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comment Is it possible that black holes are also neutron stars, but so dark that we cannot see them?
Why the downvote?
Apr
29
comment Dipole moment of a single point charge
If the moments up to order $n-1$ vanish, the moment of order $n$ is independent of coordinate system. Perhaps you should add "in general" to the first sentence?
Apr
28
comment Converting between matrix multiplication and tensor contraction
@tfb Some months ago I came across this, which reminds me of your story.
Apr
28
comment Unitary operators evolving the set of Pauli matrices
The paper mixes two conventions and probably defines $\vec{\sigma}(t) = \sigma_1(t) \vec{e}_1 + \sigma_2(t) \vec{e}_2 + \sigma_3(t) \vec{e}_3$, where the vector space is defined by the Pauli matrices, i.e. $\vec{e}_1 = \sigma_x$, etc.
Apr
28
comment Unitary operators evolving the set of Pauli matrices
Check Frobenius's answer below, it's more explicit. Does the paper define $\vec{\sigma} = (\sigma_x, \sigma_y, \sigma_z)$?
Apr
28
comment Unitary operators evolving the set of Pauli matrices
For example, $\vec{\sigma}_{11}$ is $\vec{e}_z$. Probably, the notation confuses you because you interpret $\vec{e}_i$ in terms of coordinates. Vectors are elements of a vector space and it is an extra step to assign coordinates to them. The vector spaces where $\vec{e}_i$ and $\sigma_{i}$ belong are not the same in general!
Apr
28
comment Unitary operators evolving the set of Pauli matrices
$\sigma_{ij}$ is the $i,j$-th element of the matrix $\vec{\sigma}$. A common convention is to interpret $i$ as the row and $j$ as the column.
Apr
28
comment Unitary operators evolving the set of Pauli matrices
This notation can be used even if the coefficient matrices are linearly dependent!
Apr
28
comment Unitary operators evolving the set of Pauli matrices
$\vec{\sigma}$ is a $2\times 2$ matrix, $\sigma_x \vec{e}_x + \sigma_y \vec{e}_y + \sigma_z \vec{e}_z$.
Nov
15
comment Why do we obtain classical physics by taking the limit of Planck's constant to zero?
Well, one obvious reason is that $h$ does not appear in the equations of classical physics. If $[x,p]=0$, every state is defined simultaneously by position and momentum, which differentiates between the classical and the quantum theory. This answer does not explain certain details, which appear in chapter VI, §1 of Quantum mechanics by Messiah.
Nov
13
comment Conservation of energy and Killing-field
Also, it is not necessary that the Killing vector is timelike, it is necessary that it is asymptotically timelike, and this is for the reason described in my answer below, namely that the scalar defined using it has the appropriate asymptotic behaviour. In Kerr space-time, $K^a$ may be space-like!
Nov
13
comment Conservation of energy and Killing-field
"Physically, asymptotically flat space-times represent isolated systems", cf. Robert Geroch and Jeffrey Winicour. Linkages in general relativity. Journal of Mathematical Physics, 22(4):803-812, 1981.
Nov
12
comment Conservation of energy and Killing-field
What do you mean by "isolated system"?
Nov
12
comment Conservation of energy and Killing-field
Isn't the reason asymptotically flat space-times are preferable to model physical systems that they can be considered as isolated?
Nov
12
comment Does gravity acting on a resting object produce any heat?
Yes, you're right, I missed that line.
Nov
11
comment Does gravity acting on a resting object produce any heat?
If the object is not a point particle, as you imply, then tidal forces can produce heat.
Nov
6
comment How is hydrostatic pressure overcome when a star is formed?
By gravity you mean the acceleration of gravity, right? How did you derive $P \propto 1/r^3$? By using $PV = NkT$ and keeping $T$ constant? But the temperature changes, too. You need to assume a realistic equation of state for the cloud to collapse.
Nov
6
comment How is hydrostatic pressure overcome when a star is formed?
@user2800708 What do you mean by these numbers?
Nov
5
comment How is hydrostatic pressure overcome when a star is formed?
The OP assumed certain equation of state in his question and wonders whether the cloud will collapse or not, therefore the virial theorem must be expressed in terms of the thermodynamic variables of the cloud. I'm not going further than this, because I'm not certain what the OP is claiming.
Nov
5
comment How is hydrostatic pressure overcome when a star is formed?
Yes, I noticed that, and that's why I added the last sentence.