Invariant and covariant in special relativity

1. Jun 11, 2007

bernhard.rothenstein

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) ?

2. Jun 11, 2007

neutrino

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") :)

3. Jun 11, 2007

bernhard.rothenstein

invariant covariant

Thanks. xx+yy+zz-cctt=x'x'+y'y'+z'z'-cct't' an example of covariant expression?

4. Jun 11, 2007

neutrino

Oh, that! I guess it is. But the quantity s^2 is invariant.

5. Jun 11, 2007

masudr

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).

6. Jun 11, 2007

masudr

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).

7. Jun 11, 2007

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).

8. Jun 11, 2007

smallphi

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.

Last edited: Jun 11, 2007
9. Jun 11, 2007

pmb_phy

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
In Chapter-IX Classical Mechanics, page 292, Lanczos goes on to say
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
Best regards

Pete

Last edited: Jun 11, 2007
10. Jun 11, 2007