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.

Filter by
Sorted by
Tagged with
1
vote
1answer
230 views

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) ...
2
votes
1answer
186 views

Is Newton's third law Lorentz-covariant?

Let $(x,y,z,t)$ be a Lorentz frame equipped with the Minkowski metric. Assume 2 particles interact, without external forces applied to them. The total 4-momentum $p_1+p_2$ is therefore conserved. If ...
0
votes
2answers
35 views

Autocorrelation and variance: can the fluctuation-dissipation theorem actually be written in terms of fluctuations?

I am considering the theorem in a statistical mechanics context, but I suppose the question could be extended to other fields where it applies as well. If we have a system with property $A$ and apply ...
0
votes
0answers
47 views

Why do the author define energy using one-form? [duplicate]

I was reading the book First course on GR from Schurz. In the latter chapters the author is going to calculate how does the motion of a photon is affected by a spherical symmetric metric. See define ...
12
votes
4answers
5k 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 ...
3
votes
2answers
88 views

How do we know that solving Euler-Lagrange equation gives us the correct equation of motion in any coordinate?

As far as I understand it: If we defined a quantity called the Lagrangian as the difference between the kinetic energy and potential energy for a particle in one dimension in Cartesian coordinates, ...
0
votes
1answer
43 views

Co-ordinate invariance in Lagrangian form of equations

I have read that in his Mecanique Analytique [1788], Lagrange sought a “coordinate invariant expression for mass times acceleration”. The discussion regarding this is given in 'Marsden and Ratiu [15, ...
1
vote
1answer
62 views

Has the Klein-Gordon equation in curved spacetimes the same form as in flat ones? [duplicate]

The KG equation in curved geometries has the following form: $$\frac{1}{\sqrt{-g}}\partial_\mu(\sqrt{-g}~g^{\mu \nu}\partial_\nu\phi) + m^2\phi = 0,$$ where $g$ is the determinant of the metric tensor ...
16
votes
3answers
17k views

D'Alembertian for a scalar field

I have read that the D'Alembertian for a scalar field is $$ \Box = g^{\nu\mu}\nabla_\nu\nabla_\mu = \frac{1}{\sqrt{-g}}\partial_\mu (\sqrt{-g}\partial^\mu). $$ Exactly when is this correct? Only for $...
1
vote
2answers
86 views

Change of coordinate vs change of reference axes

Does basis vectors change opposite to coordinate scaling ? For example, suppose I have some oblique coordinate system, and I decide to scale up both 'axes' by a factor of $a$ and $b$ respectively. The ...
0
votes
0answers
15 views

Is there a difference between base vectors and projection method of calculating coordinates? [duplicate]

I'm trying to grasp the idea of co- and contra-variance. During my study I meet something like that: Let's say we have a vector $\vec{v}$ and want to calculate its coordinates in base $\hat{b}$. I've ...
1
vote
2answers
222 views

Geometrical representation of Contravariant and covariant vectors

After cruising through a lot of material online, and answers over here, my understanding of contravariant and covariant vectors are, in a finite-dimensional vector space, suppose we have a vector, ...
1
vote
2answers
335 views

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}...
3
votes
2answers
249 views

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^...
13
votes
6answers
2k views

Vectors as functions?

In my study of general relativity, I came across tensors. First, I realized that vectors (and covectors and tensors) are objects whose components transform in a certain way (such that the underlying ...
0
votes
2answers
44 views

Covariant or contravariant nature of Gradient

I've been having this confusion regarding the gradient being a covariant vector. Intuitively I seem to have understood the concept. However, mathematically, I'm unable to show this, in a single ...
2
votes
3answers
111 views

Vector representation in dual space

I'm new to tensor analysis, and came across the topic of vectors and duals, and faced a massive confusion. Are vectors and duals different representations of the same object ? I had another doubt ...
1
vote
0answers
93 views

Does general covariance really imply that there is no conserved stress energy tensor in gravity?

There are many questions on this website which ask similar questions to this, but none I have found have asked this exact question. First, a bit of history on the origin of Noether's theorem. David ...
0
votes
0answers
22 views

Covariance of Partial Derivative with Specific Coordinate Transformation

In a context of differentiable manifolds partial derivative are not "covariant" in the sense that if applied to a tensor the result is not a tensor anymore: $\partial^{'}_{\mu}=\frac{\...
2
votes
1answer
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 ...
0
votes
2answers
134 views

How are these two different definitions of covariant vector related?

Definition 1 If under a coordinate transformation $x^i\to \bar{x}^i(x^i)$ certain objects $A^i$ transform as $$A^i\to \bar{A}^{i}=\sum_{j}\frac{\partial \bar{x}^{i}}{\partial x^j}A^j,$$ those objects ...
0
votes
0answers
38 views

Diffeomorphism invariance in tetrad formalism

How do we show general coordinate invariance of Einstein-Hilbert action in the tetrad formalism or rather the Einstein-Cartan formalism where the frame field and the spin-connection are independent?
4
votes
2answers
379 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 (...
0
votes
0answers
33 views

Transform two-mode squeezed state (TMSS) to its covariance matrix

The two-mode squeezed state can be written as: ${\left| \chi \right\rangle _{AB}} = \sqrt {1 - {\chi ^2}} \sum\nolimits_n^\infty {{\chi ^n}} {\left| n \right\rangle _A}{\left| n \right\rangle _B}$ ...
5
votes
3answers
3k 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 ...
21
votes
3answers
17k views

Galilean covariance of the Schrodinger equation

Is the Schrodinger equation covariant under Galilean transformations? I am only asking this question so that I can write an answer myself with the content found here: http://en.wikipedia.org/wiki/User:...
0
votes
2answers
60 views

Are inner product equations invariant everywhere in spacetime?

