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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Physical meaning of Hawking's Singularity theorem

I'm studying O'Neill's "Semi-Riemannian Geometry with applications to Relativity". I know that the following theorems are related to the Big Bang, but I don't understand how. Let $M$ be a ...
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19 views

The Hawking mass is non-decreasing during “jumps” (Penrose Inequality)

I am reading $\textit{The Penrose Inequality}$ by H. Bray and P. Chrusciel and I am stuck at one of their statement. The question regards the fact that the Hawking mass is non-decreasing during a ...
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30 views

How does one determine whether a coordinate basis is orthogonal or not?

Apologies for what is perhaps a very basic question, but I have been studying differential geometry with a view to gain a deeper understanding of general relativity and I have hit a stumbling block. ...
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15 views

Weyl Transformations and Group actions

I have the following question. Let $(M,g_{ab})$ be a Riemannian manifold $M$ with metric $g$, and with an action of a Lie group $G$. Moreover, the Riemannian metric $g_{ab}$ is taken to be invariant ...
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1answer
57 views

Book recommendation for rigorous multilinear algebra , tensor analysis, manifolds.

I am looking for recommendation on books about multilinear algebra, tensor analysis, manifolds theory, basically everything to be able to understand basic concepts of general relativity. I am ...
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1answer
21 views

Differentiate with respect to $x^{a}(t)$

having a little difficulty with this conceptually. Can someone quickly walk through this differentiation please? $K=\frac{1}{2}g_{ab}\dot{x}^{a}\dot{x}^{b}$ Find $\frac{dK}{dx^{a}}$ In this case, ...
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1answer
33 views

Invert tensor expression involving Levi-civita symbol

I have to prove that $$ \omega_{\mu \nu} = \epsilon_{\mu \nu \lambda \kappa} \omega^\lambda u^\kappa $$ given the relations: $$ u_k u^k = -1 $$ $$ \omega_{\mu \nu} u^\nu = 0 $$ $$ \omega^\mu = ...
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45 views

Why is the transformation law for tangent space coordinate basis vectors “easily seen” using the chain rule?

In Spacetime & Geometry, Carrol immediately posits that, given that you have some tangent space vector (in the coordinate basis representation): ...
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1answer
40 views

Where does the minus sign come from in this expression?

I'm trying to solve a problem from a course on General Relativity. Consider the change of coordinates $x^\mu \to x'^\mu = x^\mu + \varepsilon \xi^{\mu} (x)$ and for a tensor $T$ define $\delta T(x) = ...
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0answers
57 views

Definitions of riemann curvature tensor

Could someone please, in a concrete and non-circular way, explain what is mean by terms such as $ \nabla_u\nabla_v w\\ \nabla_{\nabla_u}v \text{ whoops, this one should probably be }\nabla_{\nabla_u ...
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40 views

Basis of differential one-form confusion

I'm trying to understand one-forms in the context of general relativity. Lee (Introduction to Smooth Manifolds) says that at a point $p$ and with a vector field $X$ we define a covector field ...
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25 views

Finding diffeomorphism given vector fields

Given a vector field how do you find the associated diffeomorphisms? Say I am given a vector field in Minkowski space, $\textrm{d}s^2 = -\textrm{d}t^2 + \textrm{d}x^2 + \textrm{d}y^2 + \textrm{d}z^2$, ...
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19 views

Integral curves and null geodesics

Let be $(M^{n+1},g)$ a spacetime (Lorentz manifold, connexe and time-oriented), $n\ge 2$, and $S\subset M$ a null hypersurface (codim $S=1$ and the restriction of $g$ to each tangent space $T_p ...
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0answers
55 views

Finding null geodesic that intersects a point and a time-like geodesic

I'm trying to find the right way to calculate the point of intersection of the null hypersurface emitted by a point $S$ event in a spacetime manifold with metric $g_{\mu \nu}$, with another time-like ...
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0answers
42 views

Weyl transformation of geodesic distance

Consider a Riemannian manifold $M$ with a metric $g$. For two points $x,y \in M$ the geodesic distance $d(x,y)$ is defined in the usual way. I would like to know if there is a formula expressing how ...
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0answers
21 views

Integral curves in null hypersurfaces

Let be $(M^{n+1},g)$ a spacetime (Lorentz manifold, connexe and time-oriented), $n\ge 2$, and $S\subset M$ a null hypersurface (codim $S=1$ and the restriction of $g$ to each tangent space $T_p ...
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0answers
11 views

What is an isometry in terms of the metric tensor, not distances?

I get that an isometry preserves distances, but I'm a bit confused about how to determine if something is an isometry using the metric only. Consider in 1D for simplicity: $$ ds^2 = \frac{dx^2}{x^2} ...
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14 views

Find how a vector changes from parallel transport around a spherical triangle?

So I'm trying to find the components of a unit vector, initially on the sphere's equator parallel to the $\phi=0$ line, after parallel transport to $\phi=\phi_0$, then to the pole and then back to the ...
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1answer
29 views

Calculating the line element after a change of coordinates

1. The problem statement, all variables and given/known data Consider $\mathbb{R}^3$ in standard Cartesian co-ordinates, and the surface $S^2$ embedded within it defined by $(x^2+y^2+z^2)|_{S^2}=1$. ...
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0answers
13 views

Higher order derivatives than Riemann Tensor

Does anyone know of any meaningful tensors that are related to the derivative of the riemann tensor? i.e. in the following picture we can consider Given arbitrary Pseudoriemannian manifold and metric ...
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11 views

The crhonological future set of $a\in M$, $I^{+}(a)$, is open in $M$.

