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).

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Maximizing properties of geodesics on Pseudo-Riemannian manifold [closed]

I'm studying some GR and my book says that in Pseudo-Riemannian manifolds geodesics may even maximize the path locally. That's what happen to the timelike geodesics, for example. My first question: Is ...
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1answer
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Relativistic velocity transformation law

I'm looking for some clarification on what each of the terms in the relativistic velocity transformation law are. The formula is: $s = (v+u)/(1 + uv/c^2)$ It would be really great if you could give ...
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0answers
29 views

Vector Relations in Minkowski Space

Consider $\mathbb{R}^4$ equipped with the Lorentz inner product: $$\eta(X,Y)=x^0y^0-x^1y^1-x^2y^2-x^3y^3$$ Let $X,Y\in\mathbb{R}^4$, $X\not=0$ and $Y\not=0$, two future-causal (this means: ...
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2answers
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Killing vector field along a geodesic

I was trying to show that a Killing vector field satisfies the Jacobi Equation for a geodesic, just by assuming that \begin{equation} \nabla_\mu X_\nu + \nabla_\nu X_\mu=0 \end{equation} Indeed, if I ...
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1answer
34 views

Chronology condition and metric perturbations

Let $(M,g)$ be the quotient of the 2-dimensional Minkowski space-time by the discrete group of isometries generated by the map $f(t, x) = (t + 1, x + 1)$. Show that $(M, g)$ satisfies the ...
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How can we represent the 3 space coordinates uniquely by one point that can be put on a single axis?

In the Space-Time diagram (in the rest frame) we often take the Space axis as the horizontal axis and the time axis as the axis perpendicular to it as in the given figure. While there are other ...
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1answer
31 views

Constructing a bilinear form on $\mathbb{R}^2$ that gives rise to a particular matrix

As the title says, I'm trying to create a bilinear form $B(\cdot, \cdot)$ on $\mathbb{R}^2$ with some particular constraints (which I do not know as yet) related to the Lorentzian space ...
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2answers
26 views

Orthogonal vectors in case of Lorentz Metric

Let us consider $\mathbb{R}^4$ equipped with the Lorentz metric $$\eta(X,Y)=x^0y^0-(x^1y^1+x^2y^2+x^3y^3)$$ Let $X\in\mathbb{R}^4$ a time-like vector, that is $\eta(X,X)>0$. I want to show that all ...
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1answer
35 views

Is there a general coordinate transformation perserving the components of an Euclidean metric?

In the Euclidean space (or Lorentz spacetime, if you are interested in relativity), there is one orthonormal coordinate system $\{x^\mu\}$ such that the distance squared is given by ...
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1answer
47 views

Does the Riemann tensor encode all information about the second derivatives of the metric?

In answer to this question I suggested the following as a motivation for the definition of the Riemann tensor: Let two $\mathcal M$ and $\mathcal N$ be two dim-$n$ Lorentzian manifolds with ...
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Proving path of motion is a Geodesic in general reletivity.

I am studying the work of Miguel Alcubierre, in particular his warp drive metric. A consequence of his metric is that the ship will travel on a geodesic and this is what I am trying to prove. I ...
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1answer
19 views

Tensor equations. Can I change an equation from covariant to contravariant?

Say I have a tensor equation like $G_{ab}=R_{ab}-\frac{1}{2}g_{ab}R$. Does this also imply that $G^{ab}=R^{ab}-\frac{1}{2}g^{ab}R$?
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0answers
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Proving Lorentz invariance given an action functional

Given an action integral: S = $\int_{t1}^{t2} L[q(t), \dot q(t); t]dt$ How does one prove whether or not it is Lorentz invariant?
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2answers
52 views

Is this the correct way to think about this exterior derivative?

If we have, expression (1) with the $\star$ sign used for Hodge star $$\star(d(\alpha))$$ where $\alpha$ is a complex function. We are speaking in 3 dimensions (x,y,z) that is expression(1) can be ...
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1answer
43 views

Frame acting on a curve/Geodesic eqution

I have a technical question about the geodesic equation. Assume we have a frame $(E_{1},E_{2},E_{3},E_{4})$ (not necessarily a coordinate frame). Assume we have a parametrized curve $\gamma(s)\in M$ ...
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0answers
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Wald General relativity…regarding problem 2

In proving theorem 2.2.1 Wald uses the result: $F(x)=F(a)+\sum_{\mu}(x^\mu-a^\mu)H_{\mu}(x)$ this result is later posed as problem 2. My question is whether $H_\mu(x)$ really independent of $a$. If ...
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170 views

