5
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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 ...
5
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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:
$$\Gamma_{\rho\mu\nu} = \frac{1}{2} (g_{\rho \mu,\nu}+g_{\rho\nu,\...
- 1,258
4
votes
Accepted
What is Dirac's reasoning when showing the curvature vanishing implies we can choose rectilinear coordinates?
Suppose that we have $n$ independent variables $x^\mu$ and $m$ "dependent variables" $y^i$ and an overdetermined system$$ \frac{\partial\phi^i}{\partial x^\mu}(x)=F^i_{\mu}(x,\phi(x)) \qquad(...
- 10.3k
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 ...
- 1,607
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 ...
- 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 ...
- 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 ...
- 2,362
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 ...
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\...
- 27.3k
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=...
- 14.3k
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 ...
- 119k
1
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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 ...
- 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 ...
- 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 ...
- 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 ...
- 549
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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.
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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 ...
- 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 ...
- 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 $[P_{;abcd}]$, ...
- 686
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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 ...
- 188k
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