Questions tagged [invariants]

This tag is for questions relating to invariant, a property of a system which remains unchanged under some transformation. In physics, invariance is related to conservation laws.

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59 views

Curvature and Symmetries of spacetime

Is there any relation between symmetries of spacetime and the curvature invariants? For example is spherical symmetric spacetimes, necessarily have positive curvature? Could we define any spherical ...
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1answer
203 views

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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164 views

Chern-Simons action as a topological invariant

It is stated that the Chern-Simons action is a topological invariant that is proportional to the Chern-Simons form. But the latter is just a conformal invariant. How do we reconcile these views? Both ...
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2answers
42 views

What are the analogues of momentum, inertia and angular momentum for energy?

If Energy and mass are the same thing, then is it logical to look for analogous (duals) of properties of one in another? or there is there any conceptual framework that such questions make any sense? ...
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207 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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Classifying all symmetries of a mechanical system [duplicate]

Given a newtonian mechincal system with $n$ objects, we may think of it as living in $\mathbb{R}^{6n+1}$ ; one dimension is time, $3n$ dimensions for velocities, and $3n$ for positions. We then have ...
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Geometrical interpretation of curvature invariants

Consider a Riemannian manifold. It is possible to describe it by curvature invariants. Now, is there any geometrical description (intuition) for simple invariants such as scalar curvature, Ricci ...
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2answers
127 views

Why is it necessary that different observers agree on the value of the spacetime interval $ds^2$?

What's the physical reason that all (inertial) observers agree on the value of the spacetime interval $$ds^2 = (c dt)^2 - dx^2 - dy^2 -dz^2 \, ?$$ What would be the physical implications if different ...
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How could 'rest mass' and 'invariant mass' be the same?

The terms rest mass and invariant mass are often interchanged, however i cannot reconcile these concepts: Consider a photon ...
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1answer
53 views

time invariance for “Translations” versus “Galilean transformations”

Why would the time coordinate (t) be NOT invariant under translations, but invariant under Galilean transformations? I thought it should be invariant under both Here is what I'm tying to understand:
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Building the Lagrangian of electromagnetism from the Lorentz invariant?

The definition of the relativistic Action is $$ S=\int_a^b ds $$ The Lorentz invariant of electromagnetism is $$ s^2=\frac{1}{c^2}||\mathbf{E}||^2-||\mathbf{B}||^2-2i\frac{1}{c}(\mathbf{B}\cdot \...
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Why are certain quantities so fundamental to physics? [closed]

Apart from having a qualitative description of quantities such as momentum, work and energy, why are these quantities considered so fundamental? What is the reason to define them in the first place? ...
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297 views

Why helicity for massless particles is Lorentz invariant?

By definition helicity is projection of spin onto the 3 momentum. $$h={\bf J} \cdot {\mathbf{P }} $$ where ${\mathbf{P }}=(P_1,P_2,P_3)$ is the momentum operator and ${\mathbf{J }}=(J_1,J_2,J_3)$ ...
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115 views

General definition of symmetry in physics? [duplicate]

I've looked at a number of questions on what symmetries are in physics, such as this one, this one and this one. However, I found the questions and answers to be not completely satisfying because they ...
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1answer
256 views

Is Hamiltonian a scalar or tensor in Quantum Mechanics?

According to Wikipeida, a scalar operator is invariant under rotations, and the Hamiltonian satisfies this definition. But at the same time, a Hamiltonian can be written as a matrix, which means it is ...
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Antisymmetric matrices in effective field theory

I'm trying to construct a nonlinear $d$-dimensional version E&M as an effective field theory. Let $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ be the field strength. The most general action ...
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1answer
271 views

Gravitons and self-interaction

In the book quantum field theory and standard model by Schwartz, there is a problem 9.4 that says by considering lorentz invariance of Compton scattering, you can prove that for spin 1 massless field ...
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Eigenvalues of quadratic Casimirs of simple Lie groups

I want to find a generic formula for calculating eigenvalue of quadratic casimirs of Lie groups, in terms of Dynkin labels. For a simple example if we take $SU(2)$, with $[R]$ indicating the highest ...
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2answers
98 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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56 views

How to construct invariant forms under the effect of an arbitrary group?

