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2 votes

Cone vs. small circle parallel transport

This is in a way a Riemann-geometric analogue of the Aharonov--Bohm effect in quantum mechanics. Consider a manifold $M$ equipped with a linear connection $\nabla$. It is well-known that the parallel ...
Bence Racskó's user avatar
2 votes
Accepted

Isn't the induced electric field vector always tangent to the loop?

But isn't the field $\mathbf E$ which is responsible for the induced emf also along the contour (unit tangent vector $\hat{\mathbf t}$)? No, the equation is true for any contour you choose in a given ...
The Photon's user avatar
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2 votes
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Two contradictory derivations of Killing equation

As an overall comment, I stress that conservation of $Q$ is valid for the Killing vector $\xi$ if the considered curve is a geodesic. Let us come to the issue. First of all, generally speaking, the ...
Valter Moretti's user avatar
2 votes

Two contradictory derivations of Killing equation

Both approaches are fine. In the first approach, the analysis is done at the coordinate/component level of the equations. Simply asking the question how does $Q$ very with $\tau$ if we write ...
TimRias's user avatar
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1 vote

Two contradictory derivations of Killing equation

I think the problem lies in the notation. I guess Tong is treating the components $\xi_\mu$ as scalars for which we have $\frac{\mathrm{d}}{\mathrm{d}\tau}=u^\alpha\partial_\alpha$. And the ...
Silas's user avatar
  • 425
1 vote

Isn't the induced electric field vector always tangent to the loop?

Do you mean "which is responsible" or "that is responsible"? (See Strunk and White, Which Hunts: https://clubtroppo.com.au/2008/11/30/the-great-which-hunt/). The former is ...
JEB's user avatar
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1 vote
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How to compute the vector field from a potential in the complex plane?

What did you plot exactly, I rather obtained something like this: which is compatible with the lecture. The velocity field is: $$ \dot z = w := \sqrt{E-V(z)} $$ with $\min V<E<0$. Since it is ...
LPZ's user avatar
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1 vote

The conservative force

While @basics has the math right, I understand your question is about the physical interptetation. To answer this, we must understand what a force field is, since the definition of the rotation ...
paulina's user avatar
  • 1,897
1 vote

The conservative force

An example might be the best way to see it. Set up some electrodes that create a horizontal electric field. Make it so the field strength is proportional to height. This is a non-conservative field. ...
mmesser314's user avatar
  • 41.1k
1 vote

The conservative force

Remark the curl it's the measure of the rotation of the vector field around a specific point Before answering your question, I'd replace (local) "rotation" with (local, or elementary) &...
basics's user avatar
  • 10.7k
1 vote

Killing tensor in the Kerr metric

The Killing equation is a overdetermined system of PDEs of finite type, as a result, there is a algorithm to compute all Killing tensors for a given metric. See for example arXiv 1704.02074.
liyiontheway's user avatar

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