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

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5
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1answer
79 views

Momentum is a cotangent vector?

Imagine we have a particle described by $x \in M$, where $M$ is some manifold, then it is very intuitive I think that a velocity is an element of the tangent space at $x$, so $x' \in T_{x}M.$ Thus, by ...
2
votes
1answer
72 views

Proof that 4-potential exists from Gauss-Faraday field equation

This is a problem concerning covariant formulation of electromagnetism. Given $$\partial_{[\alpha} F_{\beta\gamma]}~=~ 0 $$ how does one prove that $F$ can be obtained from a 4-potential $A$ such ...
1
vote
1answer
31 views

Show that a solenoidal field is always a curl of a vector field

Can someone prove that: $$\nabla \cdot \mathbf{B} = 0 \implies \mathbf{B} = \nabla \times \mathbf{A}~?$$ I know that $$\nabla \cdot (\nabla \times \mathbf{A}) = 0$$ identically. But can one prove ...
0
votes
0answers
28 views

Killing vector field along geodesic [on hold]

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 ...
3
votes
0answers
142 views

Curvature and spacetime

Suppose that it is given that the Riemann curvature tensor in a special kind of spacetime of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a ...
6
votes
2answers
332 views

Sewing together flat spacetime pieces = flat spacetime?

I'm trying to imagine the geometry "operations" here: Angular deficit and Conical spacetime of cosmic string If we sew flat spacetime pieces together, what is the requirement for the sewing to not ...
1
vote
1answer
101 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 ...
8
votes
1answer
97 views

What is the physical interpretation of the Poisson bracket [duplicate]

Apologies if this is a really basic question, but what is the physical interpretation of the Poisson bracket in classical mechanics? In particular, how should one interpret the relation between the ...
0
votes
0answers
36 views

Jets and vertical differential [migrated]

For a vector bundle $(E,\pi, M)$ let $\phi :M\mapsto E$ be a section of $\pi $, $x\in M$ and $u=\phi (x)$. The vertical differential of the section $\phi$ at point $u\in E$ is the map: ...
2
votes
2answers
48 views

Under what representation do the Christoffel symbols transform?

I often read the statement, that the Christoffel symbols aren't tensors. But then, under which representation do they transform?
2
votes
1answer
43 views

Local translations in curved spacetime

A global Poincare transformation on a scalar field induces $$\delta(a, \lambda)\phi(x) = [a^{\mu}+\lambda^{\mu\nu}x_{\nu}]\partial_{\mu}\phi(x). \tag{11.46}$$ In curved spacetime we replace $a^{\mu} ...
0
votes
0answers
31 views

Perturbations and Chronology condition [on hold]

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 ...
2
votes
2answers
70 views

Differential operators in curvilinear coordinates

In the appendix A of Griffith's Electrodynamics text, he cites Spivak's Calculus on Manifolds as a reference more a more complete treatment of taking the gradient, curl, divergence, and Laplacian in ...
1
vote
1answer
88 views

Covariant derivative of Levi-Civita tensor [on hold]

I'm currently studying Carroll's GR book Spacetime & Geometry, and ran into some trouble understanding the text at page 99 which says: By using metric compatibility $$(\nabla_{\alpha}g)_{\mu ...
7
votes
1answer
110 views

What does Ricci tensor do with two vectors?

I have found it easier to understand the meaning of a particular tensor if I can find out what does it do if I cancel all its lower indices with vectors in short: $g_{ij} u^i v^j$: dot product of ...
1
vote
2answers
79 views

Phase space Lagrangian?

Reading out of this lecture series we define a phase space Lagrangian $\mathcal L$ to be a function of $4n+1$ variables namely $q,\dot q,p,\dot p,t$. My question is, what space is this function ...
0
votes
0answers
81 views

Integrability in classical mechanics

An integrable system in classical mechanics is defined by action-angle variables and closed loop trajectories in phase space. I have also heard that the flow lines of an integrable system are ...
1
vote
0answers
131 views

Can the universe be round but still infinite?

Can the universe still be infinite in space if its curvature is > 1? Is a manifold of positive curvature necessarily compact? Does the Tarski paradox have any bearing on the finite or infinite ...
1
vote
0answers
49 views

Is the Weyl Postulate correct?

