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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102 views

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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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-...
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3answers
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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 ...
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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?
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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 ...
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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 \...
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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 ...
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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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Invariance of action integral under point transformations

I am reading Goldstein classical mechanics chapter 2 p. 35. Here the author states that the action integral $$\int L(q,\dot q,t)dt$$ is invariant under change in generalized coordinates $$q_i=q_i(s_1,...
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1answer
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Commutator of covariant derivatives to get the curvature/field strength

For notation and convention, please see Gauge theory formalism and Generalizing the covariant derivate for gauge theory. The covariant derivative can be used to construct curvatures (called field ...
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Covariant and contravariant components of a vector in curvilinear coordinate system

I'm reading a Quora answer on an intuitive explanation of covariant/contravariant components of vectors. If we have a coordinate system with straight coordinate axes, the geometric explanation given ...
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Generalizing the covariant derivate for gauge theory

Concrete example gauging the complex scalar field $\mathcal{L}=(\partial_\mu \phi)(\partial^\mu \phi^*)+m^2 \phi^*\phi$ $\phi(x) \rightarrow e^{-i\Lambda(x)}\phi(x)$ $A_\mu \rightarrow A_\mu + \...
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What are covectors in special relativity?

In special relativity the purpose of vectors makes fairly intuitive sense, they represent a point in spacetime: $$x^{\mu}=\begin{pmatrix}x^0 \\ x^1 \\ x^2 \\ x^3\end{pmatrix}$$ and we can define the ...
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Why do we write this tensor notation of space-time gradient contravariant tensor?

Why is $\partial^\mu=\frac{\partial}{\partial x_{\mu}}$ the contravariant component of space-time gradient four vector instad of $\partial^{\mu}=\frac{\partial}{\partial x^{\mu}}$?
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1answer
132 views

Prove $D_\mu\phi^\dagger=(D_\mu\phi)^\dagger$

In gauge transformation, $D_\mu$ was defined to be $\partial_\mu-igA_\mu$. However, I have hard time to see that $D_\mu\phi^\dagger=(D_\mu\phi)^\dagger$ without ambiguity. (A comparable example in QED ...
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What is the difference between invariance and covariance? [duplicate]

In relativistic physics, paricularly in General Relativity and Quantum Field Theory, we often find the use of the two terms 'invariance' and 'covariance'. But I couldn't find any mention of the ...
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153 views

In electromagnetism, how do we know that either $F^{\mu\nu}$ or $A^\mu$ is a tensor?

In special relativity the partial derivative $\partial_\mu$ is a tensor. Now if some function $A^\mu$ was a tensor, then also the quantitiy $F^{\mu\nu}=\partial^\mu A^\nu - \partial^\nu A^\mu$ would ...
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Why metric tensor can be not covariantly constant?

I learn GR now, and there is a strange thing that I discovered. It is well-known, that the condition $\nabla_{\mu}g_{\alpha \beta}=0$ is specified, when we choose specific metric-compatible Levi-...
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351 views

What is the difference between a dual vector and a reciprocal vector?

I am familiar with the concept of a dual space $V^*$ as the set of all linear functionals $\tilde{\omega}: V \rightarrow \mathbb{R}$. The inner product on $V$ is usually used to define the dual of a ...
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436 views

Is every Lorentz invariant a Lorentz scalar?

All examples of lorentz invariant quantities that I have come across seem to be scalars: rest mass, proper time, spacetime interval,dot product of two 4 vectors etc. Another thing is that these are ...
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Does completely antisymmetric tensor act on a tensor always produce a tensor or not?

So completely antisymmetric tensor $\epsilon$ act on a tensor can produce a new object. i.e. $G_{\alpha\beta}=\frac{1}{2}\epsilon_{\alpha\beta\mu\nu}F^{\mu \nu}$. However, According to Landau's ...
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Covariance of the perfect fluid's stress tensor

In Special Relativity, for a perfect fluid (i.e. without heat transference or viscosity) we have a stress tensor $T_{\mu \nu}$ $$ T_{\mu \nu} = -p\eta_{\mu \nu} + (\rho + p)u_\mu u_\nu $$ It is said ...
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What if the Lagrangian $\mathscr{L}$, a Lorentz scalar, is replaced by a Lorentz vector?

