The covariance tag has no wiki summary.
6
votes
2answers
146 views
What's the basic premise of General Relativity?
What is the basic assumption(s) required to explore general relativity?
For example, if one merely assumes that the speed of light $c$ is the same for all observers, and the laws of physics are the ...
0
votes
1answer
66 views
Vectors on Different Coordinate Systems? [closed]
Consider the product of vectors coordinated relative to a given coordinate frame, defined by
$$\vec{a}\square\vec{b}=((a_{1},b_{1})\square(a_{2},b_{2})):=(a_{1}b_{1},a_{2}b_{2})$$
Explain why ...
2
votes
5answers
98 views
Why define four-vectors to be quantities that transform only like the position vector transforms?
A four-vector is defined to be a four component quantity $A^\nu$ which transforms under a Lorentz transformation as $A^{\mu'} = L_\nu^{\mu'} A^\nu$, where $L_\nu^{\mu'}$ is the Lorentz transformation ...
0
votes
1answer
51 views
Covariant derivative-Differential
I was trying to prove that the derivative-four vector are covariant. This can be proved only if you consider the time and space derivatives to be
$\dfrac{\partial}{\partial ...
1
vote
1answer
65 views
Arbitrary tensor covariant derivative
what are the rules for performing covariant derivatives on tensors of arbitrary rank?
I found a few examples of Tensor derivatives:
$$\nabla_{c} T^a {}_{b} = \partial_{c}T^a {}_{b}+ \Gamma^a{}_{cd} ...
2
votes
1answer
69 views
Invariance, covariance and symmetry
Though often heard, often read, often felt being overused, I wonder what are the precise definitions of invariance and covariance. Could you please give me an example from quantum field theory? ...
0
votes
0answers
60 views
How to solve following equation (Yukawa field)?
By using Lagrangian for Yukawa interaction,
$$
L = -\frac{1}{c}A_{\alpha}j^{\alpha} + \frac{1}{8 \pi c}(\partial_{\alpha}A_{\beta})(\partial^{\alpha}A^{\beta}) + ...
1
vote
2answers
131 views
Lorenz gauge fixing
Is it always possible to define function $\psi$ satisfying the Lorenz gauge equation
$$
\partial_{\mu}\partial^{\mu} \psi + \partial_{\mu}A^{\mu} = 0?
$$
0
votes
0answers
17 views
quantifying interaction between variables in an equation [closed]
What do I need to measure interaction between variables in a particular equation?
For e.g.
Me just taking 50 grams of protein everyday will help me health wise.
Me just doing exercise for 1 hour ...
2
votes
0answers
66 views
Is a solution to the Klein-Gordon equation homeomorphic (or even diffeomorphic) to a solution of an equation with a different covariance group?
Consider some solution $\psi(x,t)$ to the linear Klein-Gordon equation: $-\partial^2_t \psi + \nabla^2 \psi = m^2 \psi$. Up to homeomorphism, can $\psi$ serve as a solution to some other equation ...
0
votes
3answers
222 views
Relativistic basic question - four vector, Lorentz matrix
I have heard relativistics only very compressed during my student time. Now I looked up the definitions again and a question comes into my mind:
A contravariant vector is transformed like this: ...
1
vote
1answer
113 views
Covariant derivative with upper index
I just need clarification, that is, to see that I'm doing the right thing.
When calculating central charge for certain metric, I need to solve an integral that contains Lie brackets etc. And I have ...
0
votes
1answer
134 views
How to find the intrinsic covariant derivative component?
How to find the intrinsic covariant derivative component?
In general relativity the elements of the acceleration four-vector are related to the elements of the four-velocity through a covariant ...
2
votes
1answer
386 views
Levi Civita Symbol and contravariance vs covariance
I have a question regarding the Levi-Civita symbol and contravariance vs covariance. Some of this was asked in a previous post, but I think I need more clarification.
Consider the magnetic field:
...
3
votes
2answers
205 views
Why is the “canonical momentum” for the Dirac equation not defined in terms of the “gauge covariant derivative”?
The canonical momentum is always used to add an EM field to the Schrödinger/Pauli/Dirac equations. Why does one not use the gauge covariant derivative? As far as I can see, the difference is a factor ...
3
votes
3answers
196 views
First Postulate of Special Relativity: What does it mean?
Wikipedia has this quote:
Special principle of relativity: If a system of coordinates K is
chosen so that, in relation to it, physical laws hold good in their
simplest form, the same laws hold ...
4
votes
1answer
565 views
Covariant derivative and Leibniz rule
I read the Wikipedia page about the covariant derivative, my main problem is in this part:
http://en.wikipedia.org/wiki/Covariant_derivative#Coordinate_description
Some of the formulas seem to lead ...
1
vote
2answers
238 views
fourth rank tensor for stress energy
The Weyl tensor equates the Riemann tensor in vacuum
$$ C_{\mu \nu \eta \lambda} = R_{\mu \nu \eta \lambda} $$
So it makes me wonder about the tensor
$$T_{\mu \nu \eta \lambda} = C_{\mu \nu \eta ...
2
votes
0answers
69 views
quadripolar moment in curved space
So, i'm going over the Thorne's derivation of the quadrupolar radiation term, and they write the core term as:
$$ \frac{3 r_i r_j - 2 r^2 \delta_{ij}}{4 r^5} $$
But if i try to obtain this term by ...
4
votes
1answer
227 views
failing to see the conundrum in the Einstein hole argument
I've been reading about the Einstein hole argument, and i fail to understand what makes active diffeomorphisms "special" compared to passive diffeomorphismsm also known as good old coordinate ...
0
votes
1answer
621 views
Autocorrelation Functions <---> Pair Correlation Functions
Are there any ways to convert an autocorrelation function to a pair correlation function, and vice versa?
6
votes
4answers
494 views
Are there controversies surrounding the principle of general covariance in GR?
I'm a physics graduate now working with computers. I study GR in my spare time to keep the material fresh. In the Wikipedia article about the mathematics of GR, one can read the following:
The ...
2
votes
0answers
249 views
composition of space expansion and movement as a gauge invariance
suppose i have a space-time where we have one point-like object* which we will call movement space probe or $\mathbf{M}_{A}$ for short, and it will be moving with constant velocity $V^A_{\mu}$ in ...
7
votes
3answers
786 views
Definitions and usage of Covariant, Form-invariant, Invariant?
Just wondering about the definitions and usage of these three terms.
To my understanding so far, "covariant" and "form-invariant" are used when referring to physical laws, and these words are ...
6
votes
4answers
970 views
covariant derivative for spinor fields
scalars (spin-0) derivatives is expressed as:
$$\nabla_{i} \phi = \frac{\partial \phi}{ \partial x_{i}}$$
vector (spin-1) derivatives are expressed as:
$$\nabla_{i} V^{k} = \frac{\partial V^{k}}{ ...
