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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Geodesic equation

I have a technical question about the geodesic equation. Assume we have a frame $(E_{1},E_{2},E_{3},E_{4})$ (not necessarily a coordinate frame). Assume we have a parametrized curve $\gamma(s)\in M$ ...
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1answer
42 views

Understanding the covariant derivative and its relation to parallel transport

I have been reading section 3.1 of Wald's GR book in which he introduces the notion of a covariant derivative. As I understand, this is introduced as the (partial) derivative operators $\partial_{a}$ ...
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The Covariant Spinor Derivative in the Locally Supersymmetric Generalisation of the Polyakov Action and Potential Mistakes in the Literature

Questions) I recently came upon the thesis The Landscape of Free Fermionic Gauge Models by D. G. Moore and G.B. Cleaver. On pages 20 and 21 they explain that the locally supersymmetric action, ...
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2answers
44 views

Identifying a scalar function

We know that a scalar is invariant under rotations. What about a scalar function? Should it also be invariant under rotations? Therefore, under rotation $\phi(x,y,z)$ must be equal to ...
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2answers
122 views

Is a vector field not a vector quantity?

I'm trying to make sense of Poisson bracket relation $$\{L_i,A_k\}_{PB}~=~\epsilon_{ikl}A_l,\tag1$$ where $L_i$ is $i$th component of angular momentum, $A_k$ is $k$th component of an arbitrary ...
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2answers
115 views

Is there an accepted axiomatic approach to general relativity?

I am reading Steven Weinberg's book Gravitation and Cosmology. He makes a big deal out of the equivalence principle and showed a bunch of deductions you can make based on it. This surprised me since ...
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0answers
28 views

Is energy-momentum of curvature a boundary/holographic density?

Since the beginnings of General Relativity, we have had this awkward, unholy separation of the universe in marble versus wood. divergence of the stress-energy momentum holds at all points of ...
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1answer
41 views

Transformation of $q_k$ and $p_k$ from invariance of Hamiltonian

This is a step in Nakahara's Geometry, Topology and Physics, 2nd edition, 2003, on pages 7-8: Given that $q_k ' = q_k +\epsilon f_k(q)$, we have that $$\Lambda_{ij} = \frac{\partial q_i'}{\partial ...
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7answers
339 views

What does the statement “the laws of physics are invariant” mean?

In the first paragraph of Wikipedia's article on special relativity, it states one of the assumptions of special relativity is the laws of physics are invariant (i.e., identical) in all inertial ...
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2answers
143 views

Metric tensor in SRT

I just read on this webpage that we have (click me) $g_{\alpha \beta} = g_{\alpha}^{\beta} = g^{\alpha \beta}.$ Now, although I understand that the first and the last one are equal, I don't think ...
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0answers
54 views

How did Einstein know tensors would be needed in the EFEs? [migrated]

I'd really never studied tensors until I started studying the Einstein Field Equations. Since then, I have realized they are fairly common tool in physics and pretty basic to understanding many areas. ...
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1answer
77 views

Naturalness of tensor fields in general relativity?

In the third chapter of the book The Large Scale Structure of Space-Time, the authors say regarding the matter fields in general relativity: These fields will obey equations which can be expressed ...
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1answer
57 views

Why does choosing a time break covariance?

I'm reading that in EM theory, in hamiltonian formalism, we choose a specific reference frame with a specific time, and that this breaks covariance. Why? Surely it's simple because it's just stated ...
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1answer
45 views

What is the point of defining the lie algebra of the proper Lorentz group in a “covariant” way?

In Muller-Kirsten's book Introduction to Supersymmetry, the author first defines the proper Lorentz group's lie algebra basis in the standard manner - antisymmetric matrices consisting of $0$s and ...
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2answers
50 views

Transformation of spinors due to Lorentz group

Assume we have a Dirac spinor $\psi(x)$ which satisfies the Dirac equation: $$(i\gamma^{\mu}\partial_{\mu} - m)\psi(x) = 0.$$ If we boost our spacetime coordinates to a new system with a Lorentz ...
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0answers
35 views

Expressing the gauge field strength tensor in terms of covariant derivatives of the vector potential

Writing the covariant derivative as $$ \tag{1} D_\mu = \partial_\mu -ig A_\mu $$ it is easy to show that (in the non-abelian case) $$ \tag{2} [D_\mu,D_\nu] = -ig (\partial_\mu A_\nu - \partial_\nu ...
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1answer
71 views

Proof of the relation $d^4 \xi = \sqrt{|g|} \,\, d^4x$ switching between local and non-inertial coordinates

Denoting with $d\xi^m$ and $dx^\mu$ respectively flat and non-inertial coordinates, we have the following relation between the volume elements in the two coordinate systems: $$ d^4 \xi = \sqrt{|\det ...
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2answers
116 views

What does it mean to transform as a scalar or vector?