The Weyl postulate in cosmology states that worldlines do not intersect but it can be shown in GR that using Raychaudhuri equation that geodesics can intersect if there is curvature so I'm really ...
2
votes
1answer
56 views

When can an autonomous system be written using a Hamiltonian?

If I have an autonomous series of differential equations $$\tag{1} \frac{dx_i}{dt} ~=~ A_i(x_1,...,x_n)$$ with the condition that $$\tag{2} \sum_{i=1}^n\frac{\partial A_i}{\partial x_i}~=~0$$ in all ...
1
vote
1answer
119 views

Prove Christoffel Symbol Identity

In a book I am reading, the following identity is claimed and then "left to the reader to prove." $g_{ij}$ is the metric tensor, and $\Gamma$ is the Christoff symbol of the second kind with the ...
0
votes
0answers
28 views

How do you get the curvature tensor of the Schwarzschild Solution? [closed]

So, on the Wikipedia page on the derivation of the Schwarzschild solution, I get everything up to the part about the Ricci tensor. What were the components of the tensor that were used? Could ...
6
votes
1answer
130 views

Non-trivial scalar quantity

Is there any scalar quantity made of only the Christoffel symbols, determinant of a metric and tensors, not derivatives? In other words, can we construct a scalar quantity which cannot be written in ...
2
votes
7answers
427 views

Curvature of Spacetime

I have been exploring for some time both the Special and General Relativity, hoping to glean at least a conceptual grasp of their basic tenets. In reading the book "Gravitation" by Misner, Thorne and ...
3
votes
0answers
62 views

Maxwell's equation in curved spacetime - how come? And experimental evidence?

I'm trying to understand the generalization of Maxwell's equations to curved spacetime. In FLAT (Minkowski) SPACETIME: If we define the "four-potential" as $$\ ...
0
votes
1answer
56 views

EM Field tensor of a point charge [closed]

If I say the Reissner-Nordstrom metric $$ ds^2=-\left(1-\frac{2m}{r}+\frac{e^2}{r^2}\right)\text d t^2 + \left( 1-\frac{2m}{r}+\frac{e^2}{r^2}\right)^{-1}\text d r^2 + r^2 \text d \Omega^2 $$ is the ...
1
vote
0answers
45 views

How is Infinitesimal coordinate transformation related to Lie derivatives?

I am reading the book "Gravitaion and Cosmology" by S. Weinberg. In section 10.9, while discussing Lie derivatives of tensors of different ranks, he makes a general comment: The effect of an ...
1
vote
1answer
82 views

Distributions (generalized functions) over manifolds

I have asked a similar question on the math stackexchange website, but since this type of question might have an answer that is known to physicists better than mathematicians I'm posting the question ...
0
votes
0answers
17 views

Atlas 2 For Mathematica. Need to calculate Riemann tensor, etc [migrated]

Is anyone familiar with Atlas 2 for Mathematica to calculate the Riemann Tensor, Ricci Tensor, and scalar I have a metric that I need to calculate these things for. Can anyone help? I'm not too up on ...
3
votes
1answer
63 views

What is the physical cause that circulation on a closed surface is zero?

This is quoted from Feynman's Lectures: We would like to see what happens when the loop shrinks down to a point, so that surface boundary disappears - the surface becomes closed. Now, if the ...
1
vote
1answer
97 views

What is the covariant basis around a Schwarzschild black hole?

First of all, I'm not interested in time for this question. So lets consider a 3-manifold whose metric is the spatial part of the Schwarzschild metric, so it has the event horizon and the singularity ...
2
votes
2answers
6k views

Derive vector gradient in spherical coordinates from first principles

Trying to understand where the $\frac{1}{r sin(\theta)}$ and $1/r$ bits come in the definition of gradient. I've derived the spherical unit vectors but now I don't understand how to transform ...
1
vote
1answer
58 views

Question about basic formalism of GR and the metric tensor

I really don't know much about GR, but I've come across a few rough sketches of its formalism in my DG books. I'm trying to piece it together to get a very basic intuition of what spacetime is in GR. ...
1
vote
0answers
25 views

SR: vector field and change of reference [closed]

If $U$ and $V$ are vector fields, then the derivative of $U$ along $V$ is the vector field $\nabla _V U$ with components $$\nabla _V U^a=V^b \frac{\partial U^a}{\partial x^b}.$$ I would like to verify ...
0
votes
1answer
91 views

How to find a metric of a general observer?

