# Difference between lorentz invariant and lorentz covariant

1. Dec 3, 2004

### tiger_striped_cat

title says it all. I've heard these two phrases.

Lorentz invariant: Equation (Lagrangian, or ...?) takes same form under lorentz transforms.

Lorentz covariant: Equation is in covariant form.

I'm don't think I know what I mean when I say the latter. Can someone elucidate the difference between these two. Is there some necessary/sufficient condition that relates the two.

Thanks for your help!

2. Dec 3, 2004

### dextercioby

By "Lorentz invariant" i understand any expression (that is product) of tensors on the flat manifold $$M_{4}$$ which has the the same form in every inertial reference frame.So it is a scalar wrt to the Lorentz transformations/group.
For example,the D'Alembert operator on flat spacetimes is a scalar:$$(\partial^{\mu})'(\partial_{\mu})'=\partial^{\mu}\partial_{\mu}$$,where,obviously:
$$(\partial^{\mu})'=\Lambda^{\mu}\ _{\nu} \partial^{\nu}$$.

Any expression written correctly wrt to suffices' position and containing a finite tensor product of tensors defined on the same flat manifold (space-time) is a Lorentz covariant.Arbitrary (but finite) rank tensors are Lorentz covariants,including scalars obtained through a finite number of contraction of suffices in a tensor product.

3. Dec 4, 2004

### tiger_striped_cat

I'm sorry I don't understand that explaniation. Can you give me two examples, and say:

1) Formula , "look at the formula" this is what i mean by lorentz invariant
2) Another Formula, "this is what i mean by lorentz covariant"

What I mean is just give an example without so much mathematical jargon

4. Dec 4, 2004

### jcsd

A Lorentz invaraint is a quanirty decsrivbed by a single number and is the same for all inertial observers, an example of this would be mass.

A Lorentz covaraint is a quantity described by 4^n (n = 0,1,2,3,...) numbers whose componets may change unbder transformation but essientially remain the same quantity (to be non-technical), Lorentz invaraints are alos Lorentz covaraint, any four-vector like four-momentum is Lorentz covariant as indeed is any Lorentz tensor.

5. Dec 4, 2004

### Janitor

The Lorentz invariants are a proper subset of the scalar quantities in physics. The Lorentz invariants are also a proper subset of the Lorentz covariants.

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