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.
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How many photons exist in another dimension spaces?
As I understand, there are 2 types of photons in our (3+1) space with photon helicity $\pm 1$. How many photons exist in another spaces like (2 + 1) or (1 + 1)? Can we apply the same for gravitons?
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Does the variation of $I$ yield Bach tensor？
For $$I_1=\int \sqrt{-g}C_{abcd}C^{abcd}d^4x,$$ where $C_{abcd}$ is the Weyl tensor. If we neglect the Gauss-Bonnet term this can be reduced to $$I_2=2\int \sqrt{-g}(R^{ab}R_{ab}-\frac {1}{3} R^2)d^4x....
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Does the Casimir effect only occur between flat plates?
What happens to the strength of the Casimir effect when the Casimir plates are curved instead of being completely flat. Does this have an effect on the negative vacuum pressure at different points ...
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Is there any physical interpretation of the curvature in electromagnetism?
Electromagnetism can be modeled as a $U(1)$-principal bundle over Minkowski spacetime. The strength of the electromagnetic field is given by the 2-tensor $F_{\mu\nu}$. In differential geometry this is ...
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Definition of asymptotically flat spacetime
Following the definition in Wald's book on general relativity, in page 276 asymptotically flat spacetimes are defined using conformal isometry with conformal factor $Ω$.
Then one of the requirements ...
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How does a curvature in time equate to Newtonian gravity? [duplicate]
I often read that a curvature in time (the rate at which clocks tick) near a massive object, is considered to be the source of Newtonian gravity.
This got me wondering, does General Relativity use the ...
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Verifying whether the vanishing term in the Quasi-minkowskian metric is indeed a tensor
In Weinberg's Gravitation and Cosmology, on page 165 he notes that $h_{\mu \nu}$ is lowered and raised with the $\eta$'s since unlike $R_{\mu\kappa}$ it is not a true tensor (or at least implies it). ...
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Cosmological perturbation theory and relationship to Taylor series?
In cosmological perturbation theory, it's hard to find papers that would expose the general principle to perturb physical quantities (metric, fluid pressure and density, speed...) up to the $n$th ...
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Is space-time curvature same for all observers?
Since tensors are invariant under coordinate transformations, and a moving observer is just in another coordinate system, he should measure the same Riemann curvature tensor (with different components)...
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Curvature expression using Levi-Civita symbol
I've seen the following expression to curvature tensor
$$R^j_{ab}=2 \partial_{[a}\Gamma^j_{b]}+ \epsilon_{jkl}\Gamma^k_a \Gamma^l_b $$
both on Thiemann (equation $4.2.31$) and Rovelli's (equation $3....
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Would continuously increasing curvature of space explain seeming expansion of the universe? [closed]
Could expanding universe phenomenon be explained by slowly but continuously increasing curvature of space around masses like galaxies and galaxy clusters? In other words, I am curious whether the ...
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Curvature tensor and parallel transport around an infinitesimal loop (quadratic terms)
Given a manifold endowed with a connection $(M, \nabla)$, I want to see how the curvature tensor appears parallel-transporting a vector around a closed loop. To avoid complications with holonomies, I'...
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Holes in spacetime
Suppose a take a standard 1+1 Minkowski spacetime. Then I "make a hole in it", in the sense that I remove set of points that satisfy $x^2+t^2 \leq 1$ in some inertial frame. Can the ...
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Given an electric potential in curved spacetime, how to calculate the total electric charge?
In general $d$ dimensional setup ($d-1$ spatial coordinates and $1$ temporal), with a nontrivial metric $g_{\mu\nu}(t,\vec{r})$, having for simplicity a static electric potential $\phi(\vec{r})$.
How ...
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I can't wrap my head around the idea of matter interacting with spacetime. How is the interaction taking place?
I have tried Googling this for a long time. I have read many forums on this. But still, it doesn't make sense.
General relativity says that space-time is bent/changes when a massive object is there. ...
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How to find the double covariant derivative of a general vector?
I have been reading through Carroll's GR textbook and there is a line in the derivation of the Riemann tensor that I do not understand.
$$\nabla_\mu \nabla_\nu V^\rho=\partial_\mu(\nabla_\nu V^\rho) - ...
