Questions tagged [general-relativity]
Questions related to the mathematical aspects of Einstein's theory of relativity. For the physics and its interpretations, please ask at the physics.SE. You may also consider the tags (differential-geometry) and (pde).
798
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Writing the explicit form of antisymmetrising the metric and Ricci tensor
Whilst going through the solutions to a GR worksheet, I struggled to understand a lie in the solutions. The line is:
$g_{\sigma[{\mu}}\nabla_{|\rho|}R_{\nu]}^\rho=\frac{1}{2}(g_{\sigma\mu}\nabla_\rho ...
- 1
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Formulating $Spin^c(3,1)$ Connection and Curvature on a $GL+(4,R)/Spin^c(3,1)$ Structured Manifold
I am exploring a geometric framework where the usual metric tensor role (as in $GL^+(4,\mathbb{R})/\text{SO}(3,1) $) is replaced by a structure defined by the quotient $ GL^+(4,\mathbb{R})/\text{Spin}^...
- 2,551
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Confused by the path I am asked to follow in order to solve the killing equation on S2.
I am asked to find the Killing vector fields on $S^2$ where the line element is given by $ds^2=d\theta\otimes d\theta +\sin^2\theta d\phi\otimes d\phi$.
I know how to solve this problem by considering ...
3
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1
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Understanding the Geometry Spawned from Quotient Spaces $GL^+(4,R)/SO(3,1)$, $GL^+(4,R)/Spin(3,1)$, and $GL^+(4,R)/Spin^c(3,1)$
I'm working on a theoretical framework where I explore different quotient spaces formed with GL$^+$(4,R) and various groups. Specifically, I'm interested in the types of geometry that arise from the ...
- 2,551
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39
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Is there an easy proof of $d \star S = s \cdot \epsilon$ for two forms S
In https://arxiv.org/pdf/1906.08616.pdf eq. 3.51 the following identity is proposed to hold for two-form $S$ on a manifold with metric $g_{\mu \nu}$.
$d \star S = s \cdot \epsilon$,
where $d$ is the ...
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1
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Naive questions by a beginner in general relativity.
I want to ask some very naive questions in general relativity. I have the background of PDE and few Riemannian geometry.
After Schwazchild gave a solution, people study its singularity and predict the ...
- 443
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How to differentiate a light-cone integral in relativity?
Consider the following integral
$$
\int_{V_X(v)}I(Y)\,dY\tag1
$$
In light cone coordinates $x_\pm=x\mp vt$, where $t$ is time and $v\in\mathbb{R}^+$, $V_X(v)$ is the past light cone, given, by
$$
V_X(...
- 3,095
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Geodesics with a normal distribution
I'm working with the Desmos 3D calculator and I want to find the geodesic across some manifold currently I have a generalized equation for a generalized 3D distribution curve: $$f(x, y) = ab^{-{\left(\...
- 23
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How to know if a set of equations are Lorentz Invariant Spinors
I'm currently busy with a course in QFT and am completely baffled by Spinors. In particular there are two parts, that while I mostly understand the theory, struggle to show mathematically (especially ...
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1
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How can I transform L_V W^μ to another coordinate system and show that it transforms like a tensor, L is the lie derivative
I have reached $L_V*W^μ=(dx^{ν'}/dx^ν)*V^{ν}*(dx^{ν}/dx^{ν'})*d_ν*(dx^{μ}/dx^{μ'}*W_μ)-dx^{ν}/dx^{ν'}*W_*dx^{μ}/dx^{μ'}*d_μ*dx^{ν'}/dx^ν*V^ν$ and I am stuck. I know i have to find that
$L_V*W^μ= dx^{μ}...
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Is it possible to find a coordinate chart associated to a change of basis in the tangent space?
I am following Lee's Introduction to Smooth Manifolds, but this question is motivated from studying bases of vectors in the context of general relativity. Let $M$ be a smooth $n$-manifold. In the ...
