Covariant entities transform in a specific way under Lorentz transformation. Take the rotation in space as an example to make it easy. If you rotate the coordinate system, the observer who rotates with it will see an object that was sitting in the old frame of reference at position (x,y,z,t) "moving" in the oposite direction to position (x',y',z',t). We say that the position vector of that object transforms covariantly, the transformation matrix that transforms the coordinate system must be inverted and then applied to the vector (x,y,z,t) to leave the objetc at its unchanged position in space, which in the new frame of reference is (x',y',z',t). Actually any entity that has space-time indices and transforms in this way is co-variant (aligned with the variance), in general called a tensor. Only entities that do not have any index, for example the length of a vector, are invariant under that transformation. They are called scalars. (NB: also pseudo-scalars exist that behave like a scalar, except for changing sign under a parity inversion such as improper rotations.) 

A contravariant entity on the other hand is subject to the reverse transformation. The product of a covariant with a contravariant vector delivers a scalar.

Lorentz Invariance is a special subset of Poincare' Invariance.

the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime, introduced by Hermann Minkowski.

Notice the subtle difference between invariance and covariance

with respect to a set of transformations.

 • Invariance means that the physical observables of the

system are not changed by the transformation (think

of the rotation of a featureless sphere).

 • Covariance means that the system is invariant, and

that this invariance is manifest in the formulation of

the theory (can be “seen at a glance”).

Thus, a system that is not formulated in a covariant way

might still be invariant, but it may not be obvious that this

is so (not manifest) without detailed examination.

http://eagle.phys.utk.edu/guidry/astro616/lectures/lecture_ch16.pdf

   Bhushan, according to you, covariance is what other people call "manifest covariance".  I wonder, then, if there is also a form of covariance that is not manifest?  Or is that what you call invariance?

Electromagnetic field tensor is  Lorentz co-variant but not manifest co-variance because it contains partial derivatives and pure manifest co-variance equation is one in which all expressions are tensors.

Clear enough. Thanks, Bhushan.

The eom and action can be Lorentz invariant when one uses vector potentials A^\mu.  Covariant is often used poorly by physicists.  It comes from the mathematical notion of "covary" so that one object varies opposite another so they can combine an invariant when combines as A^\mu C_\mu.  Covariant can also mean opposite of contravariant so that an object transforms as a gradient.  

When you say equations are covariant one usually means manifestly invariant.  It is a lousy use of the terminology that is all too popular.  

Covariant entities transform in a specific way under Lorentz transformation. Take the rotation in space as an example to make it easy. If you rotate the coordinate system, the observer who rotates with it will see an object that was sitting in the old frame of reference at position (x,y,z,t) "moving" in the oposite direction to position (x',y',z',t). We say that the position vector of that object transforms covariantly, the transformation matrix that transforms the coordinate system must be inverted and then applied to the vector (x,y,z,t) to leave the objetc at its unchanged position in space, which in the new frame of reference is (x',y',z',t). Actually any entity that has space-time indices and transforms in this way is co-variant (aligned with the variance), in general called a tensor. Only entities that do not have any index, for example the length of a vector, are invariant under that transformation. They are called scalars. (NB: also pseudo-scalars exist that behave like a scalar, except for changing sign under a parity inversion such as improper rotations.) 

A contravariant entity on the other hand is subject to the reverse transformation. The product of a covariant with a contravariant vector delivers a scalar.

An equation that is covariant under a particular transformation will retain its form in any coordinate system. In that sence it is invariant. That applies BTW to all kinds of indices, not only to space-time.

Thank you all for your contributions.

A Lorentz co-variant quantity has a specific transformation property under the action of Lorentz transformations, which is that they should transform under a particular representation of the Lorentz group, such as scalar, vector, spinor and so on. A Lorentz invariant quantity does not transform under the action of a Lorentz transformation at all. Therefore this quantity must be a Lorentz scalar, example: Lagrangian density. 

Interestingly, thus we can see why both co-variant and contra-variant vectors under general coordinate transformations are Lorentz co-variant.

A brief answer to your question is all Lorentz invariant quantities are automatically Lorentz co-variant; but the contrary is not true.

I was once obsessed enough with this topic that I worked and found that one can have noninvariant actions that generate covariant equations of motion.  These led me to a new set of free field nonlocal conservation laws for electromagnetism.  In some sense, the whole subject of nonlocal conservation laws is a bit of a curiosity but might lead to something one day.  

Article Automorphism Induced Nonlocal Conservation Laws