Difference between lorentz invariant and lorentz covariant

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Lorentz invariants are quantities that remain unchanged under Lorentz transformations, such as mass, which is the same for all inertial observers. In contrast, Lorentz covariant quantities, like four-vectors, consist of multiple components that may change under transformations but represent the same physical entity. Any Lorentz invariant is also Lorentz covariant, making invariants a subset of covariants. The distinction lies in the fact that invariants are scalars, while covariants can be tensors with varying components. Understanding these concepts is crucial for grasping the behavior of physical laws in different inertial frames.
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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!
 
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tiger_striped_cat said:
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!

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.
 
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
 
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.
 
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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