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.

learn more… | top users | synonyms

0
votes
0answers
18 views

Commuting of the covariant derivative: Menzel's Mathematical Physics

Menzel defines covariant differentiation as equivalent to partial differentiation with respect to the general coordinates. “To indicate the covariant nature of the differential operator, set ...
0
votes
2answers
71 views

Deriving $A^{\mu}_{;\nu}$ from $A_{\mu ; \nu}$

We have a covariant derivative of a covariant tensor: $$ A_{\mu ; \nu} = A_{\mu , \nu} - \Gamma^{\alpha}_{\mu \nu} A_{\alpha} $$ The covariant derivative of a contravariant tensor is: $$ ...
2
votes
2answers
222 views

How is 4-current a 4-vector?

I am looking at Jackson sec 11.9, where he states that the $\rho,\bf{J}$ form the 4-current $$J^\alpha=(c\rho,\bf{J})$$ Jackson says this is from the invariant of the 4-divergence $\partial^\alpha ...
2
votes
2answers
90 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 ...
1
vote
1answer
50 views

Behaviour of Dirac Bilinears

Dirac bilinears transform in the Lorentz indices as, $\bar{\psi}\psi$ scalar $\bar{\psi}\gamma^\mu\psi$ vector $\bar{\psi}\sigma^{\mu\nu}\psi$ 2nd rank (antisymmetric) tensor ...
3
votes
2answers
519 views

Is Newton second law covariant or invariant?

Is Newton second law covariant or invariant between two inertial frames, moving with uniform traslational motion with respect to each other? If it is invariant then, indipendently from the frame, ...
1
vote
1answer
55 views

Killing equation manipulation

Why does the killing equation $$K_{\mu;\nu}+ K_{\nu;\mu} = 0$$ equal $$K_{\mu,\nu}+ K_{\nu,\mu} -2\Gamma^{\rho}_{\mu\nu}K_{\rho} = 0 $$ when in general a covariant derivative $V_{\beta;\alpha} = ...
1
vote
1answer
52 views

Why is the spatial term for contravariant 4-gradient negative, whereas for other 4-vectors it is the covariant part that is negative spatially?

The contravariant 4-displacement is: $${x}^{\alpha} = (ct,\mathbf{r})$$ And the contravariant 4-gradient is: $${\partial}^{\alpha} = (\frac{1}{c}\frac{\partial}{\partial{t}},-\nabla)$$ From what I ...
3
votes
1answer
60 views

Is time a vector in Minkowski space? [duplicate]

I am arguing about this topic with my school teacher in so long time, I want to finish this debate. My teacher's opinion is "Yes, Time is vector" because four-vector has $t$ component, and mine is ...
0
votes
3answers
88 views

Tensor manipulation and showing equality

Im taking GR for the first time and it is definitely throwing me for a loop. The question I am working on is this: Prove that if a contravariant tensor $A^{uv}$ is symmetric, then it remains ...
4
votes
0answers
51 views

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 = ...
3
votes
1answer
62 views

contravariant and covariant vectors and their orthogonality in Euclidean space

I am reading this paper Sigma Coordinate - Contravariance and covariance and I understand how covariant and contravariant vectors are defined mathematically Covariance and Contravariance and I had ...
0
votes
2answers
93 views

What does coordinate invariance mean?

I would like to really understand what the mathematical as well as Physical meaning of coordinate invariance is. I have pretended to know what this means, but upon thinking a little harder today, I am ...
2
votes
1answer
54 views

Is there any way to justify or derive the form of the Lorentz force from relativity theory?

Lorentz force is in this form: $$\vec{F}=q[\vec{E}+\vec{u}\times\vec{B}]$$ As we know, it is Lorentz-invariant. Is there any way to justify or derive its form from relativity theory?
6
votes
0answers
112 views

Canonical second quantization vs canonical quantization with multisymplectic form in AQFT

First of all, I'm a mathematician that knows less than the basics of QFT, so forgive me if this question is trivial. Please, keep in my mind that my background in physics is very poor. 1) The usual ...
1
vote
2answers
62 views

How does one show Maxwell's equations in vector calculus form describe the same motion in all reference frames?

