Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.

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60 views

Particle on $S^1$ and $U(1)$-principal bundle

I have a question arisen from a simple QM problem: let consider a boson on $S^1$ minimally coupled with a constant gauge field $A$. Taking the stationary Schrödinger (S) or Klein-Gordon (KG) equation ...
0
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1answer
16 views

Relationship between fracture of solid plates and its curvature

I do some school projects about fracture of glass, my assumption is this phenomenon related to the curvature of these plates, specifically, fracture starts from the point which is having “maximum” ...
2
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1answer
20 views

Two Robertson-Walker observers, at what time will a light signal be received?

Here is a question I have that is inspired by this question here. The spacetime metric of a radiation-filled, spatially flat ($k = 0$) Robertson-Walker universe is given by$$ds^2 = - dT^2 + T[dx^2 + ...
0
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1answer
115 views

Torsion-free, symmetric connection and non-coordinate basis

The torsion tensor is defined as (Hawking p.34) \begin{equation} \mathbf{T}(\mathbf{X},\mathbf{Y}) = \nabla_{\mathbf{X}}\mathbf{Y} - \nabla_{\mathbf{Y}}\mathbf{X} - [\mathbf{X},\mathbf{Y}]. ...
1
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1answer
42 views

About periodicity of coordinates given a metric

If I am given a metric how do I decide which coordinate is periodic? Eg. can I look at metric in plane polar coordinates and tell that θ direction is periodic. Also How do I calculate the period of ...
3
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1answer
47 views

Difference between Fermi and Riemann normal coordinates

What is the difference between Fermi normal coordinates and Riemann normal coordinates? Which one of them is related to the vanishing of the Christoffel symbols?
3
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1answer
52 views

From states in string theory to differential forms

This question is probably really easy and isn't usually discussed in textbooks. So suppose when we construct states in string theory we obtain something like $\left|- \right\rangle \otimes \left|+ ...
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2answers
78 views

Using $\sqrt{-g}$ in integrals of proper volume

I am a little confused over integration using proper volume element. When do we use $\sqrt{-g}$ in calculations? For example, in many calculations involving stars, say when using TOV equation, this ...
6
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2answers
181 views

What manifold is spacetime?

In General Relativity, spacetime is a $4$-dimensional manifold with one Lorentzian metric tensor defined on it. In the Special Relativity case what manifold is spacetime is quite clear: it is ...
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29 views

Geodesic tangent vector in a Riemannian 4-space

I am doing a question in Lewis Ryder's introduction to General relativity. I am very close to the answer but not quite there. The question is: A Riemannian 4-space has metric $$ds^2 = ...
3
votes
1answer
49 views

Is this equation $\nabla_a\sqrt{-g}=0$ correct? [duplicate]

Is the equation $$\nabla_a\sqrt{-g}=0$$ correct? Here $\nabla_a$ is the Levi-Civita connection, and $g$ is the determinant of metric $g_{ab}$. Apparently, we have $\nabla_ag_{bc}=0$, but I am not sure ...
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1answer
31 views

Conformal Transformation: Minkowski sheet to cylinder

What conformal transformation can I make to 2d Minkowski with metric $ds^2=-dt^2+dx^2$ to show that it is conformal to a cylinder?
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0answers
32 views

What is the Metric Tensor? [duplicate]

I was studying Einstein's Field Equation, and this was the most common symbol. Can you explain what it is and how it could be used?
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0answers
32 views

Conformal Connections in Physics [closed]

For my diploma thesis in Mathematics I investigate conformal connections (as an example of Cartan connections). All in all the thesis should deal with geometric aspects (associated bundles, ...
1
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1answer
68 views

When and where to check the formal definition of a manifold

In most texts on GR we are first introduced to a formal and rigorous definition of a manifold. We then learn the point that in GR "any coordinate system" might be used for the 4D spacetime metric. ...
4
votes
2answers
117 views

Why is the covariant derivative of the determinant of the metric zero?