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Equivalent definition of Hawking quasi-local mass
I actually asked the following question at MathSE but didn't receive any response. My question is really about why the definition (2) below can be derived from the definition (1). Specifically, I don'...
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Is the universe closed or flat?
Apparently there is a tension in the measuring of the curvature of the universe (https://arxiv.org/abs/2307.07475) as apparently in 2018 the Planck collaboration got a series of results consistent ...
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How is the Ricci scalar the trace of the Ricci tensor?
The Ricci scalar is the uncontracted version of the Ricci tensor $R=R^{\mu}_{\mu}=g^{\mu\nu}R_{\mu\nu}$. Carrol describes the Ricci scalar as being the trace of the Ricci tensor, but I do not ...
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Can $\mathbb{R}^4$ be globally equipped with a non-trivial non-singular Ricci-flat metric?
I'm self-studying general relativity. I just learned the Schwarzschild metric, which is defined on $\mathbb{R}\times (E^3-O)$. So I got a natural question: does there exist a nontrivial solution (...
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Eigenvalues of the geodesic deviation equation, curvature invariants, and singularities
The geodesic deviation equation tells us what tidal forces freely falling observers experience in a local Lorentz reference frame. The tidal deformation tensor is
$$E^{\alpha}_{\gamma}=R^{\alpha}_{\...
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Same curvature but different orientation of light cones? [duplicate]
Can there be two regions of spacetime which have the same curvature, but with their light cones oriented in different directions?
In the Stack Exchange question "General Relativity via light ...
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Can there be black holes in a 1+1D spacetime? [duplicate]
In 2D Einstein tensor is always zero, that means no mass (or cosmological constant or other stress energy tensor component) is allowed. Nevertheless, we can get nonzero Riemann, even nonzero Ricci ...
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What is the minimal count of dimensions in which our curved 3D space can be embedded into? [duplicate]
As I know, GR does not need to assume anything about a >3D space, where our 3+1 spacetime can be embedded into. However, I think the curved 3D space (more clearly, the spacelike cuts of the 3+1D ...
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Ricci tensor proportional to Ricci scalar in 2D
How can we prove that in 2D the Ricci scalar is proportional to the Ricci tensor? I started with the second Bianchi identity, set 2 axes equal - and it led me to the fact that in 2D the covariant ...
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How is the curvature of a vibrating string defined in classical mechanics?
I think maybe under Fourier analysis the curvature of the string is not definable. Does a sine wave have a curvature?
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How to solve nonlinear diff eq for a general Schwarzschild metric?
So I have a general form of a spherically symmetric metric:
$$ds^2 = -g(r)_t \, dt^2 + g(r)_r \, dr^2 + g(r)_s (d\theta^2 + \sin^2 \theta \, d\Phi^2)$$
$$ R_{\theta\theta} = \frac{-g'_s g'_t g_r + ...
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Zero Einstein Tensor in 4D
In 2D the Einstein tensor is always zero, and we can easily get solution with non-zero Ricci tensor but zero Einstein tensor. But is it possible in 4D? Can we get a space-time with zero Einstein ...
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How do I conceptualize the difference between the Weyl tensor and Riemann Curvature tensor?
Currently, I am studying General Relativity from Sean Carroll's An Introduction to General Relativity: Spacetime and Geometry. In chapter 3 of this book, he develops the Riemann Curvature Tensor. I ...
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The apparent dilatation of time in General Relativity
Maybe this a dumb question, but, is the gravitational dilatation of time caused because a particle travelling through a geodesic in a curved space-time must cover a larger distance than the one ...
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Is source of space-time curvature necessary?
Einstein field equations have vacuum solutions that (probably) assumes the source of curvature (either energy-momentum tensor or the cosmological constant term or both) is elsewhere. Like, in ...
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Ricci Scalar Curvature under conformal transformation
Consider the Klein-Gordon equation in curved spacetime with metric $g$
$$\square_g \phi - \xi R \phi = 0$$
and consider a conformal transformation
$$g \longmapsto \tilde{g} = \Omega^{2} g \quad , \...