First I would like to mention that I do not know that should I post this question here or in the math community, but since my background is in physics and this kind of question is usually asked by ...
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1answer
112 views

An invariant for transformations of Lorentz

Exist a physical demonstration why $$E^2- p^2c^2 =m^2c^4=E'^2- p'^2c^2 $$ is an invariant for transformations of Lorentz? N.B.: $m$ is mass; $E$ is the energy and $p$ is momentum in the frame $\...
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Invariant mass in special relativity [closed]

I'm following a special relativity course and I'm trying to understand how the invariant mass works. In particular I don't get how the following passages work. We have a collision between two ...
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1answer
125 views

Issue showing that the phase of a harmonic wave is invariant under a Galilean transform

The phase $Φ$ of wave is defined as $kx-wt$. It should be the case that all observers moving relative to each other in the non relativistic case will agree on this. So given the transforms $x'=x-vt$ ...
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1answer
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Invariance of length [closed]

Invariance of interval in Minkowski space under coordinate transformation was proved by the postulates of special relativity. (https://physics.stackexchange.com/a/453536/213658 .see this answer) Is ...
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1answer
546 views

Newton's theory of gravity is covariant under Galilean transformations

We know from classical mechanics that the gravitational field equation for the scalar potential takes the form $$\nabla^2\phi=4\pi \rho,$$ where $\rho$ is mass density (which, can depend on time and ...
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398 views

How length is an invariant in Euclidean space?

The special theory of relativity shows that intervals are invariant under Lorentz transform in the Minkowski space -time. But how can we prove (any postulates or theory) that the length is an ...
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A Scalar Function Tranformation — Question on Notation in 't Hooft Document

I started reading a document by Gerard 't Hooft which can be found here. Right at the start I am puzzled by a simple expression. It is equation 2.2 showing how a scalar function transforms. I ...
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315 views

Why do we differentiate a 4 vector with respect to proper time to obtain 4-velocity?

The coordinates of an event in spacetime are given by the 4-vector $(ct, \mathbf{r})$, where $\mathbf{r}$ is the spacial coordinates of the event. This 4-vector can be seen as 4-displacement of a ...
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399 views

Invariance of the relativistic interval

Imagine we have two events, $E_1, E_2$ in the coordinate systems $K, K'$ (with coordinates $(x,y,z,t),\ (x',y',z',t')$), whilst $K'$ ist moving with the speed $\vec v$ in regard to $K$. From the ...
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1answer
64 views

Converting an invariant matrix to a non-invariant tensor

I'm working on the following problem: In 4-dimensional notations, given a transformation matrix Calculate the matrices $\Lambda_{\mu\nu}$, $\Lambda_\mu^\nu$ and $\Lambda^{\mu\nu}$ The matrix $\...
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591 views

How can zero point energy vacuum be Lorentz invariant?

What distributions of electromagnetic oscillating fields are Lorentz invariant? How can oscillating electromagnetic fields look the same regardless of inertial frame? According to Marshall and Boyer ...
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1answer
124 views

Special relativity: I arrive at a contradiction regarding the Lorentz invariance of certain quantity

I want to show the Lorentz invariance of $d^3 p/E$ (Eq. 8.11 of Mandl-Shaw), where $E$ is the relativistic energy. Peskin-Schroeder gives sort-of, a proof in section 2.3 which I am convinced of. But ...
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How does one prove the channel independent inequality satisfied by the product of the three Mandelstam variables?

How does one prove the following equation (67.5) from the BLP Quantum Electrodynamics book? The q's are the 4 momenta, and h is the sum of all four masses. Two q's written after one another in the ...
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The Lorentz-invariant particle spectrum

Before the question, I need to mention some necessary definitions. The rapidity is defined as: $$y=\frac{1}{2}\ln\frac{E+p_z}{E-p_z}=\frac{1}{2}\ln\frac{1+v_z}{1-v_z}=\tanh^{-1}(v_z)$$ where $v_z=...
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Why is the scalar product of two four-vectors Lorentz-invariant?