The covariant form of Maxwell's equations is Lorentz invariant. $$\partial_{\alpha}F^{\alpha\beta} = \mu_{0} J^{\beta}$$ $$\partial_{\alpha}F_{\beta\gamma} + \partial_{\beta}F_{\gamma \alpha} + ...
1
vote
3answers
179 views

Covariant and contravariant 4-vector in special relativity

I've just learned about contra- and covariant vector in the context of special relativity (in electrodynamic) and I'm struggling with some concept. From what I found, an intuitive definition of ...
1
vote
2answers
134 views

Understanding Tensor-operations, covariance, contravariance, … in the context of Special Relativity

I'm currently learning about special relativity but I'm having a really hard time grasping the Tensor-operations. Let's take the Minkowski scalar product of 2 four-vectors: $$\pmb U . \pmb V = ...
2
votes
0answers
37 views

Is there a general procedure for covariantizing equations?

I am currently attempting to derive covariant forms of equations whose domains are D=3 space. I am considering Lorentzian $(\mathbb{R}^4, \Omega, x, \nabla)$, where $\Omega \subset\mathbb{R}^4$ has a ...
0
votes
0answers
52 views

solutions of wave equation with cubic term

Does the following equation $$ \nabla^\mu \nabla_\mu \psi + a \psi^3 = b \psi $$ where $\psi$ is a real function, $a$ and $b$ are real constants, have other solutions that extend beyond a one ...
1
vote
1answer
227 views

Gradient, divergence and curl with covariant derivatives

I am trying to do exercise 3.2 of Sean Carroll's Spacetime and geometry. I have to calculate the formulas for the gradient, the divergence and the curl of a vector field using covariant derivatives. ...
1
vote
2answers
74 views

Invariance and conservation

Why in a collision between particles is the four-momentum conserved within a frame of reference but not invariant between frames of reference?
0
votes
1answer
111 views

Proving the invariance of the inner product

If we define the inner product as ${\textbf{u}\cdot\textbf{v}=g_{ij}u^{i}v^{j}}$, where ${g_{ij}}$ is the metric tensor, ${S}$ and ${T}$ are transformation matrices, ${S}$-for covariant indices and ...
2
votes
1answer
183 views

Difference between symmetry and invariance

I'm wondering what's the real difference between symmetry and invariance in Physics? I believe that sometimes the two words are given the same meaning and some other times they are used in a different ...
2
votes
2answers
81 views

What does it mean to differentiate a spinor-valued field?

Peskin and Schroeder, equation 3.28, states that the Klein-Gordon equation $$(\partial^2+m^2)\psi=0 \tag{3.28}$$ is a valid choice of equation for a Dirac spinor field. Their explanation makes sense ...
1
vote
2answers
600 views

What does the first postulate of specially relativity really say?

I know these two versions of the same postulate is saying the same thing. But I failed to connect them. Please help me understand the links between them. version1 The laws of physics are the same ...
0
votes
1answer
44 views

Lorentz Symmetry

Quick question about Lorentz symmetry. From the wiki page the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through ...
3
votes
1answer
328 views

When can two quantities be added together?

Whenever two things are to be added together, one typically needs to check whether this actually makes sense, and an addition is said to make sense, in principle, when the units match up. Yet, ...
1
vote
2answers
110 views

What does it mean by saying the generators of translations transform as vectors under the Lorentz Group?

The commutator of generators of Lorentz transformation and translation is as follow: $$[M^{\mu\nu},P^\sigma]=i(P^\mu\eta^{\nu\sigma}-P^\nu\eta^{\mu\sigma} ).$$ Then from this we usually say that the ...
4
votes
1answer
192 views

Why are densities not fields?

I have read (in Statistical mechanics of lattice system 2: exact, series and renormalization group methods by D.A. Lavis and G.M. Bell pg 2 ), that intrinsic variables are either fields or densities. ...
1
vote
2answers
90 views

Clarification on meaning of scalar in math and scalar in physics

When a mathematician says something is a scalar, say on the plane, they mean that it associates to points on the plane real numbers. When a physicist says something is a scalar, they mean that if we ...
0
votes
2answers
48 views

Parameterisation of the equation of motion for a relativistic massive point particle

The equation of motion for a relativistic massive point particle is given by: $$\frac{dp_{\mu}}{d \tau} = 0,$$ where $p_{\mu}$ is the four-momentum defined by $p_{\mu} = m \frac{dx_{\mu}}{ds/c}$, ...
36
votes
5answers
2k views

Why do we need coordinate-free descriptions?