This question, metric determinant and its partial and covariant derivative, seems to indicate $$\nabla_a \sqrt{g}=0.$$ Why is this the case? I've always learned that $$\nabla_a f= \partial_a f,$$ ...
0
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1answer
61 views

Straight lines in general relativity

This question stems from a possibly misguided attempt to understand General Relativity. I am about to leave High school for college, I do however have a rudimentary understanding of tensors, and I ...
6
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1answer
174 views

Continuity Equation in Differential Geometry

I'm looking for a derivation of the mass continuity that applies in general on symplectic manifolds. In particular the "the amount of change in the mass in a volume is just amount that flows in or ...
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0answers
71 views

Euler-Lagrange equations in General Relativity

When obtaining the Euler Lagrange equations for a scalar field with higher order derivatives in curved space is it the same to use $$ -\partial_\nu\partial_\mu\frac{\partial \sqrt{-g} ...
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0answers
148 views

Curvature Invariants in General Relativity and Singularities

Suppose that I want to check if a given metric is singular or not. I'm interested in curvature singularities, not coordinate singularities, so I can look to scalars made with Ricci, Riemann and Weyl ...
0
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1answer
38 views

Causal structure, time orientability and equivalence classes

Quoting from this Wikipedia article, if $(M,g)$ is a Lorentzian manifold then the tangent vectors at each point in the manifold can be classified into three different types. Using a $(+,-,-,-)$ metric ...
3
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1answer
116 views

Diffeomorphism invariance and geodesic action

I'm trying to understand the role of diffeomorphism and isometry invariance in the geodesic action in GR: $$ S = \int_{\tau_1}^{\tau_2} \! d\tau~ g_{ab}(x(\tau)) \frac{dx^a}{d\tau} \frac{dx^a}{d\tau} ...
2
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0answers
42 views

Books about non-euclidean geometry [duplicate]

I study physics and want to know the best books to learn non-euclidean geometry in an easy way.
8
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1answer
143 views

Acceleration of particle “held in place” at $x = 1$ [closed]

The metric components in a two-dimensional spacetime are given in terms of the coordinates $(t, x)$ by$$ds^2 = -\cosh x\,dt^2 + dx^2.$$Consider a particle that is "held in position" at $x = 1$. What ...
4
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1answer
270 views

What's the difference between the diffeomorphism invariance and reparametrization invariance?

Can somebody tell me what's the difference between the diffeomorphism invariance and reparametrization invariance?
3
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1answer
61 views

Equivariant cohomology and Mayer-Vietoris sequence

I'm reading this article upon topological field theory and I'm a bit confused about the way he compute equivariant cohomology of $S^2$ wrt $\mathrm{U}(1)$, i.e. $H^\bullet_{S^1}(S^2)$. You can find ...
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0answers
36 views

GR - curve (in)completeness & (in)extendibility

Seeking clarification of the distinction between completeness of geodesics/extendibility of curves in GR spacetimes? (Confirm: not the geodesic completeness of a spacetime but the completeness of an ...
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0answers
61 views

Relation between second covariant derivative of Killing vector and Riemann tensor [closed]

I need to prove that $$D_\mu D_\nu \xi^\alpha = - R^\alpha_{\mu\nu\beta} \xi^\beta$$ where D is covariant derivative and R is Riemann tensor. $\xi$ is a Killing vector. I have proved that $$D_\mu ...
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0answers
35 views

Help needed to understand Kerr coordinate transformation

The (uncharged) Kerr metric for a black hole of mass $M$ and angular momentum $Ma$ takes the form $$ds^{2} = \Sigma\Big(\frac{dr^{2}}{\Delta} + d\theta^{2}\Big) + (r^{2} + a^{2})\text{sin}^{2}\theta ...
6
votes
1answer
115 views

Two “Robertson-Walker observers,” velocity of baseball as seen by second observer right before it's caught?

The spacetime metric of a spatially flat ($k = 0$) radiation dominated FLRW universe is given by$$ds^2 = -dT^2 + T[dx^2 + dy^2 + dz^2].$$Consider two "Robertson-Walker observers," i.e., observers with ...
1
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1answer
57 views

Spacetime manifold surgery: is this result still a valid etc. spacetime?

Given a valid classical GR spacetime manifold $M$ (i.e. 4D, Lorentzian, Hausdorff, paracompact, ?etc.), and $B\subset M$, a closed spatial subset (e.g. a closed ball at fixed $t$) to be excised, ...
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0answers
24 views

Book recommendations on geometrical methods for physicists (like Topology, Diff. Geometry) [duplicate]

I would like to obtain a book that has to do with geometrical methods/subjects for physicists. When I say geometrical methods/subjects I mean things like Topology, Differential Geometry, Lie ...
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0answers
116 views

Did Gauss anticipate topological field theory?

The book Quantum Field Theory I: Basics in Mathematics and Physics makes the claim ...
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1answer
58 views

If we live on the surface of Earth then why Earth images shows maps around it? [closed]

If you visits google map and go to earth we see the image as attached below. My question is if the earth is round like sphere ball and if we live on the surface of this ball (point me if i am ...
4
votes
1answer
91 views

Why do we need connections, if we have the Lie derivative?

