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

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3
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1answer
107 views

Why do we do partial and not covariant differentiation with $x^{\nu}$?

Why when taking the velocity vector we make $$u^{\nu}=\frac{d}{d\tau}x^{\nu}$$ and not $$u^{\nu}=\frac{\nabla}{d\tau}x^{\nu}$$ where in the last equation I meant the covariant derivative. Why?
4
votes
2answers
238 views

Can Minkowski spacetime be redefined as a non-flat riemannian manifold?

Minkowski space time is defined in terms of a flat pseudo-Riemannian manifold. I have wondered if it can be redefined as Riamannian manifold and in the case what type of curvature would there appear. ...
3
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0answers
85 views

Is the “Force” of Gravity Simply Hamilton's Principle on a Curved Spacetime?

It's my understanding that General Relativity abstracts away the concept of gravity as a force, and instead describes it as a feature of spacetime by which massive objects cause curvature. Then it ...
1
vote
1answer
102 views

Interpretations of (r,s) tensors [duplicate]

A tensor of type (r,s) on a vector space V is a C-valued function T on V×V×...×V×W×W×...×W (there are r V's and s W's in which W is dual space of V) which is linear in each argument. We take (0, 0) ...
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1answer
140 views

The commutator of Killing vectors

I'm going over an assignment for my general relativity course. My solution to the question below strikes me as too short, considering that it appeared in the "longer questions" section of the ...
23
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4answers
1k views

Is topology of universe observable?

There is an idea that the geometry of physical space is not observable(i.e. it can't be fixed by mere observation). It was introduced by H. Poincare. In brief it says that we can formulate our ...
3
votes
1answer
294 views

Instantons in Witten's supersymmetry and Morse theory

I'm reading Witten's paper on supersymmetry and Morse theory and am confused about the details of the instanton calculation which he uses to define a Morse complex (beginning at page 11 of the pdf) . ...
4
votes
2answers
644 views

Metric tensor in special and general relativity

I'm having trouble understanding the metric tensor in general relativity. What I've understood so far has come from my course lecture notes used in conjunction with "The Road to Reality" by Roger ...
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0answers
71 views

Avoiding Pseudo-tensors when addressing global conservation of energy in GR

Discussions about global conservation of energy in GR often invoke the use of the stress-energy-momentum pseudo-tensor to offer up a sort of generalization of the concept of energy defined in a way ...
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0answers
68 views

Laplacian in tensor [closed]

Find $\vec \nabla^2\phi $ when $$ds^{2}=-dt^{2}+a^{2}(t)[dx^{2}+dy^{2}+dz^{2}] $$ or $$g_{ij}=\begin{bmatrix} -1 & 0 &0 &0 \\ 0 &a^{2}(t) &0 &0 \\ 0&0 ...
3
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1answer
129 views

Ricci flat compact manifold with $U(1)\times{}SU(2)\times{}SU(3)$ isometry group?

As the title says, is it possible to have a Riemannian Ricci flat compact manifold with $U(1)\times{}SU(2)\times{}SU(3) $ isometry group?
3
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1answer
71 views

Show that getting parallel transported does not change angle between them- Tensors [closed]

I must tell you that I have never seen this kind of question in Tensor Analysis. Our professor had set up this question in our exam, but I don't know whether it belongs to tensors or not. The question ...
2
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1answer
326 views

How to properly construct the electromagnetic tensor in curved space-time?

How do I properly construct the electromagnetic tensor in curved space-time? I have my curved spacetime metric $(+,-,-,-)$ and my magnetic vector potential $A$. I tried two ways but not sure which is ...
2
votes
1answer
57 views

Formular for interior product, example

In Nakahara's Geometry,Topology and Physics, the interior product is defined like this : $$i_X: \Omega^{r}(M) \rightarrow \Omega^{r-1}(M).$$ Where $ X \in X(M)$ and $\omega \in \Omega^{r}(M)$ ...
2
votes
1answer
178 views

Computing the Christoffel symbols with the geodesic equation

I would like to compute the Christoffel symbols of the second kind using the geodesic equation. To practice, I have tried the Schwarzschild Ansatz $$ g_{00} = \mathrm e^\nu,\quad g_{11} = - \mathrm ...
3
votes
1answer
229 views

Is the apparent lack of (Ricci) curvature in the Schwarzschild metric due to a choice of coordinates?