Why is the scalar product of two four-vectors Lorentz-invariant? For instance, given two four-vector $A^\mu$ and $B^\mu$, so their scalar product is $A\cdot B=A^\mu B_\mu=A^\mu g_{\mu\nu}B^{\nu}$. ...
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174 views

Character expansion and Casimir

Is there a simple way to extract the quadratic Casimir of a representation from the character? I keep hearing things such as "Chern characters have an expansion that goes like" $$\chi(r) = dim(r) ...
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Can GR be reformulated in terms of invariant observables?

Question So recently I was thinking about this: How many scalars are available in $4$ dimensions in General Relativity (without being redundant)? For example, with metric we can construct the ...
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436 views

Physical meaning of the Casimir operators of Poincarè algebra

If one considers the algebra $su(2)$, it is well known that the Casimir Operator is $$ C=L_1^2+L_2^2+L_3^2. $$ It corresponds to the total angular momentum and correctly is a conserved quantity. ...
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1answer
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Value of the invariant $R_{\mu \nu}F^{\mu \nu}$

Is there a simple way to find the value of $R_{\mu \nu}F^{\mu \nu}$ (where $R_{\mu \nu}$ is the Ricci tensor and $F^{\mu \nu}$ is the electromagnetic tensor), knowing that it is an invariant? ...
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1answer
298 views

Can anyone provide a simple, inuitive explanation for Noether's Theorem? [duplicate]

I recently came across this theorem for the first time. As I understand it, what she showed was that conservation 'laws' are often simply an artifact of symmetry or invariance. For example, the ...
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1answer
112 views

How is Einstein's postulate about the invariance of the laws of physics justified? [duplicate]

According to one of Einstein's postulates related to special relativity, > "the laws of physics remain invariant in their form and nature in all inertial frames". But global inertial frames don't ...
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1answer
111 views

Restrictions on the form of a scalar-valued function imposed by Lorentz invariance

Let $f(p,q)$ be a smooth Lorentz-invariant function of 4-vectors $p$ and $q$. Should $f$ necessarily be of the form $f(p,q) = g(p^2, q^2, p_\mu q^\mu)$, where $g(x,y,z)$ is some scalar-valued ...
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Scalar versus invariant in Newtonian mechanics

I looking up coriolis transport theorem for rotating refrence frames and while reading through this derivation he wrote: In Newtonian mechanics, scalar quantities must be invariant for any given ...
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3answers
487 views

Gauge invariant scalar which is not Lorentz-invariant

I'm looking for examples of the following descriptions: A gauge invariant scalar which is not Lorentz-invariant A Lorentz covariant scalar For 1. I was thinking about the scalar potential $A$ (for ...
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1answer
502 views

What is the physical meaning of the third invariant of the strain deviatoric?

In continuum mechanics of materials with zero volumetric change, the material condition can be expressed by the strain deviatoric tensor instead of the strain tensor itself. To express the plasticity ...
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Are there Galilean scalars?

In special relativity there are scalar quantities which are invariant under any Lorentz transformation, called Lorentz scalars. For example, the magnitude of the four-velocity is a Lorentz scalar. If ...
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Formal Term for an invariant constant to all observers

I was thinking of the speed of light and realized I don't know how to quickly name the concept of "physical quantity that is measured to be the same in all reference frames". Are there examples of ...
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Using the energy-momentum invariant for a decay process

For a decay process in which particle A ----> particle B + a photon in which particle A has mass $m_A$, particle of mass $m_B$ and energy and momentum are conserved. Show that in the frame in ...
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238 views

Quantities invariant by Lorentz transform

If I rotate a triangle lying on the two-dimensional plane both the lengths of the sides and the angles formed are invariant (ie, they are the same before and after the rotation). In a 2D Lorentz ...
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1answer
144 views

Why should an action in SR be a lorentz scalar? [duplicate]

I have taken it granted that an action in the special relativity must be a lorentz scalar. However is there a fundamental reason for this requirement? I cannot think of a plausible reason for this ...