I What is the distinction between invariance and covariance?

[QuarkMaster]

TL;DR Summary

covariance of physical law

What is the difference between these two concepts? An equation is said to be "invariant" under some operation if the form of the equation doesn't change. However, isn't that exactly what "covariance" in physical laws means—that the form of the laws remains unchanged when applying an operation to them?

 

[Baluncore]

Welcome to PF.

It depends on context.
Invariance says one thing is independent of another.
Covariance is a measure of how closely two things are coupled.

 

[PeroK]

TL;DR Summary: covariance of physical law

What is the difference between these two concepts? An equation is said to be "invariant" under some operation if the form of the equation doesn't change. However, isn't that exactly what "covariance" in physical laws means—that the form of the laws remains unchanged when applying an operation to them?

Let's take a simple example of a vector quantity in 3D space. The magnitude of the vector is an invariant quantity. The components of the vector transform contravariantly. In fact, that is a more abstract definition of a vector. By contrast, the basis vectors transform covariantly.

If we move up to 4D spacetime, then the analagous concepts are (Lorentz) invariance and Lorentz covariance. Invariance means that the quantity remains unchanged by a Lorentz Transformation. Covariance means that the components transform according to the Lorentz Transformation.

You can read more about this here.

https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors

https://en.wikipedia.org/wiki/Lorentz_covariance

 

[PeroK]

So covariance would mean that the equations doesn't change form when we change coordinates.

Yes.

While invariance is more general?

Invariance is more specific. Newton's second law is exactly the same in all inertial reference frames: $$\vec F = m \vec a$$Note that $m$, $|\vec F|$ and $|\vec a|$ are invariant quantities.

 

[QuarkMaster]

Yes.

Invariance is more specific. Newton's second law is exactly the same in all inertial reference frames: $$\vec F = m \vec a$$Note that the equation involves three invariant quantities.

aha, so in this example the covariant equation was build from invariant quantities.

 

[PeroK]

aha, so in this example the covariant equation was build from invariant quantities.

Yes, although the vector quantities are actually covariant (or contravariant). It's the magnitudes that are invariant. I've corrected that statement in the post above.

 

[PeroK]

PS again, to be precise, Newton's second law is not Lorentz covariant. It's covariant in the sense that it involves 3D spatial vectors.

 

[QuarkMaster]

So invariance refers to specific quantities physical or mathematical. While covariance refers to the relationship between those quantities, the equations?

 

[PeroK]

So invariance refers to specific quantities physical or mathematical. While covariance refers to the relationship between those quantities?

Yes, although covariance refers to the components of quantities. As above, the magntiture of a vector is invariant, but its components vary contravariantly (or covariantly). Covariance is then used in a slightly different sense to refer to the equations.

 

[vanhees71]

Yes.

Invariance is more specific. Newton's second law is exactly the same in all inertial reference frames: $$\vec F = m \vec a$$Note that $m$, $|\vec F|$ and $|\vec a|$ are invariant quantities.

That's a good example! The question in #1 is pretty subtle, and it's important to understand all theoretical physics starting from classical mechanics to relativistic QFT and general relativity.

What you are here referring to is the physical equivalence of the physical laws when discribed in inertial reference frames. It means that it is impossible by any observation of Nature to define an "absolute inertial frame of reference". You can only observe the relative motion with constant velocity of two inertial frames. If you are put in a closed box, which is at rest in some inertial frame of reference you don't have a chance to figure out more than that you are in an inertial frame of reference. It doesn't make sense for you to ask about which specific inertial frame you are sitting at rest.

You can determine, how the equations of motion of bodies must look like from knowing the symmetry properties of the Newtonian spacetime model, defined by the proper Galilei group (a 10 parametric Lie group leading to the corresponding conserved quantities for closed systems, energy, momentum, angular momentum, and center-mass velocity).

Now, as any theory, you can also formulate Newtonian mechanics in a manifest covariant way under general coordinate transformations. The most elegant way is to use the action principle and derive the equations of motion in terms of the Euler-Lagrange equations, which always look the same, no matter which "generalized coordinates" you use, even if they refer to non-inertial frames of reference.

Of course, here you have no physical symmetry principle at work. It's simply rewriting the physical laws in a generally covariant way, and from this "form invariance" of the Euler-Lagrange equations you cannot in any way constrain, how the Lagrangian must definitely look. For that you need the physical symmetry, i.e., Galilei symmetry.

That's why you can formulate also special-relativistic point-particle mechanics with the action principle, leading to the Euler-Lagrange equations, which look the same as in Newtonian mechanics, but of course here you need the Poincare group as the symmetry group to figure out how the Lagrangians must look to be compatible with special relativity.

 

[haushofer]

My 2 cents:

We call quantities "invariant" when they don't change value under coordinate transformations. Examples are the mass of a particle or the magnitude of acceleration for inertial observers.

We call equations covariant when the form of these equations don't change form under coordinate transformations. The example which was given is Newton's second law. Under the Galilei group this equations transforms covariantly. For me, the most natural way to see this is to write Newtons 2nd law as "m*a - total force = 0", and note that the "0"-vector remains "0" under the Galilei group, i.e. it transforms as a vector under this group. This makes sense: components of tensors transform (multi)linearly, and the linear combination of zeroes gives zero.

But when we extend the Galilei group to other transformations, e.g. linear accelerations or time-dependent rotations, Newton's 2nd law is not covariant anymore; in switching from an inertial observer to a rotating observer the "0" in "m*a - total force = 0" becomes non-zero: a centrifugal and Coriolis force pop up. From a general relativity point of view these forces can be seen as components of a connection, and this connection transforms inhomogenously (if its components are zero in one frame, they can become non-zero in another frame).