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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Lorentz force in curved spacetime

I am trying to derive the equation for Lorentz force mentioned in the following Wikipedia article - https://en.wikipedia.org/wiki/Maxwell%27s_equations_in_curved_spacetime viz., $$ \frac{d p_{\alpha}...
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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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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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Is Lorentz-Invariant opposite to Lorentz-covariant? [duplicate]

I am having trouble understanding the meaning of these terms. Is it possible to be both Lorentz-Invariant and Lorentz-Covariant at the same time?
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132 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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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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How to prove $\nabla\vec{V}$ is a tensor without transformation properties?

In A First Course in General Relativity, Schutz asks the reader to prove that $\nabla \vec{V}$ is a $(1,1)$-tensor, where $$(\nabla\vec{V})^\alpha_{\ \ \ \ \beta} \equiv V^\alpha_{\ \ \ \ ;\beta} \...
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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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State amplitude and field operator covariance in QFT

I'm studying QFT on Bogoliubov-Shirkov's "Introduction to the theory of quantized fields" (3d edition). In $§9.3$ they discuss transformation properties of quantum states and operators in QFT. Given ...
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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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General Covariance, what does Einstein mean?

I have read the papers by Einstein and I am convinced I understand what he means completely. Given there are controversies, maybe I over understood it: It is I am convinced, can be said in two ...
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Inverting Operators, and Propagators on Curved Spacetime

I am a bit confused about inverting operators, and calculating propagators on a curved spacetime. Consider the following example: If I have a Lagrangian for a charged scalar field on a curved ...
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1answer
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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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Geodesic equation with “lowered” components of $p$

I'm reading Schutz's A first course in General Relativity, currently on Ch. 7 (Physics in a curved spacetime). Eq. 7.25 states that $p^\alpha p_{\beta;\alpha}=0$ for a free falling particle. What I ...
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2answers
211 views

What does it mean to go from a co-variant vector to a contravariant vector?

In most presentations of general-relativity I see the following statement, We can change from a covariant vector to a contravariant vector by using the metric as follows, ${ A }^{ \mu }={ g }^{ \...
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1answer
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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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Identifying Lorentz Covariant Equations

Statement: $\phi , A^{\mu}, T^{\mu \nu}$ are a Lorentz scalar, vector, and tensor. Which of the following equations are Lorentz covariant. a. $\phi = A_{0}$ b. $\phi = A^{\mu}A_{\mu}$ c. $\phi = ...
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288 views

Christoffel symbol derivation in book by Wald

In chapter 3 of Wald's General Relativity he starts by defining a covariant derivative $\nabla$ as a map on a manifold M from tensor fields $\mathscr{T}(k,l) \to \mathscr{T}(k,l+1)$ plus some required ...
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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 $...
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1answer
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Expanding a summation of covariant derivatives

I hope this is not a silly question but I am trying to understand how this part of the equation works: $$ \nabla_{\lambda} \left( \nabla_{\mu}(R_{\nu \lambda}) + \nabla_{\nu}(R_{\mu \lambda}) \right) ...
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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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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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126 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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383 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 ...
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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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Invariant quantities? [duplicate]

Every phisical quantity is tensor quantity (special cases of tensors are vectors and scalars). There are transformation rules for tensors. For example for scalar quantity F transformation rule is F'(x'...
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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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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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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^...
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Gradient is covariant or contravariant?

I read somewhere people write gradient in covariant form because of their proposes. I think gradient expanded in covariant basis $i$, $j$, $k$, so by invariance nature of vectors, component of ...
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A problem about proof of Noether's theorem in Nakahara's Geometry, Topology and Physics [duplicate]

Nakahara's Theorem 1.1 (only the first half of the proof is taken) Let $H(q_k,p_k)$ be a Hamiltonian which is invariant under an infinitesimal coordinate transformation $q_k \rightarrow q'_k=q_k+\...
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Momentum vector transformation

I am confused about the way momentum vector transforms in the following case: $$q_k \to q_k'= q_k + \epsilon f_k(q)$$ The Jacobian is thus $\Lambda_{ij} = \frac{\partial q'_i}{\partial q_j} \approx \...
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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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Background subtraction for a signal and error analysis

I use a CCD to see the split of a energy level due to Zeeman effect. I have a 1 dimensional CCD of 7926 pixel of 7μm each one. My CCD analyze a region 2 dimensional, and then it steps forward 200 ...
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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 \begin{align*...
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Christoffel symbols in general coordinates

In order to understand the meaning of covariant derivative, I have seen the following argument. Let us consider a covariant vector $V_\mu$. We would like to understand whether $$T_{\mu\nu} = \frac{\...
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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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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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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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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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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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