# 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?

## 7 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. Commented Mar 28, 2011 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. Commented Mar 28, 2011 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

They’re the same, really. In (general) relativity there is the (General) Principle of Covariance which states that the laws of physics must be invariant under (all frames) inertial frames of reference. This means that the equations describing the laws must take the same form - and thus form invariant - in any frame of reference. Equations in question are, in general, tensor equations. There’s also the principle of the invariance of the speed of light, which says that the speed of light is invariant under inertial frames of reference.

So covariance is used when talking about equations and invariance is used when talking about quantities. This is similar to how in maths function and map are the same thing but the other is preferred in one context than the other.

(i) The law "force is equal to rate of change of momentum" is invariant under rotations of coordinate frames.

(ii) The mathematical equation $${\bf f} = \frac{d {\bf p}}{dt}$$ is invariant under rotations of coordinate frames.

(iii) The mathematical equation $$f_i = \frac{d p_i}{dt}$$ on the other hand, is covariant rather than invariant. The symbols $$f_i$$ and $$p_i$$ refer to quantities (namely, components of force and momentum along coordinate directions) that are not the same as corresponding quantities $$f'_i$$ and $$p'_i$$ in some other (rotated) frame, but all the quantities vary together in such as way that in some other frame the relationship between them will be $$f'_i = \frac{d p'_i}{dt'}.$$ The equation may then be called covariant.

(iv) The word 'covariant' has a further meaning when dealing with tensor analysis, when it is used to refer to components which transform like the components of the metric tensor. This is in contrast to 'contravariant'.

(v) But when we are talking about sets of equations the term 'covariant' means that the equations retain their form while the quantities in them change, as in the example above. In this terminology Maxwell's equations for the electromagnetic field are Lorentz covariant (whether or not they are written in 4-tensor notation). If they are written in 4-tensor notation then they may be said to be manifestly covariant (meaning just 'obviously covariant').

The main difference is between invariance and covariance. They in fact refer to the same concept thought about in two different ways which is why they often crop up in tge same context.

Consider a cup in space. This is invariant. No matter which way you look at it, although your view changes, the cup itself remains the same.

Now suppose instead you are given photos of the cup from various positions. Each photo is stamped with the position from which the photo was taken. The question is then, do they show a cup? For example, I might have been sneaky and subsituted a single photo of a similar looking cup from a certain position. If I stitch all the photos together using the position info tyen I can detect this fraudulent photo.

This idea is called covariance because it co varies. From each different position we get a differemt picture. The two different kinds of info co vary.

But why would anyone go through such a bizarre process? This is usually because we cannot directly see the object in question. For example, we cannot directly see an electric field. All we xan do is set up axes and measure tye goeld with reference to these axes. Then the question arisezls, is this object a vector? We can only find out by changing the axes and seeing that the measurements change in the appropriate way. This is how the terminology, to transform as a vector arose.