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

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26
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Are black holes perfect spheroids?

What I know about black holes (correct me if I'm wrong) is that they are the most compact objects in the universe that have been discovered. Due to all that gravity, wouldn't black holes be a perfect ...
4
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1answer
171 views

Derivation of Christoffel Symbols

So I am reading a book on relativity & differential geometry and in the text, they gave the Christoffel symbols in terms of the metric and its derivatives, but I wanted to derive it myself. ...
18
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1answer
204 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 ...
1
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0answers
24 views

Sectional curvature of 3-hyperbolic space [migrated]

For a pair of $(X,Y)$ of linearly independent vectors in $T_pM$, $p\in M$, the sectional curvature is defined as $$K_p(X,Y)= - \frac{\bigl\langle R(X,Y)X,Y\bigr\rangle}{|X|^2 |Y|^2 - \langle X,Y\...
23
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5answers
4k views

Conformal transformation/ Weyl scaling are they two different things? Confused!

I see that the weyl transformation is $g_{ab} \to \Omega(x)g_{ab}$ under which Ricci scalar is not invariant. I am a bit puzzled when conformal transformation is defined as those coordinate ...
1
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2answers
54 views

Riemann Tensor and Covariant Derivative in Carroll's Spacetime

In Spacetime and Geometry, Sean Carrol defines the Riemann Tensor in terms of the commutator of covariant derivatives: $R^\rho_{\sigma\mu\nu}V^\sigma = [\nabla_\mu, \nabla_\nu]V^\rho + T^\lambda_{\...
4
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3answers
763 views

Space-Time Curvature Depends on Relative Speed

When the mass of a planet causes the curvature of space-time we see that an approaching free-falling object deviates its path towards the planet. We also see the amount of that deviation depends on it'...
1
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3answers
72 views

Ricci scalar for Black Hole

What is the value of Ricci scalar at $r=0$ inside Black Hole? Since $R_{\mu \nu}=0$ is vacuum solution and valid outside event Horizon of black hole where there is no mass energy density. But inside ...
3
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2answers
114 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 ...
11
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2answers
3k views

Why is light described by a null geodesic?

I'm trying to wrap my head around how geodesics describe trajectories at the moment. I get that for events to be causally connected, they must be connected by a timelike curve, so free objects must ...
2
votes
4answers
2k views

Calculating the Riemann tensor for a 3-Sphere

I have worked out all the connection symbols for the 3-sphere using calculus of variations, cf. this Phys.SE post. So to find the Riemann tensor I am trying to find all the nonzero components of: \...
0
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1answer
51 views

Trace of a Tensor

What is the significance of defining the trace of a tensor as $g^{\alpha\beta} R_{\alpha\beta}$ instead of $R_{\alpha\alpha}$ on a Riemannian manifold?
0
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1answer
67 views

Riemann tensor for a diagonal metric [closed]

Is it correct that for a diagonal metric tensor, the Riemann tensor with one contravariant ( upper ) index, $R^\mu_{\phantom{a}\nu\theta\phi}$, is anti-symmetric for interchange of the two first ...
1
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3answers
56 views

What is the measure of distance in higher dimensions?

In our world we are using kilometers to measure distance. What measurement is used to measure distance in higher dimensions?
1
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1answer
61 views

Covariant Taylor Series

I am reading the following lecture notes of Avramidi https://www.researchgate.net/publication/255565392_Analytic_and_geometric_methods_for_heat_kernel_applications_in_finance I want to understand ...
1
vote
1answer
38 views

Does the Komar mass density act like a four density?

The Komar mass is a means of measuring gravitational mass in spacetime. Via Wikipedia (https://en.wikipedia.org/wiki/Komar_mass) it is stated as (for a stationary metric): $$m=\int\rho d\mathrm{vol}=\...
1
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2answers
372 views

What are the mathematical models for force, acceleration and velocity?

In mechanics, the space can be described as a Riemann manifold. Forces, then, can be defined as vector fields of this manifold. Accelerations are linear functions of forces, so they are covector ...
0
votes
1answer
109 views

How to get null tetrad by metric?

How to get null tetrads ${l^a,n^a,m^a,\overline{m}^a}$ for this metric? This on is from Ryder's book (Introduction to general relativity) page 268 $g^{\mu\nu}=\begin{pmatrix} 0 & \frac{1}{c} &...
6
votes
4answers
780 views

Can general relativity be explained by equations describing a fabric of space embedded in a flat 5-dimensional Minkowski space?

