All Questions
Tagged with differentiation metric-tensor
125
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Why is $(\partial_\mu F_{\alpha\beta})F^{\alpha\beta}=F_{\alpha\beta}\partial_\mu(F^{\alpha\beta})$?
I'm trying to prove that the divergence of the energy-momentum-tensor is zero by expressing it in terms of the field strength tensor: $\partial_\mu T^{\mu\nu}=0$.
In doing this, letting the derivative ...
- 13
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1
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71
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What happens if we differentiate spacetime with respect to time? [closed]
Essentially, what would differentiating space-time with respect to time provide us with? What are the constraints associated with such operations? Is it possible to obtain a useful physical quantity ...
1
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1
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61
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How do you differentiate $F^{αβ}$ with respect to $g_{μν}$?
I want to experiment with this relation (from Dirac's "General Theory of Relativity"):
$$T^{μν} = -\left(2 \frac{∂L}{∂g_{μν}} + g^{μν} L \right)$$
using the electromagnetic Lagrangian $L = -(...
- 366
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49
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Partial derivatives of Christoffel symbols to Covariant derivatives
I wanted to express this thing: $g^{ab}\partial_c\Gamma^c_{ab} - g^{ab}\partial_a\Gamma^c_{cb}$, in terms of a covariant derivative. I figured out that if you swap $a$ and $c$ in the $\partial \Gamma$ ...
- 23
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41
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Question about the derivative of contravariant momentum 4-vector wrt proper time
I'm confused about an expression I saw without further explanation. It is the total derivative of the contravariant momentum 4-vector wrt proper time:
$$\frac{dp^{\mu}}{d\tau}=\frac{d}{d\tau}(g^{\mu\...
- 450
1
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1
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54
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Tensor equation
What is a valid tensor equation. In the book by Bernard Schutz, it is often argued that a valid tensor equation will be frame invariant. So the conclusions reached by relatively easy calculation done ...
- 137
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124
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Covariant derivative to the metric determinant?
I am reading the paper Alternatives to dark matter and dark energy, but cannot obtain one specific equation no matter how I tried. So I wrote an email to the author, the following is what he replies ...
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72
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Double covariant derivative of a mixed tensor
Let's say, we have a mixed tensor of type (2,1) denoted by $T^{mn}{}_p$ and the goal is to find the expression of $[\nabla_a, \nabla_b] T^{mn}{}_p$ in terms of fundamental tensors.
Firstly, I am ...
- 151
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74
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Covariant derivative of metric determinant with torsion
I have some troubles taking the covariant derivative of the metric determinant with torsion. Let's suppose that we take a metric such that $\nabla_\mu g_{\nu\rho}=0$. My reasoning is the following.
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- 57
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67
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Tensor Index Manipulation
I am trying to study General Relativity and I thought about starting with some index gymnastics. I found a worksheet online and I am stuck with a simple problem. I have to prove that
$$\partial_{\mu} ...
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3
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165
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What is difference between an infinitesimal displacement $dx$ and a basis one-form given by the gradient of a coordinate function?
In general relativity, we introduce the line element as $$ds^2=g_{\mu \nu}dx^{\mu}dx^{\nu}\tag{1}$$ which is used to get the length of a path and $dx$ is an infinitesimal displacement But for a ...
- 374
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318
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Covariant derivative of the Ricci tensor using pure algebra
I want to differentiate the Ricci tensor covariantly, namely without using Bianchi identities and with pure algebra, I want to prove:
$$
D _{\mu} R^{\mu\nu} = {{1}\over{2}} g^{\mu\nu} \partial_{\mu}R
$...
- 101
1
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1
answer
250
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What is the difference between $\partial_{\mu}$ and $\partial^{\mu}$? [closed]
I've seen in many books both expressions $\partial_{\mu}$ and $\partial^{\mu}$, which are the covariant and contravariant partial derivatives, respectively, and in one of Susskind's books he defined ...
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577
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Confusion on metric determinant derivative
Maybe it is a stupid confusion. I need to compute the derivative of the metric determinant with respect to the metric itself, i.e., $\partial g/\partial g_{\mu\nu}$, but I have an indices confusion in ...
- 378
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91
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Interpretation of covariant derivative of metric tensor being zero, specific problem on sphere
A question about the covariant derivative of the metric tensor being zero, example: sphere.
I understand, that the metric tensor of a (unit-)sphere is calculated via the outer product of the base ...
- 157