# Invariant and covariant in special relativity

In anglo-american literature
-a physical quantity is invariant if it has the same magnitude in all inertial reference frame,
-an expression relating more physical quantities is covariant if it has the same algebraic structure in all inertial reference frames (rr-cctt) ?

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One book I have sitting next to me(Introduction to the Relativity Principle; Gabriel Barton, Wiley) agrees with that.

(But I don't know what you mean by that "rr-cctt") :)

invariant covariant

One book I have sitting next to me(Introduction to the Relativity Principle; Gabriel Barton, Wiley) agrees with that.

(But I don't know what you mean by that "rr-cctt") :)
Thanks. xx+yy+zz-cctt=x'x'+y'y'+z'z'-cct't' an example of covariant expression?

xx+yy+zz-cctt=x'x'+y'y'+z'z'-cct't' an example of covariant expression?
Oh, that! I guess it is. But the quantity s^2 is invariant.

In anglo-american literature
-a physical quantity is invariant if it has the same magnitude in all inertial reference frame,
-an expression relating more physical quantities is covariant if it has the same algebraic structure in all inertial reference frames (rr-cctt) ?
I believe both your assertions are correct. However, the example of a covariant structure you give is in fact an invariant. A covariant object would be

[tex][ct, x, y, z][/itex]

since it transforms as a vector should. Another would be the EM field tensor, the Ricci tensor, etc. Essentially, all tensors are covariant, and all scalar tensors have the additional property that they are invariant.

Note also, that some authors mix their usage of both terms (although I can't think of an example right now).

The term covariant can also apply to equations. See the Wikipedia article on Lorentz covariance which I feel describes it well (since I wrote most of it).

Chris Hillman
A bit of background

It might help to point out that this terminology derives from invariant theory, which was a subject known to all mathematicians and physicists in the late nineteenth century, but less well known now (although it has been becoming popular again over the past decade).

Invariant quantity means 'has the same value in different coordinate systems'. Presumably, there is a definition how to calculate the quantity in different coordinate systems.

'Covariant equation' means 'has the same structure/form in different coordinate systems'. Sometimes they say the equation is 'form invariant' which is much clearer in my opinion.

Sometimes they say two quantities 'transform covariantly' when one changes coordinates. That simly means, the components of the quantities transform with the same matrices under change of coordinates.
An example of that is the older usage of 'covariant vector' and 'contravariant vector' :
'Covariant vector' has components 'transforming between two different coordinate systems just like a differential transforms under change of variables'. 'Co' means 'in the same way'. In modern usage covariant vectors are called one-forms or tensors of type (0,1) with one lower index.
'Contravariant vector' has components 'transforming opposite to the differential'. 'Contra' means opposition. In modern usage contravariant vectors are simply called vectors or tensors of type (1,0) with one upper index.

In GR, ds^2 is defined in any coordinate system through the metric and has the same value hence is invariant. The equation ds^2=xx+yy+zz-cctt is covariant/form-invariant with respect to inertial coordinate systems because it looks the same in the old and new inertial coordinates.

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Covariant means 'has the same structure/form in different coordinate systems'. An equation can be covariant if it looks the same in the old and the new coordinates.
The term "covariant" has multiple meanings in relativity. One means that the form of the equations of physics and the value of the constants in the equations, remain unchanged upon a change in coordinate. Another use that I see is in Lanczos book The Variational Principles of Mechanics. In this book Lanczos uses it in one sense to mean the following: A number is covariant if it changes with a change in coordinate system. This is how Lanczos uses the term in his book "Variational Principles of Mechanics." On page 20
For example, an invariant differential form of the first order

dw = F1 dx1 + F2 dx2 + ... Fn dx2

defines a vector. The quantities F1 , F1 , F2 ,..., Fn are the "components" of the vector: They change with coordinate system and are therefore "covariant": quanties. The whole differential form, however, is an invariant.
In Chapter-IX Classical Mechanics, page 292, Lanczos goes on to say
These equations show that the time t ceases to be an absolute quantity ..... The time t has changed from an invariant quantity to a covariant quantity, whereas the speed of light c has changed from a covariant quantity to an invariant quantity.
There is a very good explaination in a footnote in Gravitation and Spacetime - Second Ed., Ohanian and Ruffini which reads in a footnote on the bottom of 371
We have "covariant" vectors, "covariant derivatives" and now "covariant equations". The word "covariant" is unfortunately much overworked, and which of the three meanings is intended must be guessed from context.
Best regards

Pete

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samalkhaiat
Invariant quantity means 'has the same value in different coordinate systems'. Presumably, there is a definition how to calculate the quantity in different coordinate systems.

'Covariant equation' means 'has the same structure/form in different coordinate systems'. Sometimes they say the equation is 'form invariant' which is much clearer in my opinion.
Well done!

Covariance means "form invariance".
Covariant object is an object with well defined transformation law; scalars, vectors, spinors, and higher rank tensors.
Some quantities (tensors of certain rank) retain the same value in all coordinate systems, such quantites are called scalars or invariant.

regards

sam