5 votes

How to Relate the Functional Derivative to Infinitesimal Change in Noether's Theorem

One way to do a calculation similar to OP's without a notion of infinitesimal is to consider a one-parameter deformation of the field $\phi^\epsilon$ (with $\phi = \phi^0$) and define $\delta$ to mean ...
SolubleFish's user avatar
  • 5,751
5 votes
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References on Newton-Cartan Gravity

First, the MTW book has a chapter 12 on Newton-Cartan (NC) theory, but it is rather outdated in the sense that some important results were obtained after this book was published. A book by D. ...
4 votes
Accepted

How to Relate the Functional Derivative to Infinitesimal Change in Noether's Theorem

I will be quoting a lot from the functional derivative page on wikipedia, since it provides a great reference. Recall that a function is an object which takes as input a function and has as output a ...
AccidentalTaylorExpansion's user avatar
2 votes
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How to understand the relationship between Weinhold geometry and Ruppeiner geometry in thermodynamic geometry?

You have to provide more background information, with more-accessible references. From a Google search, these seem relevant and more-accessible https://en.wikipedia.org/wiki/Ruppeiner_geometry (...
robphy's user avatar
  • 11.8k
2 votes

Exponential of the metric tensor

Note that when raising an index of the metric tensor, you get a Kronecker delta (by design): \begin{align} {g_i}^j=g_{ik}g^{kj}=\delta_i^j \end{align} This is because $g^{kj}$ is shorthand for the ...
AccidentalTaylorExpansion's user avatar
2 votes

Does the divergence theorem require the covariant derivative to be metric compatible?

Suppose $M$ to be orientable and oriented and let $\mu\in\Omega^m(M)$ be a volume form ($m=\dim M$). Let $\nabla$ be a linear connection on $\tau:TM\rightarrow M$. Given a vector field $X\in\mathcal D(...
Bence Racskó's user avatar
2 votes

Stokes' theorem and vector continuity equations

As an example of the physical meaning behind such manipulations; Faraday's law is of the form of your first differential relation: $$\nabla\times\vec E+{\partial\vec B\over\partial t}=0.$$ We can ...
Albertus Magnus's user avatar
2 votes

Geodesic in flat space in spherical coordinates

You have made some major mathematical mistakes. There is actually no contradiction here at all. You have only defined $\mathbf{u}$ to be the tangent vector to a curve in the manifold $M$, which means ...
Vincent Thacker's user avatar
2 votes

How does the conformal scaling behavior of the stress-energy tensor depend on the spacetime dimension?

This is just dimensional analysis. The Hamiltonian $$ H=\int\mathrm d^D\boldsymbol x \, T $$ has dimensions of energy, so $T$ has engineering dimension $\Delta=D+1$. So for example, in $d=1+1$, $\...
AccidentalFourierTransform's user avatar
2 votes

References on Newton-Cartan Gravity

I don’t have very many in-depth references, but, here’s a brief list: Elie Cartan - On Manifolds with Affine Connections and the Theory of General Relativity (original French in 1923, translated to ...
1 vote

How does the conformal scaling behavior of the stress-energy tensor depend on the spacetime dimension?

Under Weyl transformations, the partition function of a CFT transforms as $$ Z[\Omega^2g] = e^{-S_{an}[\Omega,g]} Z[g] $$ where $S_{an}[\Omega,g]$ is the anomaly action. This is non-vanishing only in ...
Prahar's user avatar
  • 26k
1 vote

How to motivate that in presence of gravity the spacetime metric must be modified to $ds^2=g_{ab}(x)dx^adx^b$?

It's a consequence of the equivalence principle, for which a gravitational field is locally indiscernible from an acceleration effect, for any moving small body. And the corollary: a free fall in a ...
Cham's user avatar
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1 vote

Why is the 4-current a tensor rather than a tensor density?

