Questions tagged [covariance]
How a quantity behaves under a change of basis vectors. This tag covers relativistic covariance, as well as contravariant and covariant tensors not necessarily in the context of relativity. DO NOT USE THIS TAG for statistical covariance.
395
questions
3
votes
1answer
675 views
Is spacetime symmetry a gauge symmetry?
In previous questions of mine here and here it was established that Special Relativity, as a special case of General Relativity, can be considered as the theory of a (smooth) Lorentz manifold $(M,g)$ ...
1
vote
1answer
130 views
Analogy for covariant and contravariant tensors [closed]
I have been trying to grasp the difference between covariant and contravariant tensors in a somewhat qualitative way. This analogy popped into my mind and I wanted to check whether I'm on the right ...
1
vote
1answer
266 views
Tensors and derivatives
I am a maths student taking a module in (the mathematics of) Relativity so I get quite confused when looking for stuff that may help me understand where I go wrong in certain questions as I'm not ...
2
votes
2answers
123 views
Regarding the Dirac Hamiltonian's use of summation notation:
Einstein summation notation, as I understand it:
By writing $A_i B^i$ one implicitly means a sum over elements of the rank 1 tensors A and B. The key is the contraction of an "up" and a "down" index. ...
4
votes
2answers
319 views
Global symmetries of spacetime and general covariance
I am self learning GR. This is a rather long post but I needed to clarify few things about the effect of general coordinate transformations on the global symmetries of metric. Any comments, insights ...
1
vote
1answer
164 views
Raising index of variation
I know how to prove e.g. $$A^{ik}B_{lk}=A_{k}^iB^{k}_l.\tag{1}$$ (Raising and Lowering Indices Question). Today in a book, I find: $$g^{ik}\delta g_{lk}=-g_{kl}\delta g^{ki}.\tag{2}$$
$g^{ik}$ is ...
12
votes
2answers
901 views
What exactly does it mean for a scalar function to be Lorentz invariant?
If I have a function $\ f(x)$, what does it mean for it to be Lorentz invariant? I believe it is that $\ f( \Lambda^{-1}x ) = f(x)$, but I think I'm missing something here.
Furthermore, if $g(x,y)$ ...
5
votes
3answers
1k views
Confusions about Covariant and Contravariant vectors
I am trying to connect the concepts I learned from special relativity, to those of general relativity. Take a look at this example from wikipedia. They find a transformation matrix from the ...
2
votes
0answers
375 views
Notation for vectors and covectors
This is probably a very simple question, and I think I know the answer, but I cannot find a place to solidly confirm this.
So if I want to write a vector $\mathbf{V}$ in terms of its contravariant (...
0
votes
1answer
205 views
How are the *constant vectors* different from *vector fields* in terms of their respective transfomation properties?
How does one distinguish between the transformation properties of a scalar field $\phi(\textbf{r})$ or vector field $\textbf{A}(\textbf{r})$ (more generally, the tensor fields) from the transformation ...
2
votes
0answers
68 views
Where is a proof that string field theory is generally covariant?
Given a space-time coordinate of a string $X^\mu(\sigma)$ dependent on the position $\sigma$ around the string. And a string field functional $\Phi[X]$, is there a proof that the equations of motion (...
3
votes
2answers
1k views
Tensor vs. Tensor Densities
Currently I'm reading through Sean Carroll's Spacetime and Geometry: an Introduction to General Relativity. According to Carroll, the symbol
$$dx^0 \wedge dx^1 \wedge \cdots \wedge dx^{n-1},$$
...
1
vote
2answers
530 views
Covariant and contravariant permutation tensor
I have been reading up on the permutation tensor, and have come across the following expression (in 'Generalized Calculus with Applications to Matter and Forces' by L.M.B.C Campos page 709):
$$e_{i_1,\...
asked Oct 4 '16 at 8:27
Quantum spaghettification
5,4137 gold badges32 silver badges102 bronze badges
5
votes
1answer
168 views
Function form of Lorentz invariant functions
In QFT, Green function of gauge field is Lorentz invariant(i.e. $\forall \Lambda \in SO(3,1), f(\Lambda p)=\Lambda f(p)$).And according to the textbook I'm reading, The form of such functions is ...
6
votes
3answers
396 views
What is the purpose of emphasizing that an action is invariant under diffeomorphism?
When learning field theory and string theory, I always see physicists stress the fact that the action, which is an integral of the Lagrangian density $S(x)=\int L(x,\dot{x})dt$, is invariant under ...
0
votes
1answer
265 views
Why do we always need quantities to be Lorentz Invariant (LI) in relativity?
In particle physics, for example, we add gauge fields, look for the covariant derivative and so on. All to find the LI form of the Lagrangian. Why do we need the LI form? My impression is that when ...
2
votes
2answers
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Covariant Derivative of Basis Vector along another basis vector?