Yes, that's it. How to find a particular metric of an observer in general relativity? Let's say we have a static metric: ...
1
vote
2answers
53 views

Metric and the Lagrangian [duplicate]

Does the Lagrangian formalism require a metric on the configuration manifold $Q$ in order to define a Lagrangian $L$ on the tangent bundle $TQ$? Further, if we specify a metric on the tangent bundle ...
0
votes
0answers
38 views

$U(1)$ connection and spacetime basis $e^{\mu}$

When dealing with supergravity, it is said that a Kahler-Hodge manifold has a $U(1)$ bundle whose first Chern class coincides with the Kahler class, thus locally the $U(1)$ connection can take the ...
2
votes
2answers
476 views

Geodesic equation from Euler - Lagrange

There are several ways to derive the geodesic equation. One of which is the variational method which I seemed to understand it because it was written in great details. Then it was mentioned that the ...
1
vote
0answers
34 views

Ricci tensor of Metric black holes with nils and solv geometries of Horizon

The metric of black holes with nils and solv geometries of the horizon is generically represented by $$\mathrm{d}s^2=-r^{2z}\mathrm{d}t^2+\frac{\mathrm{d}r^2}{r^2}+\sum_{I=1}^3 r^{2q_I}(w^I)^2$$ How ...
0
votes
1answer
76 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 ...
1
vote
2answers
144 views

$\nabla^{\mu}\nabla_{\mu}$ in general relativity [closed]

I am trying to work out $\square=\nabla^{\mu}\nabla_{\mu}$ in the metric $ ds^{2}=-A(r)dt^{2}+B(r)^{-1}dr^{2}+r^{2}d\Omega^{2} $$ My work: when applying $\square$ to a scalar $\phi$, then $ ...
1
vote
2answers
117 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 ($\gamma$) ...
2
votes
1answer
49 views

Rindler and Minkowski space future/past infinity

In my black holes course, we are looking at the Penrose diagram for 1+1 D Minkowski space. My notes don't specifically describe $i^{\pm}$ (future/past timelike infinity) but do say all timelike curves ...
2
votes
5answers
112 views

A reference frame is any coordinate system or just a set of Cartesian axes?

In Physics the idea of a reference frame is one important idea. In many texts I've seem, a reference frame is not defined explicitly, but rather there seems to be one implicit definition that a ...
3
votes
2answers
193 views

Correct tetrad index notation

There seems to be some different conventions on the indexes of the tetrad. I am wondering which is the standard, which is correct, and which is an abuse of notation. In Sean Carroll's notes and in ...
1
vote
2answers
128 views

Good Fiber Bundles and Differential Geometry references for Physicists

I'm a student of Physics and I have interest on the theory of Fiber Bundles because of the applications they have in Physics (gauge theory for example). What are good books to learn the theory of ...
1
vote
1answer
60 views

Can the N-body problem be solved numerically using the geodesic equation of mass-distorted spacetime?

I would like to write a program that solves the trajectories of objects (think rockets) that are influenced by mass of other objects (think planets). I saw that I can do this using Newton's laws, but ...
13
votes
0answers
1k views

What is a Manifold? [migrated]

Now we always encounter definition of a manifold from a mathematical point of view where it is a topological space along with a family of open sets that covers it and the same old symphony. My ...
0
votes
0answers
43 views

Connection one-form and suppressed indices

I am reading Sean Carroll's notes on GR, which states (Page 91): Using our freedom to suppress indices on differential forms, we can write the defining relations for these two tensors as: $$ T^a ...
3
votes
1answer
80 views

Lagrangian vector field expression

The Lagrangian vector field $X_L$ on the tangent bundle is given in page 4 of Marco Mazzucchelli's "critical Point Theory for Lagrangian systems" to be; \begin{equation} ...