In Special Relativity, a Lorentz invariant is a physical quantity that is observed to have the same value by all inertial observers. It's something that observers in every frame of reference can agree on. Probably the most important Lorentz invariant is the spacetime interval between two events. If two events take place a distance Δx apart in space and Δt apart in time, then the spacetime interval between them could be measured as Δs2 = Δx2 - c2Δt2. Every observer will agree on this value, though they won't agree on Δx or Δt individually. Some other examples of Lorentz invariants are rest mass, electric charge (but not charge density), and any Klein-Gordon field. Often, Lorentz invariant quantities are given the adjective "proper", such as proper length or proper time.
Mathematically, a Lorentz invariant is a quantity that does not change under Lorentz transformations, a scalar in the language of tensor analysis. Sometime also called a Lorentz scalar or relativistic invariant, such a quantity may be formed by any N-form acting on N vectors or, indeed, any expression in terms of tensors where all covariant indices are contracted with contravariant indices leaving no free indices. Lorentz invariants are to Lorentz transformations as rotational invariants, such as length, are to spatial rotations. The spacetime interval mentioned earlier can be thought of as the true "distance" in spacetime precisely because it is that "distance" that is invariant under transformations.
If an equation is written such a way that it is true in the same form for every frame of reference, it is sometimes said to have a "Lorentz invariant form", or simply to be "Lorentz invariant", but this seems to imply that quantities in the equation are all Lorentz invariant, which is not necessarily true. It is preferred in such a case to say instead that the equation is Lorentz covariant, since in general the form will be the same for each frame of reference as long as the equation has a tensor of the same type on each side. So, we generally say a quantity is invariant if it actually has the same value in each frame, while an expression or equation is covariant if it has the same form in each frame of reference.
Compare to: Covariant
Sources:
- Classical Electrodynamics, J.D. Jackson
- A First Course in General Relativity, Bernard Schutz
- Mainly my B.S. and graduate studies in physics.