Hawking's and Ellis' derivation of the form of Einstein's field equations

On pages 72-73 of the book "The large scale structure of space-time" Hawking and Ellis show while determining the form of the field equations of general relativity that there is a relation of the form ...
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14 views

Non-linear perturbation definition

What exactly is the definition of a nonlinear perturbation when applied to a background spacetime metric? I have seen so called "linear perturbations" which look like $$ds^2 = -(1+2\Phi)dt^2 ...
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1answer
28 views

How would I relate $e^{i\omega_{\mu\nu}}J^{\mu\nu}$ with lorentz transformation matrix?

How to go from the given exponential form to given transformation matrix? Do I need to know the generators of boost and rotation? How will I find $\omega_{\mu\nu}$ and $J^{\mu\nu}$ in that case?
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0answers
29 views

What is the root structure of the Diffeomorphism Group?

Being a physicist, I think it'd be cool to have Coxeter plane projections of the root systems of the symmetry groups associated with the fundamental forces hanging on my walls (example for E8: ...
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General Relativity perturbation

Could anyone explain to me what I have misunderstood/missed when trying to understand this paper on GR perturbation? The paper is http://arxiv.org/pdf/0704.0299v1.pdf In equation 25 for $R_{00}$, ...
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1answer
72 views

Volume of a paracompact manifold

It is stated, without proof, in Wald (1984) (General Relativity) that given any connected manifold $M$ (which is by definition paracompact), one may define a volume measure $\mu$ such that $\mu[M]$ is ...
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0answers
40 views

When does a pseudo-Riemannian manifold have an always positive norm Killing field?

When does a pseudo-Riemannian manifold have an always positive norm Killing field? (you may assume that the isometry group is of the form $SO(1,n)$ if necessary) In the context of general ...
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The metric and Kronecker's delta

I am reading some lecture notes for GR and it is currently showing how we are going to derive the field equations using a metric for a massive free particle with a metric ...
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2answers
68 views

Is Relativity a specific instance of Riemannian geometry?

If I am a mathematician and do not anything about Special/General Relativity, then should I study Riemannian geometry to learn Relativity? Is Relativity just an instance/example of some particular ...
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1answer
21 views

How to isolate the variable “v” buried deep in a relativity formula?

finding v in v = L[1-(v^2/c^2)]/t closest attempt: [1-(c^2/v^2)]v = L/t I've been working on this since yesterday. I think I need some help.
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Simply-connected Lorentzian manifold and event horizon

Can a simply connected Lorentzian manifold admit an event horizon? Or does the event horizon makes it non-simply connected?
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1answer
41 views

Computing for $\theta$ component of the geodesic equation

The schwarzschild metric is given by: $ds^2=-(1-\frac {2GM}{r})dt^2+(1-\frac{2GM}r)^{-1} dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2$ Here is the well known geodesic equation: $0=\frac ...
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3answers
86 views

Prove: Gravitation operator is invertible.

Let $(M,g)$ be a Riemannian manifold and $\Gamma(S^2M)$ the space of symmetric 2-covariant tensors. Define the gravitation operator as the map \begin{align*} ...
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0answers
26 views

Maxwell’s equations in the Curved Space-time

I've been told that Maxwell’s equations in the curved space-time $(\mathscr{M},g)$ take the form $$\nabla^a F_{ab} =0 \, \,(*), \quad \nabla_a F_{bc} + \nabla_b F_{ca} + \nabla_c F_{ab} = 0 \, ...
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1answer
77 views

Is it possible to build a tensor with the following properties?

I am searching for a tensor in 4-dimensional space-time with two indices that satisfy: \begin{eqnarray} M_{;\mu }^{\mu \nu } &=&0, \\ M^{\mu \nu } + M^{\nu\mu}&=&0, \nonumber \\ ...
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1answer
105 views

Prove: $(\delta^\nabla\text{d}^\nabla+\text{d}^\nabla\delta^\nabla)h=\nabla^*\nabla h-\mathring{R}_gh+h\circ\text{Ricc}_g$

Let $(M,g)$ be a Riemannian manifold with Levi-Civita connection $\nabla$ and adjoint $\nabla^*$, and exterior derivative $\text{d}^\nabla$ and adjoint $\delta^\nabla$. For a symmetric 2-covariant ...
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56 views

Distance function under diffeomorphism of manifolds

community, I am concerned with measuring distances of systems under diffeomorphisms. Concrety, I consider a smooth diffeomorphism $\varphi: M\rightarrow N$ from the smooth differentiable manifold ...
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1answer
35 views

What is a thin loop?