I'm working through an introductory electrodynamics text (Griffiths), and I encountered a pair of questions asking me to show that: the divergence transforms as a scalar under rotations the ...
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1answer
41 views

How can we see that the Riemann curvature tensor is covariant?

The Riemann curvature tensor, using the conventions of wikipedia, is written in terms of Christoffel symbols as: $$ \tag{1} R^\lambda_{\,\,\mu \nu \rho} = \partial_\nu \Gamma^\lambda_{\,\,\rho \mu} - ...
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0answers
48 views

Explicit demonstration of the relativistic invariance of the Weyl equation

It can be demonstrated explicitly that the Dirac equation is relativistically invariant. This is a proof (borrowed from Peskin & Schroder, see the unnumbered equation after the eqn. 3.31): ...
3
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1answer
115 views

Time derivative of time-translation Killing vector

I'm working with the spherically symmetric, static black hole metric. In the problem I'm working on, I'm told that $K$ is the time-translation Killing vector, $\frac{\partial}{\partial t}$ or $K = (1, ...
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2answers
127 views

Is four-current a vector or a vector density?

According to MTW, $$F^{\alpha\beta}{}_{;\beta} = 4\pi J^\alpha$$ and we can infer that the four-current must be an ordinary vector field because the left side is tensorial. But Wikipedia says that ...
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2answers
73 views

Meaning of general covariance

Quoting from Wald's GR: In the context of special relativity, the principle of general covariance states that the spacetime metric $\eta_{ab}$, is the only quantity pertaining to spacetime ...
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4answers
152 views

How does the Lorentz transformation $\Lambda^{\mu}{}_{\nu}$ transform?

For example the Four-velocity transforms as $$U^{a'}=\Lambda^{a'}{}_{\nu}U^{\nu},$$ the Faradaytensor as $$F^{a'b'}=\Lambda_{\,\,\mu}^{a'}\Lambda_{\,\,\nu}^{b'}F^{\mu\nu}$$ or in Matrixnotation: ...
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1answer
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Invariance of action $\Rightarrow$ covariance of field equations?

Invariance of action $\Rightarrow$ covariance of field equations? Is this statement true? I have only seen examples of this, like the invariance of Electromagnetic action under Lorentz ...
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2answers
186 views

The definition of transpose of Lorentz transformation (as a mixed tensor)

In the appendix of the textbook of Group Theory in Physics by Wu-Ki Tung, the transpose of a matrix is defined as the following, Eq.(I.3-1) $${{A^T}_i}^j~=~{A^j}_i.$$ This is extremely confusing for ...
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4answers
420 views

Is partial derivative a vector or dual vector?

The textbook(Introduction to the Classical Theory of Particles and Fields, by Boris Kosyakov) defines a hypersurface by $$F(x)~=~c,$$ where $F\in C^\infty[\mathbb M_4,\mathbb R]$. Differentiating ...
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4answers
421 views

Is there a fundamental reason not to define the work vice-versa

My question arises from something which has never been really clear: in continuum mechanics, why is strain energy defined as: $$W=\int_\Omega ...
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1answer
120 views

How can we derive the gauge field Lagrangian?

I learned the gauge field Lagrangian is given in this form: $$\mathcal{L} = -\frac{1}{4} \mathrm{Tr}(F_{\mu \nu}F^{\mu \nu}).$$ But how one can derive this equation starting from defining the ...
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2answers
135 views

What is $V^\mu$ if $\nabla_{\mu} V^{\mu}$=scalar?