- 2,783
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Prove $g_{\mu \nu}= g_{\mu \nu}|_p + \frac{1}{3}R_{\lambda \mu \nu \rho}|_p x^\lambda x^\rho +O(x^3)$ from the geodesic eq in normal Riemann coord.
In Riemann normal coordinates (which, by definition, are the coordinates of a
local inertial frame) the equation of a geodesic through the origin $p$ is
$x^\mu(s) = a^\mu s$ where $a^\mu = \frac{dx^\...
- 2,987
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1
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How to show that conformal spaces share the same null geodesics?
Two metrics $ds^2 = g_{\mu\nu} dx^\mu dx^\nu$ and $d\tilde s^2 = \tilde g_{\mu\nu} dx^\mu dx^\nu$ in the same coordinate system are said to be conformally related if ̃
$\tilde g_{\mu\nu} (x) = \psi(...
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How to show equivalence of definitions ot the Rieman tensor
The Riemann tensor can be thought of as a map from a triple of vectors X, Y ,
Z to a fourth vector, written in a coordinate-free way as
$R(X, Y )Z = \nabla_X \nabla_Y Z − \nabla_Y \nabla_X Z − \nabla_{...
- 2,987
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A strange notation in Geroch's proof of the positive-energy theorem
The following passage came from a 1973 article authored by Robert Georch and titled "ENERGY EXTRACTION". The author tries to prove the positive-energy theorem. I would like to ask two ...
- 225
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29
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Variation of the Ricci Scalar with respect to a non-metric tensor in a product form
Let $ g_{\mu\nu} $ be the metric tensor of a spacetime manifold. Instead of expressing $ g_{\mu\nu} $ directly, I wish to express it as a product of two other tensors, say $ p_{\mu\alpha} $ and $ q^{\...
- 2,551
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Need help with planning self study for learning differential/Riemannian geometry and General Relativity rigorously.
I would like to learn Mathematics for understanding GR, Differential Geometry, Riemannian Geometry and related research papers rigorously.
I would like to carve out a clear path to understand these ...
- 1,104
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0
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56
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$L^\infty$ decay for the Klein–Gordon equation
By Fourier transform, the flow of the Klein–Gordon wave flow is $e^{it\sqrt{1-\Delta}}$. That is, if we have initial data $\phi(0,x)=\phi_0(x)$, then the solution will be given by $\phi(t,x)=e^{it\...
- 307
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Vaidya geometry null geodesic solution
The Vaidya geometry written in ingoing null coordinate $v$ is given by the metric,
\begin{equation}
ds^2 = -\left(1 - \frac{2 m(v)}{r} \right) dv^2 + 2dvdr + r^2 (d\theta^2+ \sin^2{\theta} d\phi^2) \...
- 435
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59
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How to show some smooth coordinates do not exist for a simple metric
I have a simple metric:
$$
ds^2=dr^2+r^2(1+kr)d\theta^2
$$
Near the origin $r=0$, I wanted to switch to (smooth) coordinates $x=r\cos\theta$, $y=r\sin\theta$, but it seems this is only possible when $...
- 23
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0
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47
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Computations in asymptotically flat coordinates
For reference, I'm following the first line in proof of Lemma 6.1 of this paper. This is my first time working with asymptotic flatness, and I would greatly appreciate it someone would tell me if the ...
- 1,699
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43
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Is every connection form $\omega$ the extremal of some functional $S[\omega]$?
Is every connection form $\omega$ the extremal of some functional $S[\omega]$ ?
Context: Palatini action $S_{Pal}$ of General Relativity is (assuming Cosmological constant $\Lambda=0)$:
$$ S_{Pal}[e,\...
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0
answers
55
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Estimates of a matrix norm in general relativity
Let $\bar{D}_{2\rho}$ be a closed disk in $\mathbb{R}^2$ of radius $2\rho$, where $\rho>1$.
Let $\psi_0$ be a smooth map：
$$\psi_{0}:[0,1]\times\bar{D}_{2\rho}\rightarrow\hat{S},$$
where $\hat{S}$ ...