So in my relativity course, we recently learned about the covariant derivative. it is defined as:
\begin{equation}
\nabla_{\mu}V^{\nu} = \partial_{\mu}V^{\nu}+\Gamma^{\nu}_{\mu,\lambda}V^{\lambda}
\...
1
vote
2answers
3k views
Proof of the invariance of the Levi-Civita tensor
My question is related with the proof of the following: the Levi Civita tensor, $\epsilon _{\mu \nu \rho \sigma}$ is an invariant tensor, that is, if we make a change between one reference frame with ...
5
votes
1answer
673 views
Explicitly show covariance of Euler Lagrange equations
I know that the Euler Lagrange equation (here only in 1D)
$$
\left(\frac{d}{dt}\frac{\partial}{\partial\dot{x}}-\frac{\partial}{\partial x}\right)L\left(x,\dot{x},t\right)=0
$$
is invariant under (...
7
votes
4answers
1k views
QFT: How would you explain to a mathematician what “transforms as” means?
I am taking an introductory course to quantum field theory. The lecturer goes on saying that some transforms as (represented by $\to$) . I tried to ask the lecturer, and he said that he means some ...
3
votes
1answer
838 views
What is the definition of invariance under Lorentz transformation?
I want to learn how to check if any scalar is a Lorentz scalar, so what is the definition of being invariant under Lorentz transformation? Is it correct to say that $\phi$ is invarant under Lorentz ...
4
votes
0answers
392 views
Is classical electromagnetism conformally invariant? (and a bit of general covariance)
The contest is a flat $4d$ Minkowsky space.
A conformal transformation is a diffeomorphism $\tilde x(x)$ such that the metric transforms as
\begin{equation*}
\tilde g_{\tilde \mu \tilde \nu} = w^2(x) ...
0
votes
0answers
55 views
How to show that the vector potential is a vector?
I've never quite well understood the definition of vector used by physicists based on transformation laws.
From a mathematical point of view a vector in space is:
A point derivation on the germs of ...
4
votes
1answer
121 views
Parametric and covariant expressions for the acceleration vector
I am reading S. Neil Rasband book about Classical Dynamics. In the first chapter, there are two different forms of the acceleration:
What he calls the "intrinsic". Given a trajectory with parameter
$...
12
votes
1answer
502 views
Do the interaction picture fields transform as free fields under boosts?
This post was originally written to ask about transformation properties of fields in the interaction picture of QFT under the Poincare transformations. Arnold Neumaier has pointed out that the ...
1
vote
2answers
373 views
The meaning of covariant but not manifestly covariant
What is the most general meaning of the expression covariant, but not manifestly covariant? Suppose I have a general (local) change of coordinates, $x^{\prime} = f(x)$, on an $(n+1)$-dimensional ...
7
votes
3answers
3k views
How to show the spacetime interval is invariant in general?
I understand how to derive the spacetime interval being invariant for Minkowski space, but I've never seen any derivation of it in general curved spacetime. Is the invariance just derived for ...
2
votes
1answer
732 views
Gauge invariance
In the Schrodinger equation, the statement of electromagnetic gauge invariance is that observables don't depend on the electromagnetic gauge. That is, if we let:
$$\partial_t \psi(\mathbf x,t) = \...
2
votes
0answers
111 views
Covariant quantization of an interacting relativistic particle
A method of covariant quantization for a free relativistic particle appears in the first part of some introductory string theory texts (Tong, Zwiebach,...). None of them (as far as I hae seen) give an ...
1
vote
3answers
528 views
Doubt in Lorentz Transformation [closed]
I've tried to do the following exercise:
Show that $\sum_{\mu} D^{\mu\mu}$ and $\sum_{\mu}D_{\mu\mu}$ are not invariant under Lorentz transformations but $\sum_{\mu} D^{\mu}_{\mu}$ are.
I've had ...
1
vote
0answers
119 views
What are the fields in a classical field theory?
Consider scalar Yukawa theory. The Lagrangian density $\mathcal{L}$ contains an interaction term $$\mathcal{L}_I=g\psi^*\psi\phi$$ where $\psi$ and $\phi$ are complex and real scalars respectively.
...
0
votes
3answers
360 views
Providing an intuitive description of scalar and vector quantities in physics [closed]
Often the standard introduction to the concept of scalars and vectors in physics is something along the lines of:
A scalar is a quantity that is completely described by a single number (it has no ...
8
votes
1answer
7k views
What is the Difference between Lorentz Invariant and Lorentz Covariant? [duplicate]
Like my title, I sometimes see that my books says something is Lorentz invariant or Lorentz covariant. What's the difference between these two transformation properties? Or are they just the same ...
1
vote
2answers
410 views
Scalar fields and general coordinate transformations
In classical mechanics, a scalar field is characterised by the fact that its value at a particular point must be invariant under rotations and reflections of coordinates. That is, one requires that $\...
