# 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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### Does General Relativity actually satisfy the General Principle of Relativity?

The “General Principle of Relativity” being “All systems of reference are equivalent with respect to the formulation of the fundamental laws of physics”. To my knowledge, this is related historically ...
874 views

### Why, when deriving the Einstein equations, do we want the energy-momentum tensor to be divergence free?

So when deriving the Einstein equation we assume $\nabla_\mu T^{\mu\nu}=0$. Now I get this is not true energy conservation but why do we assume this, it seems vital for the einstein tensor to have ...
99 views

### Tensor density and the coefficient $\sqrt{-g}$

Usually it is claimed that we use the coefficient $$\sqrt{-g}$$ for the action in the curved spacetime, to make the integrand treats as a scalar but not as a scalar density under general coordinate ...
260 views

### Measurement of the conjugate momentum in classical mechanics

In relativistic mechanics with Lagranian $L(\dot q^i,q^i)$ of a particle, the conjugate momentum of the position coordinate $q^i$ is defined as (wiki) $$p_i=\frac{\partial L}{\partial \dot q^i}.$$ ...
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### Magnitude of potential four-vector in Lorenz gauge [duplicate]

The Klein-Gordon equation is based on the relation $(E-e\phi)^2-(pc-eA)^2=m^2c^4$, which is the magnitude of the difference between the momentum four-vector and the four-potential. Since the magnitude ...
72 views

### The Laws of Physics and 4-Vectors/4-Tensors

I have been trying to understand why the physics, in general, is written in 4-vectors and 4-tensors. Like, how do they relate to the main postulates of special relativity? I am assuming that is it ...
89 views

### Special Theory of Relativity: 4-Vector and 4-velocity

We know that the four-dimensional scalar product is invariant under coordinate transformation, hence the space-time interval and proper time is also invariant. Since the 4-velocity is given by space-...
402 views

### Why does index contraction have to be done between upper and lower indices?

If I had to give a guess based on limited understanding, I would expect it to be something to do with the resulting object no longer obeying tensor transformation properties. However, if that is the ...
60 views

### Index manipulation of Dirac matrices

In several places I see that Dirac matrix indexes are treated as usual 4-vector indexes that can be changed with the metric tensor, for example $$\gamma_\mu=g_{\mu\nu} \gamma^\nu.$$ Why is it true?
146 views

### Coordinate Transformation of Vector & Tensor Fields

In the answer to the question: Coordinate Transformation of Scalar Fields in QFT by joshphysics a very nice mathematical explanation (using manifolds and charts) is given for the transformation of the ...
889 views

### Is the Four-gradient of a scalar field a four-vector?

Consider a scalar field $\phi$ as a function of spacetime coordinates $x^\mu$. The four-gradient of $\phi$ is given by \begin{equation} \frac{\partial \phi}{\partial x^\mu} = \left( \frac{\partial \...
148 views

### Lorentz non-invariance of $3$-acceleration

$3$-acceleration can not be constant in a relativistic system. Because $\vec a^2$ is not Lorentz invariant. Does it mean that Lorentz invariance works only for $4$-vectors? How this should be ...
2k views

### What is the reason to believe that the laws of physics are same in all frames of reference? [duplicate]

The first postulate of Special Relativity is that the laws of physics must be the same in all frames of reference i.e. invariant of coordinate transformations. I know this might be moot to ask but ...
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### Is the Minkowski metric coordinate independent?

Suppose I have some vector $\mathbf{P} = p^{\mu} e_{\mu}$. Now, for a flat spacetime, the contravariant components can be lowered via the Minkowski metric, $$p_{\mu} = \eta_{\mu \nu} p^{\nu}.$$ My ...
725 views

### Momentum operator in QM - scalar or vector?

The momentum operator for one spatial dimension is $-i \hbar d/dx$ (which isn't a vector operator) but for 3 spatial dimensions is $-i\hbar\nabla$ which is a vector operator. So is it a vector or a ...
107 views

### General relativity: Principle of minimal coupling computations

I have a question about computations in general relativity and transition from a Lorentz frame to a general fame just by substituting the flat metric with a general one and ordinary derivatives with ...
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### Why does the factor $\sqrt{-g}$ make the volume element invariant?

My question is an extension on this and this question. The question is, how or "in what sense" does the factor $\sqrt{-g}$ make the measure invariant? Suppose, I do not add this factor to the measure....
On pg.70 of Dalarsson's "Tensors, Relativity and Cosmology" For a mixed tensor of contravariant order 2 and covariant order 1 $(T^{mn}_{p,m})$, the divergence with respect to m is defined as:T^{...