# Definitions and usage of Covariant, Form-invariant & Invariant?

Just wondering about the definitions and usage of these three terms.

To my understanding so far, "covariant" and "form-invariant" are used when referring to physical laws, and these words are synonyms?

"Invariant" on the other hand refers to physical quantities?

Would you ever use "invariant" when talking about a law? I ask as I'm slightly confused over a sentence in my undergrad modern physics textbook:

"In general, Newton's laws must be replaced by Einstein's relativistic laws...which hold for all speeds and are invariant, as are all physical laws, under the Lorentz transformations." [emphasis added]

~ Serway, Moses & Moyer. Modern Physics, 3rd ed.

Did they just use the wrong word?

## 4 Answers

These words do have different meanings, this is a general guide to their differences. In different fields they may have slightly varying definitions. I would recommend looking them up to be certain.

Invariant means does not change at all. Everything is the same (whether physical law, quantity or anything). In terms of vectors, invariant is a scalar which does not transform.

Form-invariant means the form does not change, for example the inverse square law, will always be inverse square but the constants may differ.

Covariant, has a specific meaning when relating it to vectors, as it specifies the transformation rules. (This is as opposed to contravariant which is the other one). For more information see wikipedia, towards the end of the Mathematics of four vectors section.

To specifically answer your question on the phrase, Einsten's relativistic laws are invariant under Lorentz transformations, the laws don't change at all. The constants don't change, neither does the form.

• Right. Just linguistically, "invariant" is composed of "in" and "variant" and it just means "not change" - invariants don't change. "Form-invariant" (surely less common in fundamental physics!) means that "form doesn't change" for the same reason. The term "covariant" is linguistically "co" and "variant" which means "together change" - covariant objects change together with others i.e. in the same way as others. The Riemann tensor or Einstein's equations are "covariant" because their components transform just like tensor products of vectors. But they're not invariant - constant. – Luboš Motl Mar 28 '11 at 11:46
• I agree that the components of the Riemann tensor or Einstein's equations (as in GR) are covariant, maybe I miss-interpreted the question as to how it relates to "Einstein's relativistic laws" maybe the missing part was important to specify which laws. – Dom Mar 28 '11 at 11:48

This is a good question because I think physicists nowadays don't understand the difference between form invariant and covariant.

The equations of physics are form invariant under a Lorentz transformation, but they're not co-variant as in they don't vary with the Lorentz transformation.

I'm from the old school of physics (1970). This is what I remember: ALL laws of nature must be so constituted that it is transformed into a law of exactly the same form when, instead of the space-time variables x, y, z, t of the original co-ordinate system K, we introduce new space-time variables x', y', z', t' of a co-ordinate system K'. OR - another way of saying this is: General laws of nature are co-variant with respect to Lorentz transformations.

This is called the Heuristic value of Relativity.

To me Co-variant means that when a change is measured, the Formula to the natural law stays the same, but time and position of the object changes. Therefore, the natural law constituted works together with Lorentz Transformations.

In Geometry we can do ARBITRARY transformation of coordinates. The Points, Scalars, Vectors, Tensors are geometrical objects that defined by their transformation laws. They execute their transformation laws therefore being COVARIANT to the given transformation of coordinates. Example: the scalars are covariant by being invariant