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Riemann curvature tensor parallel transport
In the Riemann curvature tensor, we are supposed to parallel transport the vectors each step of the way. So when we take the first covariant derivative, won’t that be zero? So aren’t we then taking ...
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Do neutron stars (or really dense stars) contain more volume inside of them than the expected $V=\frac{4}{3}\pi r^3$?
If I understand correctly neutron stars are so dense that general relativistic effects are not negligible anymore. Does this mean that the volume inside of neutron stars is bigger than we would expect ...
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How the equivalence principle leads to the idea of curved spacetime? [duplicate]
In wikipedia https://en.wikipedia.org/wiki/Equivalence_principle, there are three forms of equivalence principle ( equivalence of gravitational and inertial mass ) :
Weak version (Galilean) :
The ...
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Does the Weyl tensor need to be continuous across a membrane with stress-energy that separates two different space-times?
Imagine that I have two $N$+1 dimensional space-times separated by a co-dimension 1 boundary. At the boundary there is a membrane containing stress-energy. The stress-energy tensor of that membrane ...
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Question about cosmological constant and radius of curvature of the universe
Dimensional analysis suggests that $\Lambda R^2 \sim O(1)$, where $\Lambda$ is the cosmological constant and $R$ is the radius of the universe. $\Lambda$ is measured to be around $10^{-52}$ m$^{-2}$, ...
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Is the Bianchi-identity conformally invariant?
I am trying to show that for a conformal transformation $\tilde{g}_{ab} = \Omega^2 g_{ab}$ the divergence of the non-physical Einstein-tensor $\tilde{G}_{ab}$ (i.e. the Einstein tensor corresponding ...
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Disentagling coordinates and curvature?
While trying to understand General Relativity, I'm struggeling with disentangling coordinates and curvature.
The metric tensor contains information on both: coordinates as well as curvature.
Curvature ...
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Einstein tensor in 2d [duplicate]
Is the Einstein tensor in 2D or 1+1D always zero? If so, why?
I recently installed EinsteinPy and started playing wing different metrics - for the 2D cases the result turned out to be always zero.
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Is there a way to visualise / understand intuitively the curvature in the $U(1)$ circle bundle responsible for the electromagnetic force?
In general relativity we have embedding diagrams of different slices of spacetimes. These can be quite helpful to understand the geometry of a given pseudo-Riemannian manifold (especially when the ...
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How do you show that Einstein's tensor simplifies to the unitary form in geometric algebra?
In David Hestenes' Gauge Theory Gravity paper (RG), he claims that
$$G^\beta = \tfrac{1}{2}(g^\beta \wedge g^\mu \wedge g^\nu) \cdot R(g_\mu \wedge g_\nu)$$ in geometric algebra (specifically, ...
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Expressing curvature invariants ($K_1, I_1, ... $), at any one event, through Synge's WF $\sigma$ (given of each event pair, in a suitable region)
Considering a set $\mathcal S$ of events such that for each pair $p, q \in \mathcal S$ Synge's world function $\sigma$ is defined and the corresponding value $\sigma[ ~ p, q ~ ]$ is given, and such ...
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Another dimensions [closed]
Just a science ponderer, and pretty much interested in physics. Please guide me if I am wrong.
There have been many statements made by the physicists about the existence of other dimensions (...
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Extrinsic curvature, Gauss equation and Christoffel symbol contribution
This question is in the context of geometry of hypersurfaces and ADM formalism. In a $4$-dimensional manifold, we define a $3$-hypersurface with space-like tangent basis $e_a$, $a=1,2,3$, and a normal ...
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GR Action and Ashtekar Connection
Palatini action $S_{Pal}$ is (assuming Cosmological constant $\Lambda=0)$:
$$ S_{Pal}[e,\omega]=\int e\wedge e\wedge R[\omega].$$
Motion equations (varying this action) gives us Einstein Equation and
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Why isn't the curvature scale in Robertson-Walker metric dynamic?
$$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 ...
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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-...
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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 ...
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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 ...
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