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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Accounting for signs in divergence thm. on Lorentzian manifold

I am trying to learn about integration in Lorentzian manifolds (I will use signature -+++) and have some problems. Oft quoted (in books for GR) form of divergence theorem is: $\int _U div( X ...
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2answers
15 views

The Riemannian Curvature in a solid sphere

Is the Riemannian Curvature at the centre of a solid sphere zero?
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2answers
85 views

Master's Exploration in General Relativity

just throwing a query out to the Math community. I'm about to embark on a master's in Gravitation, Cosmology and General Relativity and was looking for possible subjects to start researching. My main ...
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28 views

On geodesics in Schwarzschild spacetime

I am required to show that a circular lightlike geodesic exists in the Schwarzschild spacetime, and to find its radius. What's the best way to start this?
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70 views

Vectors that geodesically generate the same surface

Suppose that $\langle M,g \rangle$ is a complete, simply connected Riemannian symmetric space. The surface geodesically generated by a vector $\xi$ in $T_pM$ is the set of points lying on geodesics ...
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42 views

Yamabe flow, Metric times Scalar curvature?

I was watching a lecture on differential geometry on Ricci flow, when someone asked a question about "Scalar curvature being multiplied by metric" to my understanding this shall be written as ...
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0answers
36 views

Levi-Civita tensor in curved space

In the book "Gravitation and cosmology" by Weinberg at the page 99-100. He defines the Levi-Civita tensor as $\epsilon^{0123}=+1$ from which he writes ...
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47 views

Exact Similarity Solutions of System of Nonlinear Partial Differential Equations

I have been reading Self-Similarity and Beyond, by P. L. Sachdev. However, I am stuck on page 70, chapter 3, section 2. I have screen shotted the part which I am having a problem with I wonder if ...
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1answer
24 views

Deriving the Geodesic Equation

I found a derivation of the geodesic equation that includes this step as I write it: $$ \frac{d (g_{ab}\dot{x}^b)}{dt}=\frac{1}{2}\partial_ag_{bc}\dot{x}^b\dot{x}^c \Rightarrow \\ \\ ...
3
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1answer
30 views

Why should the metrical groundform on a variety be a quadratic form?

I'm learning General Relativity and I can't understand why the distance function on space time is a quadratic form $$\textrm{d}s^2=g_{\mu\nu}\textrm{d}x^{\mu}\textrm{d}x^{\nu}$$ I explain it through ...
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3answers
138 views

On Learning Tensor Calculus

I am highly intrigued in knowing what tensors are, but I don't really know where to start with respect to initiative and looking for an appropriate textbook. I have taken differential equations, ...
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0answers
29 views

When are geodesically generated surfaces everywhere spacelike?

Suppose that $\langle M, g\rangle$ is a Lorentzian manifold, and that $\xi$ is a timelike vector in $T_pM$, at some point $p \in M$. Let $S$ be a surface consisting of points that lie on some ...
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25 views

Finding the basis one forms (covectors) corresponding to a particular formulation of basis vectors

This formulation of the basis may be wrong, or I may be missing something, but I can't see a way to formulate the covectors this particular basis: \begin{align} \vec{e}_0 &= \vec{x} + \vec{y} ...
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1answer
23 views

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
35 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
138 views

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
38 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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2answers
24 views

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
42 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
33 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
41 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
51 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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0answers
37 views

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
21 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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11 views

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
54 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
44 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
13 views

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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184 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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0answers
23 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
32 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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31 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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0answers
46 views

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
81 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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64 views

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
74 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
24 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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57 views

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
88 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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27 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
81 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 \\ ...
2
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1answer
109 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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67 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
37 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
114 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
40 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 ...