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Connection between manifold curvature and specific type of coordinate transformation
I'm studying Susskind's GR TTM book, in which he gives a nice explanation of why differential geometry is needed for GR. But there is one gap that I want to fill.
The argument is: through a thought ...
- 2,479
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0
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15
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Marginally Outer Trapped Surface as a Minimal Surface
In general relativity, a marginally outer trapped surface is defined to be a spacelike, two-dimensional surface in a space-time such that outgoing null rays perpendicular to the surface are do not ...
- 1,230
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1
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76
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Help with Parallel transport of a vector
I have a question about parallel transport of a vector.
In picture A) we parallel transport a vector on a (red)curve in a flat plane. This can be clearly seen as the (blue)vector is constant in ...
- 625
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0
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64
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Interpreting a fictitious line interval (metric) with 3 extra dimensions
Consider a fictitious line interval (metric) with 3 extra dimensions as follows:
$ds^2 = -2dt^2 + a^2(t) \left( \frac{dr^2}{1 - r^2} + r^2 d\theta^2 + r^2 \sin^2\theta d\phi^2 \right) + da_{1}^2+da_{2}...
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Beginner (yet rigorous) book recommendation for differential geometry / topology for General Relativity [duplicate]
I went through a quite a lot of posts here but did not find the information I wanted. I'm looking for a beginner / introductory (yet mathematically rigorous) book recommendation for differential ...
- 595
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67
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What is solution of a homogenous second order linear differential equation?
Function $y(x)$ and $g(x)$, defined on interval $[0,1]$, are related by linear differential equation that can be read as a second order or as a first order ODE
\begin{equation}
g~y’’+\frac{1}{2}~g’~y’+...
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0
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Closed trapped surface without singularities
Closed trapped surfaces are important in general relativity in large part due to Penrose's incompleteness theorem.
The theorem states that: if a spacetime $(M, g)$ is globally hyperbolic, with a ...
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votes
1
answer
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Cosmological evolution of a hyperbolic space form
Please consider the following metric
$$ ds^2 =-dt^2+f(t)( {\frac {a^2}{r^2+a^2}}dr^2+ r^2 d\Omega_{n-1})$$
where $d\Omega_{n-1}$ is the metric of the hyper-sphere $S^{n-1}$
Making "experimental ...
- 2,187
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1
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Curvature invariants of hyperbolic space forms
Please consider the following hypersurface in $\mathbb{R}^{n+1}$:
$$-{c}^{2}{t}^{2}+\sum _{i=1}^{n}{x_{{i}}}^{2}=-{a}^{2} $$
Usin hyper-spherical coordinates the equation for the hypersurface is:
$$r^...
- 2,187
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0
answers
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Approximation of pressure gradient term in relativistic hydrodynamics equations to calculate vertical height of a thin rotating flow
In studies of rotating fluid flows around a relativistic star, the vertical height, or (half)-thickness, of the flow is usually obtained from the vertical component of the Euler equation.
The Euler ...
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Optical geometry in GR: How can the "vanishing of the line element" define a submanifold?
This question is based on a construction in this lecture from the WE-Heraeus International Winter School on Gravity and Light 2015. Most of the question is context, just the last paragraph is really ...
- 696
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Deriving an expression for the wave operator acting on the gravitational spinor: $\square \Psi_{ABCD} = 6\Psi_{EF(AB} \Psi_{CD)}{}^{EF}$
I am working through the exercises in the ‘Spinors’ section of Stewart’s book ‘Advanced General Relativity’.
Problem 2.5.7 asks to derive the spinor formula
$$
\square \Psi_{ABCD} = 6\Psi_{EF(AB} \...
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Is a globally hyperbolic spacetime past causal geodesic complete?
Let $(M, g)$ be a globally hyperbolic spacetime with $\Sigma$ a Cauchy surface. It is well known that such spacetimes can be geodesically incomplete. But the following statement seems intuitive and, ...
- 5,046
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How to relate a derivative in tortoise coordinates with the radial coordinate?