Suppose there is a quantity written as $\sum\limits_\mu \nabla_\mu V^\mu$ which is invariant under a coordinate transformation, i.e. scalar, where $V^\mu=(V^0,V^1,V^2,V^3)$ and $\nabla_\mu$ is a ...
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1answer
133 views

Vector fields and tensors in E&M

I'm confused by a very basic property of electric fields. The electric field is a vector field. Vectors are tensors. Wikipedia has the following statement in the article about the electromagnetic ...
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1answer
75 views

Gauge SU(2) with real triplet

I have come across a model of gauge $SU(2)$ with a real triplet. The covariant derivative for $SU(2)$ complex doublet is written as $$D_\mu=\partial_\mu-igT^aA^a_\mu$$ where $T^a$ are generators of ...
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2answers
170 views

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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2answers
84 views

Independent components in a 4-vector representing massless fields

In Ryder Page141, it is written "the electromagnetic field, like any massless field, possesses only two independent components, but is covariantly described by a 4-vector $A_{\mu}$". Why are there ...
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5answers
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What problems with Electromagnetism led Einstein to the Special Theory of Relativity?

I have often heard it said that several problems in the theory of electromagnetism as described by Maxwell's equations led Einstein to his theory of Special Relativity. What exactly were these ...
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2answers
57 views

Magnetic force and relative frame

The magnetic field for to a moving charge depend on its velocity (Biot Savart's law). My question is that is it then not frame dependent? If it is, it means if a man is walking and other is standing ...
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3answers
427 views

How to understand the definition of vector and tensor?

Physics texts like to define vector as something that transform like a vector and tensor as something that transform like a tensor, which is different from the definition in math books. I am having ...
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3answers
192 views

Relation between component and algebraic definition of covariant vectors

I studied contravariance and covariance concepts in following way: For any vector if we get its components by parallelogram way we achieve contravariant components, and if we want to get its ...
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4answers
346 views

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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7answers
806 views

Why should a (physical) principle be applicable to different systems in different positions in space and time?

This is a question with a philosophical, as well as physical, flavor. Why should a physical principle (or a description of one), be applicable to different systems that can be in different positions ...
5
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1answer
226 views

Levi Civita covariance and contravariance

I read some older posts about this question, but I don't know if I'm getting it. I'm working with a Lagrangian involving some Levi Civita symbols, and when I calculate a term containing ...
5
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1answer
135 views

Invariance of a tensor under coordinate transformation

I know, that a tensor is a mathematically entity that is represented using a basis and tensor products, in the form of a matrix, and changing a representation doesn't change a tensor, is kind of ...
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2answers
502 views

Why isn't invariant notation common?

In principle, one can write quantities in a manifestly invariant - rather than covariant - fashion in e.g. special relativity. For example, rather than writing just $x^\mu$, we could write the basis ...
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3answers
193 views

Why do we need a metric to define gradient?

For me, the gradient of a scalar field (say, in three dimensions) is simply (formally) $\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y},\frac{\partial f}{\partial z} ...
12
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4answers
210 views

Difference between matrix representations of tensors and $\delta^{i}_{j}$ and $\delta_{ij}$?

My question basically is, is Kronecker delta $\delta_{ij}$ or $\delta^{i}_{j}$. Many tensor calculus books (including the one which I use) state it to be the latter, whereas I have also read many ...
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2answers
167 views

Why are totally antisymmetric tensors more useful than totally symmetric tensors?

In an arbitrary number of dimensions, one can naturally define two tensors, Kronecker delta and Levi-Civita epsilon tensor. However, why isn't it advantageous to define some totally symmetric tensor ...
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1answer
140 views

Are diffeomorphisms a proper subgroup of conformal transformations?

The title sums it pretty much. Are all diffeomorphism transformations also conformal transformations? If the answer is that they are not, what are called the set of diffeomorphisms that are not ...
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3answers
145 views

Integral in different coordinate systems

In Griffiths' electrodynamics book, he uses the equation, $$\nabla^2\mathbf{A}=-\mu_0 \mathbf{J},$$ to state that $$\mathbf{A}(\mathbf{r}) = ...
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0answers
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Background subtraction for a signal ans Errors 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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2answers
352 views

The wave equation in general relativity, special relativity, and Cartesian coordinates

The relativistic wave equation is $$\square\varphi=\rho$$ where $\varphi$ is the field, $\rho$ is the source, and $\square$ is the D'Alembert operator, defined by ...