$ S = \frac{1}{2} \int d^Dx \sqrt{g} (g^{ab} \partial_a \phi \partial_b \phi + \xi R \phi^2),$$
where $\xi = (D-1)/4D$. Does ...
- 388
6
votes
0
answers
90
views
Raychauduri Equation in "The Large Scale Structure Of Space-Time" book by Hawking & Ellis
Previously to Raychauduri equation, Hawking-Ellis obtain equation (4.25) (pg. 84) namely
$$
\frac{d \theta_{\alpha\beta}}{ds} = - R_{\alpha 4 \beta 4} - \omega_{\alpha\gamma}
\omega_{\gamma\beta} ...
- 61