Since the radial coordinate and the tortoise coordinate are related by
r* = r + 2M Log (r/2M-1)
How can I write the derivative ...
User8563
2
votes
1
answer
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Is M/SO(3) a manifold when the orbits are spheres?
$M$ is a smooth 4-manifold equipped with an $SO(3)$-Lie group action, such that each orbit is diffeomorphic to the sphere $\mathbb{S}^2$. Prove or disprove that $M/SO(3)$ is a smooth manifold.
Due to $...
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Why is $\text{Ric}(g)=0$ a quasi-linear PDE in harmonic coordinates?
While studying the dynamics of the Einstein vacuum equations
$$
\text{Ric}(g)=0
$$
for $(M,g)$ unknwon, I've come across the statement that in harmonic coordinates $x^\lambda$ defined by $\Box_g x^\...
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0
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32
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Laplace-Beltrami operator in coordinates
I know that on a Lorentzian manifold the Laplace-Beltrami operator of a function $\phi\in C^\infty(M)$ is defined as
$$
\triangle_g \phi = \text{div}(\text{grad}\phi).
$$
Now I've come across the ...
1
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2
answers
70
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A fully evaluated tensor is a smooth function, so why does using the Leibniz rule give a bunch of Gammas?
According to how the covariant derivative acts on smooth functions:
$
\nabla_V(T(w, Y)) = V^k \partial_k (T^i_j w_i Y^j) = V^k [ \partial_k T^i_j w_i Y^j + T^i_j \partial_k w_i Y^j + T^i_j w_i \...
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0
answers
34
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Curvature of an affine system
Recently, I find an interesting paper that mentioned the Definition of curvature of an affine optimal control system. It suddenly reminded me that many textbooks on Riemannian geometry only tell us ...
- 223
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31
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Time derivative of Christoffel symbols
Let $\Gamma_{ij}^k$ be the Christoffel symbols of a time-varying metric $g$ on $\Sigma$, a 3-manifold (say). Let $k_{ij}$ be the second fundamental form of $\Sigma$ as embedded in $M = \mathbb{R} \...
- 5,046
6
votes
3
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572
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Necessary and sufficient mathematical structure for spacetime continuum
In physics, we often say that spacetime is a collection (set) of all events (idealized occurrences of zero extension in space-time, the "here and now"s). Moreover, spacetime is said to be a ...
- 171
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1
answer
76
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Is there a method to find the length of a geodesic length from its parameterisation and 'velocity'?
I am attempting to implement this, pages 11/12 method to plot geodesic equations on the surface of an object, beginning with the sphere. I would like to be able to guarantee that the geodetic length ...
0
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0
answers
35
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Sagemath -- finding Eigenvalues of a matrix representation of a tensor
I am using sagemath to compute Einstein tensors of a non-standard spacetime. The output is something horrid and non-diagonal. I need to find the Eigenvalues of this tensor... which is represented as a ...
- 13
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0
answers
24
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Finding total rotation angle and rotation axis
In the R3 space, we perform two consecutive rotations: in the first one we make a rotation of an angle (θ) leaving the x axis invariant, and in the second one we make a rotation of an angle (π) ...
0
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0
answers
72
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Two spacecraft leaving earth-Doppler relativistic effect
I recently begun approaching special and general relativity. I found this problem in my book and I'm trying to solve it but I'm finding some difficulties:
An observer on Earth sees two twins A and B, ...
- 307
1
vote
0
answers
57
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Geodesics and antipodal identification
The solution of Einstein field equations in case of static spherically symmetric perfect fluid sphere is given by the metric $$ds^2=f^2(r,\alpha)~c^2 dt^2-g^2(r,\alpha) dr^2-r^2 (d\phi^2+\sin^2{\theta}...
- 155
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0
answers
65
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Are Killing vector fields eigenvectors for extrinsic curvature?
A quick question I struggle with these days:
Let's consider a 2D smooth surface embedded with $\mathbb{R}^3$. Such a surface features an extrinsic curvature tensor field ($\boldsymbol{\kappa}$) as ...
- 11