Does such a set of equations exist or does our universe have an intrinsic curvature that can't be explained by an embedding in a flat Minkowski space of 1 higher dimension? Even if general relativity ...
0
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1answer
65 views

Could you give me an application on physics of Gauss Divergence Theorem for scalar? [closed]

Gauss divergence theorem for vectors can be easily explained by mass balance. But I can't think about one example for scalar gauss divergence theorem. Gauss Divergence Theorem for scalars: $$\int\...
4
votes
3answers
554 views

Locally flat coordinate and Locally inertial frame

I am having some doubts on myself regarding the above concepts in General Relativity. First, I want to point out how I understand them so far. A male observer follows a timelike worldline ($\gamma$) ...
6
votes
3answers
129 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 + ...
2
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1answer
191 views

If a point r lies in the boundary of the chronological future of another point p, why does the chronological future of r belong to that of p?

I am studying the global causality of the spacetime. Here, I come across a problem. Suppose a point $r\in \partial I^+(p)$. $I^+(p)$ is the chronological future of a different point $p$ in spacetime....
4
votes
1answer
37 views

Adjoint of the gauge covariant derivative

Suppose $A=A_1dx_1+A_2dx_2$ is a 1-form connection in $\mathbb{R}^2$ and $D_A \phi=d\phi-iA\phi$ is the gauge covariant derivative with $\phi=\phi_1+i\phi_2$ is a complex scalar field. May I ask what ...
0
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1answer
158 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}]. \end{...
3
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0answers
91 views

Learning about 4 topics in physics [closed]

This isn't really a question on any of those numerous underlying concepts behind the various sub-disciplines of physics, but hear me out: I'm still in Higher Secondary, but I'd really love to know ...
4
votes
3answers
140 views

Textbook on Differential Geometry for General Relativity [duplicate]

I'm going to start self-studying General Relativity from Sean Caroll's Spacetime and Geometry: An Introduction to General Relativity. I'd like to have a textbook on Differential Geometry/Calculus on ...
6
votes
0answers
169 views

Objective time derivative that is not a Lie derivative

Summary Led by an interest into the concept of "Material Objectivity", I am asking myself: Are there objective time rates that are not Lie derivatives? The long read I am trying to understand the ...
5
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1answer
75 views

Yang-Mills potential and principal bundles

In section 2.7.2 of Bertlmann's "Anomalies in quantum field theory", it is stated that since a non-trivial principal bundle (based on a Lie group $G$) does not admit a global section, the Yang-Mills ...
7
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0answers
112 views

Metric transformation, polygons and gravitons

I'm trying to understand the paper by Hitchin called: ''Polygons and gravitons". I'm stuck at page 471. At this point, he does some computations and obtains a metric: $$ \gamma dz d\bar{z}+\...
5
votes
0answers
124 views

Geometric point of view of configuration space and Lagrangian mechanics

Even though (I think) I understand the concept of a tangent bundle, I have trouble assimilating the idea of the configuration space being one and in relation to what that is the case. How can I ...
5
votes
2answers
354 views

Timelike Boundary

I was reading in a paper (see 1st paragraph of introduction section in http://arxiv.org/pdf/1510.00709.pdf) that in AdS space, waves can reach the boundary in finite time and, since said boundary is ...
1
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2answers
51 views

Local frame of reference

I am currently simulating particle trajectories in Kerr spacetime numerically with $M=1$ and $a=1$. In the picture above, I am calculating the geodesic in Boyer-Lindquist coordinates. I was messing ...
3
votes
1answer
114 views

Geometrical point of view of the harmonic constraints ($\Delta g_{ij}=0$) in General Relativity

What does it mean, from the geometrical point of view, use (in General Relativity) of the constraints on the metric tensor's coefficients such that $\Delta g_{ij}=0$? (where $\Delta$ is the Beltrami-...
3
votes
1answer
35 views

metric in three sphere and SU(2)

consider $S^3$ $i.e$ \begin{align} x_0^2 + x_1^2 + x_2^2 +x_3^2 =1 \end{align} note that in $\mathbb{R}^4$ with metric or $\mathbb{S}^3$ we have \begin{align} ds^2 = l^2 (dx_0^2 + dx_1^2 + dx_2^2 + ...
1
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1answer
169 views

