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.

Filter by
Sorted by
Tagged with
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,\...
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
1k views

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, $\...