I've been lightly studying GR lately. Something that has been bothering me has been the lack of (Ricci) curvature produced from the Schwarzschild metric in the few lectures I've watched, as well as ...
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0answers
107 views

Variation of the purely covariant Riemann tensor

I need to find the variation of the purely covariant Riemann tensor with respect to the metric $g^{\mu \nu}$, i.e. $\delta R_{\rho \sigma \mu \nu}$. I know that, $R_{\rho \sigma \mu \nu} = g_{\rho ...
3
votes
1answer
199 views

Geodesics in AdS3

I'm having some trouble doing an easy computation with the AdS space. I'm considering $\text{AdS}_3$ space with the Poincaré coordinates, so the metric reads $$ds^2 = \frac{R^2}{z^2}(dz^2 - dt^2 + ...
1
vote
1answer
127 views

Has a metric formulation of electromagnetism ever been attempted? [duplicate]

I understand that electromagnetic fields carry energy, and this energy curves spacetime gravitationally. That's not my question. I'm asking if anyone has tried to formulate electromagnetism in such ...
2
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0answers
50 views

How to prove that a time-oriented spacetime possesses a nowhere vanishing timelike vector field?

Penrose gave a very brief proof to this question. Since the spacetime is paracompact, there exists a positive definite metric called $h_{ab}$. Then, the nowhere vanishing time-like vector field $V^a$ ...
11
votes
3answers
425 views

What kind of manifold can be the phase space of a Hamiltonian system?

Of course it should have dimension $2n$. But any more conditions? For example, can a genus-2 surface be the phase space of a Hamiltonian system?
0
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0answers
28 views

freedom of choice of 1-form in canonical representation of generic local field corresponds to gauge choice?

So it is a question in Gravitation Wheeler, Thorne and Misner 4.2 Exercise. Given F=$dp_{i}\wedge dq^{i}$. Using canonical transformation from p to $\bar{p}$ and q to $\bar{q}$, one gets ...
2
votes
1answer
62 views

A question on 1 form [closed]

If $d\,\sigma=0$ and $\sigma$ is non trivially with basis' coefficient 0, then $\sigma$ is a exterior derivative of a scalar function. I knew $d^{2} =0$. So it seems that all I am quoting is that ...
6
votes
1answer
187 views

E&M and geometry - a historical perspective

Recently, I was contemplating the beautiful formulation of electromagnetism (specifically Maxwell's equations) in terms of differential forms: $$F=\mathrm{d} A\implies \mathrm{d}F=0 ...
2
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1answer
103 views

The Einstein-Hilbert Action On-Shell

If one consider the Maxwell action as $$S=-\int \mathrm{d^{4}}x\! \ \frac{1}{4}F_{ab}F^{ab} \,$$ one find the usual Maxwell equation $$\partial_{a}F^{ab}=0$$ Then one can simply arrive the following ...
5
votes
3answers
354 views

Global vs. local gauge group in mathematical sense - physics examples?

Upon reading about the principal bundle picture of (quantum) field theory I encountered two different definitions of the gauge group: Local gauge group $G$. Corresponds to the fibers of the ...
4
votes
4answers
205 views

What makes a coordinate curved?

Bear with me while I try to explain exactly what the question is. The question Can a curvature in time (and not space) cause acceleration? is imagining a coordinate system in which the curvature is ...
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0answers
26 views

Angular diameter distance in an inhomogeneous universe?

Computing the angular diameter distance $D_{A}$ is a well known academic exercise in an homogeneous Universe. But now suppose that we are in an inhomogeneous Universe and that I am interested in ...
3
votes
0answers
66 views

Asymtotically flat spacetime applicable for spacetimes which are not diffeomorphic to $\mathbb{R}^4$

I wanted to investigate changes on a compact 4-manifold $M$. More specifically it is the K3-surface. I follow a paper by Asselmeyer-Maluga from 2012. The idea there was to make sure that the manifold ...
1
vote
1answer
76 views

Duality in arbitrary finite dimension using the Levi-Civita tensor

In 4-D flat metric E&M context, given a rank $p$ tensor, one can construct dual of $4-p$ rank tensor by Levi-Civita tensor. Here dual is not in the same sense of mathematical dual. I do not know ...
2
votes
0answers
70 views

Sign convention with the $AdS$ metric

One would say that $AdS_n$ satisfies the equations for the scalar curvature (R) and Ricci tensor ($R_{\mu \nu}$), $R = - \frac{n(n-1)}{L^2}$ and $R_{ab} = - \frac{n-1}{L^2}g_{ab}$. But do the signs ...
3
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0answers
51 views

Scalar product of torsional forms - how are the standard identities modified?

It is known that for any smooth, orientable, compact manifold $X$ without boundary and $\alpha \in \Omega^{r}(X), \beta \in \Omega^{r-1}(X)$ it holds \begin{equation} (d\beta,\alpha)= (\beta, ...
8
votes
2answers
249 views

Fluid Mechanics with calculus on manifolds

Fluid Mechanics is a branch of physics that uses a lot of vector calculus in $\mathbb{R}^3$ to describe phenomena mathematically. Calculus on manifolds, however, is the straightforward generalization ...
4
votes
0answers
67 views

Why is the Taub-NUT instanton singular at $\theta=\pi$?