When I learned about the covariant derivative, it was motivated as a way of defining a good differentiation operation on tensors. To do this, we had to define a connection on the manifold, which was a ...
5
votes
1answer
100 views

What do we exactly mean by a “topological object” in physics?

I have been working on topological defects like monopoles, etc. for some time. One think that I have not been able to understand is the physical meaning of the phrase "topological object". I have ...
4
votes
1answer
75 views

Negative mass thin shell collapse

Suppose we have a collapsing light-like (ingoing) shell with negative mass and decreasing further. The shell is radiating and so the exterior region is that of the outgoing Vaidya solution. $$ds^2 = ...
4
votes
2answers
152 views

When does causal separation imply no spacelike separation?

(See here for notation.) In Minkowski space, if $p\prec q$, then there is no spacelike curve $c:[0,1]\to \mathbb{R}^{n-1,1}$ with $c(0)=p$ and $c(1)=q$. This is obvious from a spacetime diagram. Here ...
4
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0answers
202 views

How does one determine if a spacetime is globally hyperbolic?

A spacetime $M$ is said to be globally hyperbolic if it is strongly causal and if the sets $J^+(p)\cap J^-(q)$, for all $p,q\in M$, are compact. (For more information, see the Wiki article on causal ...
1
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1answer
40 views

Vierbeins in General Relativty; degrees of freedom?

I am self-learning GR. I want to ask if vierbeins $e^b_{\ \ \nu}$ need to satisfy any relations or if I am free to choose any type of vierbein I like So I have been looking into tetrads again. I ...
2
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1answer
132 views

Quotient space in the book The Large scale structure of space-time

On page 79, the author states One is thus concerned only with $\mathbf{Z}$ modulo a component parallel to $\mathbf{V}$, i.e. only with the projection of $\mathbf{Z}$ at each point $q$ into the ...
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2answers
80 views

What are world lines as opposed to arbitrary curves in spacetime?

In GR the spacetime manifold is equipped with a metric which makes it a Lorentzian manifold. It is the metric that is doing the separation of space and time (so that we end up with three dimensions of ...
2
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0answers
41 views

Could a 3+1 space be embedded in a 4+1 space and retain its 3+1 characteristics? [closed]

I'm confused because I can conceptualize this embedding scenario in two seemingly incompatible ways. Which of the following scenarios are possible?: 1) 4+1 space automatically enforces 4 dimensions ...
1
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1answer
40 views

ADHM construction and Momentum Map

while I was reading about ADHM construction I had some troubles with precise geometrical identification of the various quantities. My doubts is well manifest in these two Wikipedia pages 1) ADHM ...
1
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1answer
55 views

Killing equation manipulation

Why does the killing equation $$K_{\mu;\nu}+ K_{\nu;\mu} = 0$$ equal $$K_{\mu,\nu}+ K_{\nu,\mu} -2\Gamma^{\rho}_{\mu\nu}K_{\rho} = 0 $$ when in general a covariant derivative $V_{\beta;\alpha} = ...
3
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1answer
141 views

Fermi-Propagated Jacobi equation in the book The Large scale structure of space-time

On page 81, equation (4.6), the author use the Fermi derivative to write the Jacobi equation \begin{equation} \tag{4.6} \frac{{D^2}_\text{F}}{\partial s^2} {}_{\bot}Z^a = ...
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1answer
94 views

The Lie derivative of the metric $g_{ab}$ and index notation

I don't quite know where to start this question. I'm essentially not understanding how to compute the Lie derivative of a given metric and vector. So I have the following definition: $$ ...
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1answer
61 views

Extrinsic curvature components

I'm trying to understand how to derive the extrinsic curvature (in order to understand some calculation on fluid/gravity dynamics). But I hit a wall in my progress. I stuck at trying to verify the ...
6
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3answers
220 views

How do we measure Schwarzschild coordinates?

In special relativity, we make a big fuss about setting up inertial frames of reference, and then constructing coordinate systems using networks of clocks and rulers. This gives an unambiguous ...
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1answer
73 views

A question regarding $f(R)$ Lagrangians

Consider the class of Lagrangian known as $f(R)$ Lagrangians where the Lagrangian is some function $f(R)$, \begin{equation} S=\int\sqrt{g}d^4x\ f(R) \end{equation} assuming there are no (or ...