As an answer to this post, I made an impression that if $\mathscr{L}$ were not a Lorentz scalar in Eq.$(1)$ (see below), then Eq.$(1)$ would not be covariant. But now I think that is wrong! I state ...
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Tensor analysis: confusion about notation, and contra/co-variance

I'm learning about tensors in the context of special relativity, and I'm a bit confused some notation. I understand a four-vector is a four dimensional vector, which is written in the form $(ct, x, y,...
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63 views

Derivative of energy in General Relativity

I have found this equality that is the derivative of the energy for a local observer in GR, with energy defined as $E=-p^\mu u_\mu$: $$dE/d\tau = -p^\mu p^\nu\nabla_{(\mu}u_{\nu)}$$ trying to derive ...
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What's the covariant derivative of a normalized, timelike Killing vector?

I'm reading The large scale structure of spacetime and in page 72 the author says: A static metric admits a timelike killing vector $K$. We define the timelike unit vector $V$ as $V=K/f$, where $f^...
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184 views

Covariant form of Green's function for wave equation

In J. D. Jackson's "Classical Electrodynamics", page 614 in the 3rd edition, he states that you can write the Green's functions for the wave equation in covariant form using the fact that \...
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1answer
105 views

Which Frame of Reference is Correct?

I am reading a lot about the theory of special relativity, but I have a very basic question about this theory I still don't understand. Consider a particle in two inertial reference frames $\Sigma$ ...
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1answer
182 views

Transformation to tetrad frame

I have some vector components as measured in the comoving tetrad frame $V^{(\mu)}$. This vector exists at coordinates $x^{(\mu)}$, which is different from the origin of the tetrad coordinate system. ...
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81 views

Change of basis in a Euclidean space

I am trying to compute the change in the contravariant components of a vector when the basis is changed from Cartesian (standard basis) to spherical polars. I understand that a general vector $\...
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54 views

Do gauge fields not transform like functions of the coordinates under translations?

By "transform like a function of the coordinates," I mean that under an infinitesimal translation $x^\mu \to x^\mu + \epsilon^\mu$, to first order in $\epsilon^\mu$ the function $f(t,\mathbf x)$ ...
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2answers
63 views

Frame-dependence of the gravitational field pseudotensor

What seems to be the common consensus in physics is that a gravitational field does not have a stress energy tensor due to the equivalence principle, but rather a pseudotensor. Is this pseudotensor ...
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179 views

How does the stress energy tensor change in different reference frames?

Is the Stress-Energy tensor invariant in all RFs? If not (which is highly probable) how does it change? EDIT: does the Einstein equation help? Since (without $\Lambda$) $$ R_{\alpha\beta} -\frac{1}{2}...
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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 ...
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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 ...
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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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118 views

Are Lagrange's equations physical laws?

Well, I studied that a physical law is an equation between tensors that are function of position and time because when the frame is changed tensors change in order to leave the equation true: $$T_1(\...
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318 views

Covariant derivative with respect to commutator

I have some confusion with the notion of $\nabla_{[A, B]}\bf{v}$, that expression, with a commutator of vector fields as the subindex of the connection appears for instance in the definition of the ...
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443 views

Is there any meaning of tensor contraction?

Is there any meaning behind tensor contraction. Or is it just randomly getting rid of some components by only selecting those with same index and sum them up? For example, I know tensor is ...
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2answers
606 views

In which sense equations of motion are covariant?

I read lots questions about what covariance is and I found out that, according to this topic Lorentz invariance of the Minkowski metric, we say an object is covariant if it doesn't take the same value ...
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118 views

Choice of metric breaks diffeomorphism invariance?

In Weinberg's paper on the cosmological constant problem (CCP), he states that diffeomorphism invariance is always broken by the presence of any given metric $g_{\mu\nu}$. He then goes on to say that ...
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Raising & lowering indices of 3-pseudovectors?

Now, let space tmie metric is $$\eta_{\mu\nu}=\text{diag}(+,-,-,-)$$ now $$x_{\mu}=(x^0,-\mathbf{x})$$ and $$x^{\mu}=(x^0,\mathbf{x})$$ and $$x^{\mu}=\eta^{\mu\nu}x_{\nu}$$ also $$\partial_\mu=(\...
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1answer
832 views

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....
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1answer
349 views

Divergence of a tensor

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^{...
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142 views

Why do we need invariants to represent real life quantities?

Often it is said that one of the most useful properties of eigenvalues of a matrix is that they are invariant under change of basis. This in turn is said to be useful in physics because real, physical ...

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