5 votes

Partial derivatives vs Covariant derivatives in polar coordinates

Covariant derivatives take into account for both component and basis changes, thereby applicable for curved spaces - where partial derivatives only take component changes into account - is this ...
Lenard Kasselmann's user avatar
5 votes

Pre-requisites for V.I. Arnold's mathematical methods for classical mechanics

It is difficult to answer if the question is how to make intuitive the content if that book. The point is that, the goal of that book is just to make rigorous some important arguments and topics of ...
4 votes

Does the sign of the connection coefficients $\Gamma^\lambda_{\mu\nu}$ change with the signature of a metric?

Might as well write an answer for this, though I wrote the idea in a comment: The Christoffel Symbols of the first kind are given by: $$\Gamma_{\rho\mu\nu} = \frac{1}{2} (g_{\rho \mu,\nu}+g_{\rho\nu,\...
Amit's user avatar
  • 1,258
4 votes
Accepted

What is Dirac's reasoning when showing the curvature vanishing implies we can choose rectilinear coordinates?

Suppose that we have $n$ independent variables $x^\mu$ and $m$ "dependent variables" $y^i$ and an overdetermined system$$ \frac{\partial\phi^i}{\partial x^\mu}(x)=F^i_{\mu}(x,\phi(x)) \qquad(...
Bence Racskó's user avatar
3 votes

Partial derivatives vs Covariant derivatives in polar coordinates

As OP correctly points out connections introduce a concept of differentiation of tensor fields or more in general of sections of vector bundles that takes into account how the bases of the fibers ...
Mr. Feynman's user avatar
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3 votes

In general relativity, are light-like curves light-like geodesics?

Adding to the other answers, this is just to give an overall explanation. Case of 1+1 flat spacetime- We can first make the choice of co-ordinates and units such that geodesics are straight lines of ...
Ryder Rude's user avatar
  • 5,994
3 votes
Accepted

How does the wavefunction transform under an arbitrary change of variables?

TL;DR: As the overall phase of the wavefunction is not physical, OP's question has a non-unique answer that ultimately comes down to a choice of convention. Within a given class of situations we often ...
Qmechanic's user avatar
  • 188k
3 votes
Accepted

Classical systems with compact phase space

I could comment that a Hamiltonian system with compact symplectic phase space could appear after Lie-Poission reduction of a left (or right) invariant Hamiltonian defined on the cotangent bundle of a ...
Futurologist's user avatar
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3 votes

What does it mean for a quantum field theory to "live" on a manifold?

There are various layers to how a quantum field theory lives on a manifold. The simplest one is to note that a (classical) field is a mapping between a spacetime manifold and some target space like ...
QuantumFieldMedalist's user avatar
3 votes
Accepted

Simple distance calculation in General Relativity

The only general notion of total distance along a curve in general relativity is the proper time (or interval) $$\Delta \tau = \int \sqrt{-g_{\mu\nu} \frac{dx^\mu}{d\lambda} \frac{dx^\nu}{d\lambda}} d\...
Javier's user avatar
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2 votes

The "geometry" of thermodynamics

The surface and its differential geometry that pops up in the Caratheodory-style thermostatics and one you are alluding to is related to either of two functions $S=S_1(T,X_1,X_2,...,X_N)$ or the $S=...
hyportnex's user avatar
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2 votes
Accepted

Precise definition of a string worldsheet as a manifold in string theory

That's not how anyone uses the term "worldsheet", even though you are correct that $\mathbb{R}\times I$ and $\mathbb{R}\times S^1$ are the "worldsheets" one would assign to a ...
ACuriousMind's user avatar
  • 119k
1 vote

What is Dirac's reasoning when showing the curvature vanishing implies we can choose rectilinear coordinates?

If you are looking for a deeply mathematical explanation, please refer to @Bence Racskó's answer, I think it really is very good and insightful. I want to do something different here, I want to show ...
Amit's user avatar
  • 1,258
1 vote
Accepted

Difference between $R^{a}_{bcd}$ and $R_{abcd}$ Riemann tensor types

There is no deep intuitive geometrical meaning behind a Riemann tensor with some indices moved up/down. You could say that the two variants are "dual" to each other, loosely speaking. The ...
Avantgarde's user avatar
  • 3,513
1 vote

Equating 2 sides of EFE

First, note that for scalar functions, the covariant derivative reduces to the partial derivative. So for scalar functions, it is true that if the covariant derivative is zero at a point, then the ...
Andrew's user avatar
  • 42.9k
1 vote

Why if the metric tensor components are constant then SR applies?

Let me summarize some ideas in the comments and the other answer and add an important point regarding frames that I think is interesting to keep in mind. In Lorentzian signature, if you go to an ...
Gravitino's user avatar
  • 549
1 vote

Why if the metric tensor components are constant then SR applies?

The metric is diagonalizable, yes, and then with further coordinate transformations it can be converted to Minkowski; see, for example, Schutz exercise 6.3 where he guides you through the steps.
Lewis Kirby's user avatar
1 vote

Covariant derivative of gauge theory in curved space

It depends on what sort of field $\phi$ is. If $\phi$ is a scalar field, you can use the ordinary derivatove, but if $\phi$ is a Dirac spinor, for instance, then you will need to include the spin ...
mike stone's user avatar
  • 48.2k
1 vote

General Relativity manipulating tensors, tensor indices meaning

About the first part, you nearly answered your own question: it is indeed exactly because the definition of tensors arises naturally out of multilinear maps such as $t(e^a,e^b)$ that the order of ...
Amit's user avatar
  • 1,258
1 vote

Derivatives of parallel transport operator

In general, published papers don’t show all the steps of a tedious but straightforward calculation; they just give the final result. You might as well calculate the more general result $[P_{;abcd}]$, ...
Ghoster's user avatar
  • 686
1 vote

What is the definition of a Brachistochrone curve in a non-Euclidean space?

A non-Euclidian space here presumably means a Riemannian manifold $(M,g)$. Let us also assume there is given some gravitational energy distribution/profile $V:M\to\mathbb{R}$. So the brachistochrone ...
Qmechanic's user avatar
  • 188k

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