Generalized spin connection and dreibein in higher spin gravity

I am studying 3D higher spin gravity and I would like to know the mathematical and physical meaning of generalized spin connection and generalized dreibein that appear in this theory. It is well known ...
2
votes
1answer
119 views

Parallel Transported Orthonormal Basis

The following argument results in a conclusion that I find strange, and makes me suspect there is something wrong with the reasoning. First, consider a timelike geodesic $\gamma$ with normalized ...
5
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0answers
38 views

Confusion with conclusion to positive mass theorem

I am trying to understand the positive mass theorem as it is presented in the survey paper by Corvino and Pollack http://arxiv.org/abs/1102.5050 I am fundamentally confused by the structure of their ...
6
votes
1answer
252 views

Geometric formulation of the equivalence principle

Let $(M,g)$ be a $4$-dimensional Lorentzian manifold. It is well know that given $(U,\psi=(x^1,\ldots,x^4))$ local chart around some $p\in M$, it is posible to find a change of coordinates given by $(...
0
votes
1answer
36 views

Interpretation of the operation $v^\alpha \nabla _\alpha v^\mu$

In general relativity, we can write the geodesic equation as a contraction $v^\alpha \nabla _\alpha v^\mu = f(\lambda)v^\mu$ along a path defined by coordinates $x^\mu(\lambda)$, and where $v^\mu = \...
2
votes
1answer
635 views

The role of the affine connection the geodesic equation

I apologise in advance that my knowledge of differential geometry and GR is very limited. In general relativity the equation of motion for a particle moving only under the influence of gravity is ...
2
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0answers
40 views

Construction of vector bundles of relativistic fields by Mackey's method of induced representation

I recently stumbled on Sternberg's book on group theory and physics. The ideas expressed in the book are really great, but the detailed reasoning is very hard to follow, I find. I am kind of stuck ...
1
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0answers
47 views

What is the difference between intrinsic and extrinsic curvature? [migrated]

In general relativity, energy bends spacetime. However, this doesn't mean that a fifth dimension for spacetime to "bend into" exists." That is, spacetime isn't embedded in a higher dimensional space, ...
0
votes
1answer
41 views

Time reversal symmetry for non-orientable manifold

From a recent paper by Kapustin(https://arxiv.org/abs/1406.7329), he argued that for non-orientable manifold with spin structure $Pin^{\pm}$, the corresponding time reversal symmetry $T$ squares to ${...
21
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5answers
2k views

Does curved spacetime change the volume of the space?

Mass (which can here be considered equivalent to energy) curves spacetime, so a body with mass makes the spacetime around it curved. But we live in 3 spatial dimensions, so this curving could only be ...
0
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0answers
31 views

What is the minimum math level required to comprehend general relativity? [duplicate]

I am currently working on a B.S. In chemistry and only need to go up to Calculus II, though I like to dabble in physics. What other math classes could I take to gain greater understanding of higher ...
4
votes
1answer
256 views

How can I mathematically describe the parallel transport in the Roman soldiers example?

I've been trying to understand parallel transport. Many of the descriptions present a mathematical version: $\nabla_V X = 0$. And/or they present an example involving soldiers (usually Roman) ...
3
votes
1answer
169 views

A question on the Chern number and the winding number?

Let $\mid \psi(x,y) \rangle$ be a normalized wavefunction living in a $d$-dimensional Hilbert space and depend on two real parameters $(x,y)$ that belong to a closed surface (e.g., $S^2, T^2$, ...). ...
2
votes
1answer
136 views

Interesting Hamiltonian System [duplicate]

The definition of a Hamiltonian system I am working with is a triple $(X,\omega, H)$ where $(X,\omega)$ is a symplectic manifold and $H\in C^\infty(X)$ is the Hamiltonian function. I am wondering if ...
3
votes
1answer
488 views

Stokes' theorem in complex coordinates (CFT)

I am studying CFT, where I encounter Stokes' theorem in complex coordinates: $$ \int_R (\partial_zv^z + \partial_{\bar{z}}v^{\bar{z}})dzd\bar{z} = i \int_{\partial R}(v^{z}d\bar{z} - v^{\bar{z}}dz). $$...