# 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.

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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 $(M,g)$ ...
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 ...
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### 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 ...
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### 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. ...
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 ...
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 ...
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### 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 $g(x,y)$ ...
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### 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 ...
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 (...
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### 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 ...
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### 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 (...
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### 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 $$dx^0 \wedge dx^1 \wedge \cdots \wedge dx^{n-1},$$ ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $X$ ...
186 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 ...