Difference between Invariance and Covariance

[haaj86]

Hi, what is the difference between Lorentz Invariance and Lorentz Covariance?
From my lecture note (Group theory course) Invariance and Covariance where defined as follows:
Invariance: refers to the property of objects being left unchanged by symmetry operations.
Covariance: refers to equations whose form is preserved by a change of coordinate system.

From this I had the understanding that Lorentz Invariance refers to physical quantities that do not change under a Lorentz transformation and Lorentz covariance refers to equations that do not change after a coordinates transformations (of course the equation will be going from primed to unprimed after a Lorentz transformation but the structure of equations wouldn’t )
However, I have also read in physics books where the author talks about the invariance of “equations” just one example here “Lorentz Invariance of the Dirac equation”.
Is this because the above definitions are not as solid in physics as in mathematics, or is my understanding of what they meant is wrong, or is it because the authors are being sloppy?

 

[HallsofIvy]

Yes, an "invariant" quantity doesn't change when you change coordinate systems. But I would say that a "covariant" quantity is one that change homogeneously: the transformed coefficients are sums of products of the old coordinates and numbers that depend on the coordinate transformation. One important consequence of that is that if v is a covariant quantity and v= 0 in one coordinate system them v= 0 in every coordinate system. From that it follows that if you have an equation involving covariant quantities, say u= v with u and v both covariant, then if u= v in one coordinate system, then u-v= 0 in that coordinate system so u- v= 0 or u= v in any coordinate system. That is the idea of a "covariant equation".

But notice that "covariant equation" is not the same as "invariant equation"! An equation is "covariant" if it being true in one system implies it is true in every coordinate system. An equation is "invariant" if it has exactly the same form in every coordinate system. For example, Laplace's equation is "invariant" in Euclidean coordinate systems- that is translations and rotations only.

 

[haaj86]

Thank you for your reply,

But notice that "covariant equation" is not the same as "invariant equation"! An equation is "covariant" if it being true in one system implies it is true in every coordinate system. An equation is "invariant" if it has exactly the same form in every coordinate system. For example, Laplace's equation is "invariant" in Euclidean coordinate systems- that is translations and rotations only.

Ok, what do we say about Maxwell's equations, are they Lorentz invariant or Lorentz covariant? And can you please give me examples of Lorentz covariant equations in physics.

 

[Bob_for_short]

Ok, what do we say about Maxwell's equations, are they Lorentz invariant or Lorentz covariant? And can you please give me examples of Lorentz covariant equations in physics.

The Maxwell's equations are Lorentz covariant: the values in them transform. Its form is "invariant": the same in different reference frames. These equations include the EM fields (as a four-tensor) and the charge four-current. Their Lorentz covariance was first established by the French Academician H. Poincaré (1905), as well as the Lorentz covariance of the mechanical equations. In fact, H. Poincaré had advanced the principle of relativity and had built the relativistic theory before A. Einstein did. He found the main invariants of the realtivity theory - the interval and the field invariants (E*H and E^2-H^2). He showed that this all could be understood a four-geometrical features.

Bob.

 

[haaj86]

Thank you Bob, I really now understand what invariance means. But I have one remaining point regarding to covariance

The Maxwell's equations are Lorentz covariant: the values in them transform

Are you referring here to the fact that the E and B field themselves transforms between a pair of inertial frames? Or are you referring to the Lorentz transformations of the space and time variables of the Maxwell’s equations?

Regarding to what you said about Poincare, I have read it recently that he was the one to come up with the principle of relativity and it was three months before Einstein’s paper on the electrodynamics of moving bodies, and he called it “Lorentz’s principle of relativity” . It’s quite interesting how Einstein had to put all the pieces together, but I'm not sure if he was aware of POincare's work.

 

[Bob_for_short]

Thank you Bob, I really now understand what invariance means. But I have one remaining point regarding to covariance

Are you referring here to the fact that the E and B field themselves transforms between a pair of inertial frames? Or are you referring to the Lorentz transformations of the space and time variables of the Maxwell’s equations?

Yes, E and B transform between a pair of inertial frames. The space and time coordinates X transform also between a pair of frames. The Lorentz transformations apply to all the variables while changing the reference frame. Some variables transform as components of a tensor (E and B) , some other do as components of a vector (space-time X, Momentum-energy P), some do as scalars or invariants (mass m and charge e). A Lorentz transformation always means calculation of the searched variables in a new reference frame if they are known in an old one.

Bob.