First of all, the electromagnetic wave equation given in the question is valid (assuming the Lorenz gauge condition) only in flat or at least Ricci-flat spacetime. In general curved spacetime there ...
Brian Bi's user avatar
  • 6,511
1 vote

Geodesic in flat space in spherical coordinates

The real question is, "what exactly is a divergence?" The physical meaning of the divergence of a point particle's geodesic doesn't make sense. A divergence is meaningful for a vector ...
Paul T.'s user avatar
  • 7,105
1 vote

Derivation of Wald's general relativity equation 7.5.8 ; conformal transformation of Riemann tensor

I notice the point where i mis-understood the textbook. Re-writing a Riemann tensor in terms of covariant derivatives was fine. But the problem was expanding connections and collect the terms properly....
phy_math's user avatar
  • 3,562
1 vote

How is this deduced? (Differentiation of tensors)

The "abstract" covariant derivative $\nabla$ maps tensors of type $(i, j)$ to tensors of type $(i, j+1)$. More precisely, the covariant derivative adds one dual slot to the tensor. Using the ...
Vincent Thacker's user avatar
1 vote

What is the relation between gauge field and Levi-Civita connection?

Both are connections on a principal fibre bundle, but on different principal fibre bundles. For Yang--Mills theory with structure group $G$, there is a principal $G$-bundle $\pi:P\rightarrow M$ over ...
Bence Racskó's user avatar
1 vote

Does the Weyl tensor amount to tidal effects of gravity?

A nice way of understanding why the Weyl tensor is of importance in this context is to see that you can define a ``gravitational entropy" (NOT to be confused with the area term in generalized ...
VaibhavK's user avatar
  • 442
1 vote

How to Relate the Functional Derivative to Infinitesimal Change in Noether's Theorem

It is not entirely clear what OP is trying to achieve. If OP is trying to avoid using infinitesimal variations and infinitesimals, one can in principle equivalently use vector fields and Lie ...
Qmechanic's user avatar
  • 202k
1 vote
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Gauge transformation with harmonic one-form

You can add any vector field whose curl is zero (any one-form whose differential is zero). If you are working in a simply connected domain (no holes) then these vector fields can only be the gradients ...
Jahan Claes's user avatar
  • 8,100
1 vote

Gauge transformation with harmonic one-form

Harmonic forms are solutions of the Laplace equation $$ (\delta d +d\delta)\psi=0. $$ In infinite flat space a harmonic form would be a $C_mdx^\mu$, where the $C_\mu$ are numerically constant. With, ...
mike stone's user avatar
  • 53.2k
1 vote

Inverstion matrix as parallel transport in CFT

The $x \mapsto x'$ map (if it's conformal) can be undone by the Weyl rescaling you write. Namely \begin{equation} g_{\mu\nu}(x) \mapsto \Omega(x)^{-2} g_{\mu\nu}(x) \end{equation} for a suitable $\...
Connor Behan's user avatar
  • 7,246
1 vote

Lie derivative: moving boat on a flowing river

I'm not quite sure what you mean here, but it looked initially like maybe you were taking $$ V = V_x \hat x + V_y \hat y,\\ \xi = x_0 \hat x + y_0 \hat y, $$ with none of $V_x, V_y, x_0, y_0$ ...
Steven Dorsher's user avatar
1 vote

Why future infinity have no future end points?

In simple terms, if you imagine tracing the paths of light rays or gravitational waves as they propagate through spacetime, those paths never end in the future; they keep extending outwards towards ...
Adversing's user avatar
  • 161
1 vote

Contracting the metric tensor with its inverse yields Kronecker delta

The statement $g_{\mu \nu} g^{\nu a} = \delta _{\mu}^{a} $ Is a convenience in tensor notation for writing out that matrix $g_{\mu \nu}$ indexed by $\mu, \nu$ multiplied by the matrix $g^{\nu a} $ ...
Sidharth Ghoshal's user avatar
1 vote
Accepted

Carroll's interpretation of 1-forms

I know that this post is pretty old, but I came across this sentence lately while skimming through Carroll's book, and there is one possible interpretation of this that hasn't been discussed yet. ...
Valh's user avatar
  • 26

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