2
votes
1answer
2k views
Trace of a Tensor
What is the significance of defining the trace of a tensor as $g^{\alpha\beta} R_{\alpha\beta}$ instead of $R_{\alpha\alpha}$ on a Riemannian manifold?
1
vote
1answer
313 views
Covariant Taylor Series
I am reading the following lecture notes of Avramidi
https://www.researchgate.net/publication/255565392_Analytic_and_geometric_methods_for_heat_kernel_applications_in_finance
I want to understand ...
1
vote
3answers
430 views
Why is speed defined as coordinate derivative over proper time rather than observer's time in STR?
In special theory of relativity, why is 4-velocity defined as:
$$
u^\mu = \frac{dx^\mu}{d\tau}
$$
and not as
$$
u^\mu = \frac{dx^\mu}{dt}
$$
where ${\tau}$ is proper-time and t is time in some ...
1
vote
1answer
489 views
Lorentz Transformations in Minkowski space
If $\Lambda$ represents the Lorentz transformation matrix, then the transformation of contravariant components $x^\mu$ is given by $$x'^\mu=\Lambda^{\mu}{}_{\nu} x^\nu$$ and that of the covariant ...
0
votes
1answer
299 views
Relation between differentiation of one-form basis and Christoffel Symbols
If I want to covariantly differentiate a one form then I can write:
$\nabla_\beta \tilde p = \dfrac{\partial p_\alpha}{\partial x^\beta} \tilde \omega^\alpha + p_\alpha \dfrac{\partial \tilde \omega^...
7
votes
3answers
938 views
What is “a general covariant formulation of newtonian mechanics”?
I am a little confused: I read that there are general covariant formulations of Newtonian mechanics (e.g. here). I always thought:
1) A theory is covariant with respect to a group of transformations ...
5
votes
2answers
2k views
Einstein Summation Convention: One as Upper, One as Lower?
My question refers to the often specified rule defining Einstein Summation Notation in that summation is implied when an index is repeated twice in a single term, once as upper index and once as lower ...
0
votes
2answers
195 views
Four Vectors in SR and QFT
I'm covering both special relativity and quantum field theory in the summer. I'm currently using Spacetime Physics by Taylor and Wheeler to cover SR. Since I'm covering SR on the side with QFT, I'm ...
12
votes
2answers
1k views
Is Young's Modulus a Lorentz Scalar?
If a spring is at rest and lies along $X$ axis in a frame $O$ with a spring constant $k_{0}$ then its spring constant in a frame $O'$ which is moving with a speed $v$ at an angle $\theta$ with the $X$ ...
1
vote
1answer
186 views
Second covariant derivative, computation problem
I am having a question on the wikipedia article http://en.wikipedia.org/wiki/Second_covariant_derivative
Using the notation therein I don't get why
$(\nabla_{u}\nabla_{v}w )^a=u^c\nabla_{c}v^b\nabla_{...
0
votes
1answer
107 views
Commuting of the covariant derivative: Menzel's Mathematical Physics
Menzel defines covariant differentiation as equivalent to partial differentiation with respect to the general coordinates. “To indicate the covariant nature of the differential operator, set
$$\frac{\...
0
votes
2answers
114 views
Deriving $A^{\mu}_{;\nu}$ from $A_{\mu ; \nu}$
We have a covariant derivative of a covariant tensor:
$$
A_{\mu ; \nu} = A_{\mu , \nu} - \Gamma^{\alpha}_{\mu \nu} A_{\alpha}
$$
The covariant derivative of a contravariant tensor is:
$$
A^{\mu}_{;\nu}...
1
vote
3answers
2k views
How is 4-current a 4-vector?
I am looking at Jackson sec 11.9, where he states that the $\rho,\bf{J}$ form the 4-current
$$J^\alpha=(c\rho,\bf{J})$$
Jackson says this is from the invariant of the 4-divergence $\partial^\alpha ...
4
votes
2answers
956 views
Using $\sqrt{-g}$ in integrals of proper volume
I am a little confused over integration using proper volume element. When do we use $\sqrt{-g}$ in calculations?
For example, in many calculations involving stars, say when using TOV equation, this ...
1
vote
1answer
574 views
Behaviour of Dirac Bilinears
Dirac bilinears transform in the Lorentz indices as,
$\bar{\psi}\psi$ scalar
$\bar{\psi}\gamma^\mu\psi$ vector
$\bar{\psi}\sigma^{\mu\nu}\psi$ 2nd rank (antisymmetric) tensor
$\bar{\psi}\gamma^{\mu}\...
4
votes
2answers
1k views
Is Newton second law covariant or invariant?
Is Newton second law covariant or invariant between two inertial frames, moving with uniform traslational motion with respect to each other?
If it is invariant then, indipendently from the frame, $\...