For example, in Minkowski space, the energy of a massive particle is given by $$E=-P_{\mu}U^{\mu},$$ where the sign depends on the metric convention, $P$ is the particle 4-momentum and $U$ is the 4-...
0
votes
0answers
40 views

Proca Field Hamiltonian density

Having the most general Lagrangian of the Proca Field given by $$\mathcal{L}=C_1(\partial_\nu A_\mu)(\partial^\nu A^\mu)+C_2(\partial_\nu A_\mu)(\partial^\mu A^\nu)+C_3 A_\mu A^\mu$$ the canonical ...
0
votes
1answer
28 views

Converting covariant objects into non-covariant

I need to rewrite expressions of the type $(\partial_\nu A_\mu)(\partial^\mu A^\nu)$ from the "covariant" form, to non-covariant form (so with roman indices). Here the greek indices run from ...
0
votes
0answers
19 views

How to calculate the variance on the ratio of 2 angular power spectra?

In the context of Survey of Dark energy stage IV, I need to evaluate the error on a new observable called "O" which is equal to : $$ O=\left(\frac{C_{\ell, \mathrm{gal}, \mathrm{sp}}^{\prime}...
16
votes
10answers
5k 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 ...
0
votes
1answer
99 views

What is meaning of the phrase “The laws of physics are the same in all inertial frames of reference.”? [duplicate]

One of the postulates special relativity is that "The laws of physics are the same in all inertial frames of reference." What is meaning of this statement? If it is talking about physical ...
0
votes
0answers
42 views

How to compute the covariance error term in cosmology context?

Below the error on photometric galaxy clustering under the form of covariance : $$ \Delta C_{i j}^{A B}(\ell)=\sqrt{\frac{2}{(2 \ell+1) f_{\mathrm{sky}} \Delta \ell}}\left[C_{i j}^{A B}(\ell)+N_{i j}^{...
1
vote
1answer
234 views

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 ...
0
votes
0answers
63 views

Transformations in General Theory of Relativity

In our lecture we dicussed that certain elements in GTR transform as certain other things, i.e. Christoffel symbol transforms as an affine connection covariant derivative of a contravariant vector ...
1
vote
1answer
82 views

Why do basis vectors transform covariantly?

I have some trouble understanding transformation rules of basis vectors. My question/goal is to obtain a mathematical derivation to see why basis vectors transform covariantly and other vectors (...
0
votes
1answer
45 views

Curve is geodesic iff $\nabla_k g(V,V)$ vanishes

Let $V$ be a Killing vector field and let $s \longmapsto x^i(s)$ be a curve such that $$\dot{x} \enspace \equiv \enspace \frac{dx^i}{dx}(s) \enspace = \enspace V^i\big(x(s)\big)$$ Show that $x^i(s)$ ...
3
votes
3answers
442 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 ...
0
votes
1answer
66 views

Solving for $\Gamma^c_{ab}$ Christoffel symbols from the metric $g_{ab}$

I am trying to compute the Christoffel symbols $\Gamma^c_{ab}$ to have a metric-compatible covariant derivative $\nabla_a g_{bc}=0$. I worked out $$\nabla_a g_{bd}=\partial_a g_{bd}-\Gamma^{c}_{ab}Y_{...
1
vote
1answer
72 views

Contravariant tensor definition must be incorrect?

Both my textbooks (Schaums Tensor calculus and Neuenschwander's Tensor calc for physics) define a contravariant tensor of order one (which they also state is synonymous with a simple vector) as ...
1
vote
1answer
234 views

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 ...
13
votes
2answers
3k views

Covariant vs contravariant vectors

I understand that, in curvilinear coordinates, one can define a covariant basis and a contravariant basis. It seems to me that any vector can be decomposed in either of those basis, thus one can have ...
20
votes
2answers
1k views

Lagrange's equation is form invariant under ANY coordinate transformation. Hamilton's equations are not under ANY phase space transformation. Why?

When we make an arbitrary invertible, differentiable coordinate transformation $$s_i=s_i(q_1,q_2,...q_n,t),\forall i,$$ the Lagrange's equation in terms of old coordinates $$\frac{d}{dt}\left(\frac{\...
0
votes
1answer
77 views

Must the action be a coordinate scalar?

I know that an action must be locally-Lorentz invariant based on physical reasons, but is there any requirement for it to be a coordinate pseudo-scalar (up to surface terms)? In particular, would an ...
0
votes
0answers
37 views

Torsion tensor and affine connection symbols

I have read tons of questions about this topic but I think my particular issue is not solved. If so, please let me know. So I want to prove that the torsion tensor $\mathcal{T}$ actually transforms ...
0
votes
2answers
81 views

Definition of an equation to be Lorentz invariant

What is the precise mathematical definition of an equation to be Lorentz invariant? Is it the same as being invariant under the maps $x \mapsto \Lambda x$, with $\Lambda$ being a given Lorentz ...
8
votes
3answers
195 views

How does Equivalence Principle imply a Curved Space-Time?

I am a bit confused as to how the Equivalence Principle implies a curved spacetime. Or if it doesn’t imply a curved spacetime, then what exactly makes it necessary to have a curved space time? I could ...
0
votes
2answers
720 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 ...
0
votes
0answers
25 views

Transformation rules for quantities

When we formulate transformation laws for vectors and tensors, the transformation rule for $x^\mu$ is calculated via arguments from total derivatives considering $x^\mu=x^\mu(x^{'\nu})$ that in turn ...
0
votes
0answers
24 views

Invariance of Euler-Lagrange equation [duplicate]

How exactly does the variational approach make it clear that the Euler-Lagrange equations of motion will have exactly the same form no matter what coordinate system we use as long as it is done with ...

1
2 3 4 5
10