Consider the following metric $$ds^2=V(dx+4m(1-\cos\theta)d\phi)^2+\frac{1}{V}(dr+r^2d\theta^2+r^2\sin^2\theta{}d\phi^2),$$ where $$V=1+\frac{4m}{r}.$$ That is the Taub-NUT instanton. I have been ...
1
vote
1answer
111 views

Second fundamental form

How do I calculate the integral of the trace of the second fundamental form on a surface? The formula used in the Gibbons, Hawking, York paper Action integrals and partition functions in quantum ...
6
votes
2answers
508 views

Why isn't invariant notation common?

In principle, one can write quantities in a manifestly invariant - rather than covariant - fashion in e.g. special relativity. For example, rather than writing just $x^\mu$, we could write the basis ...
6
votes
2answers
190 views

Killing vectors in flat FLRW metric

I have the flat FLRW metric, $$ ds^2=-dt^2+a(t)^2(dx^2+dy^2+dz^2) $$ and a geodesic $\gamma(s)=(t(s),x(s),y(s),z(s))$ with parameter $s$. Two of the Killing vectors of the metric are $ \partial_x$ ...
12
votes
1answer
251 views

Conformal/trace anomaly and index theorem

I am reading the chapters on characteristic classes and the index theorems in Nakahara. It is proven in the text that any chiral or gravitational anomaly $\mathcal{A}$ is given by $$\mathcal{A}=\int ...
2
votes
2answers
164 views

Why is the anti-symmetric tensor more important than symmetric tensors?

In differential geometry, the differential forms are anti-symmetric tensors. So, why is the anti-symmetric tensor like $ d x_1 \otimes dx_2 - d x_2 \otimes d x_1 $, more important than the ...
4
votes
1answer
127 views

The relationship between Lorentz Lie algebra and curvature

Here I transfered the question from the comment The relationship between spin and spinor curvature How $\mathcal{R}_{ab} = \frac{1}{4}R_{abst}\gamma^s \gamma^t$ is from $\Psi \mapsto \Psi + ...
2
votes
1answer
214 views

Two definitions of Riemann curvature tensor

I am relatively used to the coordinate free expression of the Riemann tensor: $$ R(X, Y)Z=\nabla_X\nabla_Y Z - \nabla_Y\nabla_X Z - \nabla_{[X, Y]} Z, $$ where $\nabla$ is the Levi-Civita connection ...
2
votes
2answers
377 views

Time-like Killing vector in FRW metric?

The spatially flat FRW metric in cartesian co-ordinates is given by: $$ds^2 = -dt^2 + a^2(t)(dx^2 + dy^2 + dz^2)$$ As I understand it there are Killing vectors in the $x$, $y$, $z$ directions implying ...
1
vote
3answers
244 views

Why do we need a metric to define gradient?

For me, the gradient of a scalar field (say, in three dimensions) is simply (formally) $\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y},\frac{\partial f}{\partial z} ...
1
vote
1answer
85 views

How big or small is a reference frame in Relativity?

What exactly is a frame of reference? Does it have an objective existence and if so what is it? What's the size of a reference frame? Is a reference frame the same size for a stationary frame of ...
2
votes
2answers
80 views

Orthogonality in curved space/spacetime

When are two vectors orthogonal in curved spacetime? From wikipedia: "In 2-D or higher-dimensional Euclidean space, two vectors are orthogonal if and only if their dot product is zero, i.e. they ...
2
votes
1answer
144 views

How to understand the curvature of this metric?

Suppose we have the metric $ ds^2 = dr^2 + \alpha^2 d\phi^2$, where $\alpha$ is a constant, $0 \leq r \leq \infty$, $ 0 \leq \phi \leq 2 \pi$ and we identify points $\phi = 0$ with points $\phi = ...
12
votes
4answers
224 views

Difference between matrix representations of tensors and $\delta^{i}_{j}$ and $\delta_{ij}$?

My question basically is, is Kronecker delta $\delta_{ij}$ or $\delta^{i}_{j}$. Many tensor calculus books (including the one which I use) state it to be the latter, whereas I have also read many ...
2
votes
1answer
552 views

Is the scalar curvature of the Schwarzschild solution 0?

The Schwarzschild solution is meant to be a solution of the vacuum Einstein equations. That is $$R_{\mu\nu}=0.$$ So, the Ricci tensor must be null for $r>0$. Now, if the scalar curvature is ...
4
votes
0answers
125 views

Lie derivative of Dirac Delta

In the setting of general relativity, I came across a source term of the wave equation of the following form: $$ \frac{1}{\sqrt{q}}\,\delta^{(3)}(p-\gamma(t)) $$ where $p\in M$ is a point in our 4d ...
2
votes
2answers
136 views

Covariant derivative of a covariant tensor wrt superscript

Is it true that when you take the covariant derivative of a covariant tensor, do you always have to do with a subscript? What if you do it wrt a superscript?Does the first term (with the partial ...