I'm studying some topics about Lorentz Geometry and I have a little problem in the proof of a Causality Condition theorem. Let be $(M^{n+1}, g)$ a spacetime (Lorentz manifold, connected and ...
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1answer
22 views

Einstein-Cartan Theory vs Metric Affine Gravitation Theory

Can anyone point out the real difference between Einstein-Cartan Theory and Metric Affine Gravitation Theory? Both of them rely on a pseudoriemannian metric $g$ and generalised affine connection ...
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1answer
14 views

Christoffel connection

I am trying to determine the correct expression when expanding a contravariant derivative acting on another contravariant derivative acting on the Ricci scalar. $\nabla^a \nabla^b R = \partial^a ...
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2answers
68 views

Riemannian geometry - worldline meets nullcone.

I've been studying the book "Semi-Riemannian Geometry" by B. O'Neill and doing some of the excersises. Chapter 6 (special relativity) includes the following one: If $p$ is an event not on the world ...
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1answer
33 views

Question about the metric tensor

From wikipedia: The metric can be written in the form $g=g_{ij}dx^i \otimes dx^j$. The metric is thus a linear combination of tensor products of one form gradients of coordinates. If we denote the ...
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19 views

Cohomology of Anti-de Sitter manifold and black hole

Anti-de Sitter manifold AdS$_n$ is a maximally symmetric pseudo-Riemannian manifold with constant negative scalar curvature. This has $\mathbb{R}^{2,n-1}$ as its embedding and is a solution to the ...
4
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1answer
62 views

Ricci tensor from Riemann tensor

I have been studying Differential Geometry and General Relativity, and I have a question about the Ricci tensor. So as I understand, to set things up: one defines the Riemann curvature operator as ...
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1answer
15 views

Deriving Relativistic Force

Newton’s law states that $F=\frac{dp}{dt}$, where $p$ is the momentum of a body. In Newtonian physics, if the body has constant mass $m$, its momentum is $mv$, and Newton’s law becomes the familiar ...
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1answer
36 views

Laplacian of coordinate function

I'm supposed to show $\nabla^\nu\nabla_\nu x^\mu = -g^{\lambda\nu}\Gamma^\mu_{\lambda\nu}$ I think the right approach is to write $\nabla^\nu = g^{\nu\sigma}\nabla_\sigma$ for the first covariant ...
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1answer
16 views

covariant derivative of ricci scalar

How to derivate (covariant derivative) the expressions which is function of Ricci scalar? Also, if R is Ricci scalar, what would be ∇i∇j F(R) ?, where ∇i is covariant derivative.
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17 views

Calculate Ricci Tensor (Axial Symmetry)

I have a problem calculating a Ricci tensor. My metric (Lorentz signature) is $\mathbf{g}=Xd\phi^2 + g_{ab}dx^adx^b$ where $X,g_{ab}$ don't depend on $\phi$, and $g_{ab}$ is a $2+1$ metric. I define ...
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1answer
56 views

Given a spacetime in terms of values of Lorentzian distance, how to determine whether it is flat?

Yesterday I learned that geometric relations between events can be characterized generally (and up to a common non-zero factor) in terms of their pairwise "Lorentzian distance, $d_{\ell}$", which ...
3
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1answer
52 views

Second fundamental form for surfaces (extrinsic curvature)

I'm trying to understand the Gibbons–Hawking term. There appears $K$- a trace of the second fundamental form, as they say. As far as I know, the second fundamental form is the following thing: ...
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0answers
19 views

Two-Component Spinor Index Placement

This may ultimately be a silly question, but a pedantic mind like mine gets tied into knots over differing notation. Let $\mathbb{W}$ be a complex two-dimensional vector space which carries the ...
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1answer
63 views

Recommendation for a Differential Geometry/General Relativity text

I'm an undergraduate Math student and I am doing a side project for which I would like to study Differential geometry and its applications in general relativity. I have taken a few proof based courses ...
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28 views

In twistor theory, what's the relation between points with dual Plucker coordinates? Also about special null lines

In twistor theory, each point $Z=[Z0,Z1,Z2,Z3]$ in the complexified Minkowski space $CM$ has a correspondent Plucker coordinate $P(Z)$ embedded in $CP^5$ and we can also find its dual $P(Z)^{*}$. My ...
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26 views

Question about $\alpha-$plane in twistor theory

In twistor theory, given the complexified Minkowski space $CM$ and the projective twistor space $PT$, an $\alpha-$plane is defined as the correspondence in $CM$ wit a point $Z \in PT$. But I found ...
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0answers
37 views

How to prove a symmetric tensor is indeed a tensor?

Our professor defined a rank (k,l) tensor as something that transforms like a tensor as follows: $$T^{\mu_1' \mu_2'...\mu_k'}{}_{\nu_1'\nu_2'...\nu_l'} ~=~ ...
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1answer
29 views

Covariant derivative notation?

I was reading up on covariant derivatives and came across this document. On the second page it says: We define a procedure called parallel transport by defining a vector $\vec A (\lambda)$ along ...
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0answers
33 views

Derive the Lorentz Transformation by solving (1).

This photo and problem is taken from Zee's textbook pg. 169, Einstein's Gravity in a nutshell. Here in the moving reference frame we're moving at a velocity $u$, and the total time it takes to go ...
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2answers
75 views

show that the product of two delta functions δ(x)δ(y) is invariant under rotation around the origin.

Show that the product of two delta functions $\delta{(x)}$$\delta{(y)}$ is invariant under rotation around the origin. This is a problem from Zee's textbook on Gravity on page 51. The book was ...
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29 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 ...
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0answers
19 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 ...
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0answers
51 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 $p\in M$ ...
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1answer
89 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 ...
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29 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 ...
3
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1answer
73 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 ...
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2answers
24 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
118 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
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1answer
43 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?