I was reading a book on differential geometry in which it said that a problem early physicists such as Einstein faced was coordinates and they realized that physics does not obey man's coordinate ...
4
votes
1answer
182 views

Invariant equations of motion under Lorentz transformations

My question regards the statement that an equation of motion may be invariant under a Lorentz transformation I just finished watching the Stanford University special relativity lectures on special ...
0
votes
1answer
119 views

Problem understanding Lorentz invariance [duplicate]

So they usually started with "...This is obviously Lorentz invariant, because of the 4-vector character of the quantity,..., (and after a two page long derivation) another quantity is also obviously ...
1
vote
0answers
41 views

What is the correct terminology for a “symplectic covariant” equation?

A Lorentz covariant equation is one that takes the same form even when a Lorentz transformation is applied to each variable. Lorentz covariance is generally made manifest by writing the equation with ...
2
votes
2answers
193 views

Making sense out of covariance and contravariance

I just read about co- and contravariant vectors and I am not sure that I got it right: If we imagine that we have a n-dimensional manifold $M$ then a tangent space is spanned by the vectors ...
0
votes
1answer
160 views

Lorentz invariance vs. covariance

I am a bit confused whether relativistic theory is Lorentz invariant or covariant. And please explain why?
3
votes
2answers
132 views

Under what representation do the Christoffel symbols transform?

I often read the statement, that the Christoffel symbols aren't tensors. But then, under which representation do they transform?
1
vote
1answer
38 views

Matrix dimensions in the spacetime interval equation don't seem to agree? (Possible notation misunderstanding)

The spacetime interval in flat space can be expressed as $(\Delta s)^2 = g_{\alpha \beta} \Delta x^\alpha \Delta x^\beta$. I understand covariant, $x_a$, and contravariant, $x^a$, vectors to be row ...
3
votes
1answer
178 views

In continuum mechanics, why is the stress vector $T=\sigma\cdot n$ not a covector?

In continuum mechanics, the stress vector (see Cauchy stress tensor) $T=\sigma\cdot n$ is the surface density of a force. Forces are covectors, since they map a displacement vector to a scalar energy. ...
1
vote
1answer
145 views

Covariant formulation of physical equations?

Is it possible to rewrite equations like the Klein-Gordon, the Dirac or the Proca equation in a generally covariant way? And if yes, how and how can the general covariance be shown? (I searched ...
1
vote
1answer
98 views

Covariance of the Dirac Equation

i want to show that the following equation holds: $$ \frac{1}{8}\left[\gamma^{\mu},\omega_{\mu \nu} [\gamma^{\mu},\gamma^{\nu} ] \right] = \omega^{\mu}_{~~~\nu}\,~ \gamma^{\nu} $$ $\gamma^{\mu}$ ...
1
vote
2answers
127 views

Index gymnastics and representing bra-kets as covariant and contravariant tensors

I am trying to figure out how to write, in Einstein notation as well as pick out elements of $$\langle A|[\mu]|B\rangle \langle X|[\nu]|Y\rangle$$ where $[\mu] = \begin{bmatrix} \mu_{11} & ...
0
votes
2answers
218 views

Off-diagonal terms in metric for 4D space-time [closed]

Consider a delta between two events in 4D space-time written as a 4-vector, $x^\mu=(dt, dR)$. The time $dt$ is a scalar difference in time. The 3-vector $dR$ points some direction in space. One ...
5
votes
3answers
262 views

How can a Physical law not be invariant?

In Relativity, both the old Galilean theory or Einstein's Special Relativity, one of the most important things is the discussion of whether or not physical laws are invariant. Einstein's theory then ...
2
votes
1answer
149 views

Transpose of (1,1) tensor

When we transpose a (1,1) tensor, shall we simply switch the two indices while keeping their upper/lower positions or switch them and also switch their upper/lower positions? In general, would the ...
11
votes
7answers
335 views

How can a set of components fail to make up a vector?

Many books in Physics insist to define vectors are objects with components with the property that the components transform in a proper way under a change of coordinates. Now, in mathematics, on the ...
1
vote
2answers
227 views

Relation between Vector space $V$ and its dual $V^{*}$ [closed]

I asked the same question in Math.SE, but I was suggested to ask it here as well. I am studying relativity, and as you know the theory extensively uses the notion of covariant and contravariant ...
0
votes
1answer
101 views

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 $\gamma(s)\in M$ ...