I read one definition of a thin loop: $\gamma$ is a thin loop if there exists a homotopy of $\gamma$ to the trivial loop with the image of the homotopy lying entirely within the image of $\gamma$. ...
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1answer
36 views

Differentiating a rank-2 tensor to some power in index notation.

If I have some rank-2 tensor $g_{ab}$ with components dependent on some coordinate system $x^a$, how do I do the following differentiation in index notation (assuming the $\dot x^d$ are independent of ...
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0answers
91 views

Some questions about synthetic differential geometry

I've been trying to read Kock's text on synthetic differential geometry but I am getting a bit confused. For example, what does it mean to "interpret set theory in a topos"? What is a model of a ...
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1answer
36 views

Parallel Transporting a vector

I want to parallel transport a vector $V^{\mu}$ with the initial condition $V^{\mu} = (V^{\theta},V^{\phi}) = (1,0)$ along a closed curve parameterized by $ \lambda \in [0,1]$ and determine the ...
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Why is sinus amplitudinus the solution to a polynomial differential equation?

So I followed the derivation in https://en.wikipedia.org/wiki/Schwarzschild_geodesics and understand how to derive the equation $$ \left( \frac{du}{d\varphi} \right)^{2} = \frac{1}{b^{2}} - \left( 1 ...
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1answer
44 views

Show $A_{ab}$ are the components of a tensor.

The question asks: "If $v_a$ are the components of a vector, show that in an arbitrary coordinate system that $A_{ab}$ are components of a rank-2 tensor, where:" $$A_{ab}= ...
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1answer
41 views

How to solve system of Differential Equations with 1 independent and 3 dependent variables

How can one solve this set of three differential equations in one independent variable "t" and three dependent variables A, B and F, which are functions of only t? $$ \frac{F(t) B''(t)+B'(t) ...
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2answers
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About the term $-\nabla_{[u,v]}w$ in the definition of Riemann curvature tensor

As we know, in the definition of Riemann curvature tensor, we require $$ R(u,v)w=\nabla_u\nabla_v w-\nabla_v\nabla_u w-\nabla_{[u,v]}w $$ Could somebody tell me why we need $-\nabla_{[u,v]}w$ ...
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1answer
108 views

Mathematical aspects of General Relativity

I was wondering what mathematical subjects are used in the study of the theory of General Relativity (black holes ...) I assume mostly differential geometry, Riemann Geometry ... but is there also ...
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1answer
146 views

Partial derivative with respect to metric tensor $\frac{\partial}{\partial{g_{kn}}}(g_{pj}g_{ql})$

$$-\frac{1}{4\mu_0}F^{pq}F^{jl} \frac{\partial}{\partial{g_{kn}}}(g_{pj}g_{ql})=+\frac{1}{4\mu_0} F^{pq} F^{lj} 2 \delta^k_p \delta^n_j g_{ql}$$ I need to know how to derive ...
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Spacelike surface lying in a null hypersurface

Why is it possible to be able to pick a spacelike 2 surface S that lies in a null hypersurface N? We know that all the tangents vectors to N are either spacelike or parrelel to the normal vector. I ...
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1answer
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Special reference for differential geometry

I am not entirely sure how to formulate the question, but here it is. I am looking to start a self study on general relativity, and of course I need a good grasp on semi-riemannian geometry (I am ...
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1answer
100 views

Divergence in terms of Levi-Civita connection

The divergence of a vector field $X$ on a manifold $M$ is defined usually as the function $\text{Div}(.)$ such that $(\text{Div} X) \;\mu =L_X \mu$ for $\mu$ a volume form. I know that there is also ...
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1answer
110 views

What is contracting a tensor actually doing?

I'm learning about tensors, and have a vague idea regarding what contracting a tensor means—but I'm still not sure of exactly what it's doing. Maybe someone here can put it in more intuitive terms. ...
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1answer
100 views

Wormhole - How to model it?

I am trying to model wormhole between two points in 3D space, but do not know/understand how to do so. A concrete example: Think of something like a game where we have a 3D world with a size of ...
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1answer
41 views

(Strong) causality condition

I'm studying the causality in Lorentz Manifold with the book "Semi-Riemann Geometry", B.O'Neill. I have the followig problem: He says that, picked a subset A of a Lorentzian Manifold M, the causality ...