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).

learn more… | top users | synonyms

1
vote
0answers
15 views

Relativistic Projective Geometry

If we assume that space-time has an extra two dimensions so that there is more symmetry between space (with 3) and time (now with 3). What would the corresponding cross ratio equation look like if we ...
1
vote
1answer
20 views

Relativity and Projective Geometry

How do you identify the cross ratio equation of projective geometry from the hyperbolic geometry of relativity? Specifically, what relativistic variables would correspond to A,B,C,D in the standard ...
1
vote
0answers
14 views

Foliation vs Coordinates in de Sitter

I'm studying de Sitter manifolds and am confused about the difference between the choice of foliation and the choice of coordinates (and how they relate to the spatial curvature). I can choose the ...
1
vote
0answers
42 views

Identity regarding the components of a dual basis

This problem is from Robert Wald's "General Relativity." The problem is 4(b) from chapter 2. Let $Y_1\cdots Y_n$ be smooth vector fields on an $n$-dimensional manifold $M$ such that at each $p\in M$ ...
2
votes
1answer
27 views

Matrix representations of tensors

I've been trying to teach myself general relativity, and I always get stuck at the same point: I don't really understand what the metric tensor is. Unless I'm incorrect, and please correct me if I'm ...
0
votes
0answers
23 views

Expanding E, B in post-Newtonian Gravitational Potential

Thanks to someone who can help me with this particular equation. I've been trying to take a stab at these equations by myself, though I realized I need to seek some help. I'm currently trying to ...
0
votes
0answers
16 views

The equivalence principle and experiments concerning it? [migrated]

Imagine that we are in a rocket accelerating with some magnitude $a_1 = dx^2/d^2y$, also imagine that we have a stationary rocket ship in close proximity to ours, stationary relative to our reference ...
3
votes
1answer
67 views

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 ...
1
vote
2answers
19 views

The Riemannian Curvature in a solid sphere

Is the Riemannian Curvature at the centre of a solid sphere zero?
6
votes
2answers
92 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 ...
4
votes
0answers
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?
6
votes
0answers
78 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 ...
1
vote
0answers
58 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 ...
1
vote
0answers
42 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 ...
0
votes
0answers
48 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 ...
0
votes
1answer
25 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
votes
1answer
31 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 ...
4
votes
3answers
183 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, ...
1
vote
0answers
32 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 ...
0
votes
0answers
29 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} ...
0
votes
1answer
30 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 ...
1
vote
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: ...
4
votes
2answers
146 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 ...
1
vote
1answer
40 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 ...
0
votes
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 ...
0
votes
1answer
43 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 ...
0
votes
2answers
37 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 ...
1
vote
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 ...
2
votes
1answer
58 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 $\mathcal M$ and $\mathcal N$ be two dim-$n$ Lorentzian manifolds with points ...
0
votes
0answers
41 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 ...
1
vote
1answer
22 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 $G^{ab}=R^{ab}-\frac{1}{2}g^{ab}R$?
0
votes
0answers
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?
1
vote
2answers
57 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 ...
0
votes
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 $\gamma(s)\in M$ ...
0
votes
0answers
16 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 ...
3
votes
0answers
190 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 ...
0
votes
0answers
27 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 ...
0
votes
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?
1
vote
0answers
32 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: ...
1
vote
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 $R_{00}$, ...
3
votes
1answer
82 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 ...
3
votes
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 ...
5
votes
0answers
68 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 ...
1
vote
2answers
75 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 ...
0
votes
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.
3
votes
0answers
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?
2
votes
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 ...
1
vote
3answers
89 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*} ...
1
vote
0answers
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 \, ...