Originally posted by Dimitri Terryn
Greetings,
In some discussions about GR, I heard the term "covariance" and covariant form (eg, covariant form of Maxwell's equations) pop up often.
I've been wondering for a while why the notion of covariance in GR is so important. I have some background in mathematical physics, so I know the difference between co- and contravariant components of a tensor a such.
Cheers,
The principle of relativity itself is that physics does not depend on frame. We model physics with equations and as such we want the equations to take an invariant form so that they may express "general laws". If the equation has a form that does not depend on frame then the physics described by that equation is described as frame independent. That equation is then a candidate for the description of a general law of physics. Take for example ordinary force
f^i = \frac{dp^i}{dt}
and coordinate acceleration
a^i = \frac{du^i}{dt}.
One observer may observe the instantaneous result from a force on a particle to relate the two as
f^i = ma^i
for example if the particle is instantaneously at rest according to his frame.
He then might propose this as a law of physics. In fact Newton did. The problem is that this equation is not frame invariant in form. Let's say another observer using a frame according to which the particle is instantaneously in motion perpendicular to the force describes the responce. He finds
f^i = \gamma ma^i.
He descides to propose this for a law of physics. It still isn't general. Consider a third observer according to which the particle is instantaneously in motion in the direction of the force. He finds
f^i = \gamma ^{3}ma^i.
All 3 are in disagreement.
Lets say they finally arrive upon an equation that reduces to all 3 cases like equation 3.2.10 at
http://www.geocities.com/zcphysicsms/chap3.htm#BM29
An accelerated frame observer would STILL dissagree with it.
Tensors are frame covariant in the literal sense of the word which guarantees that the form of the equations involving only tensors and invariants will be invariant. So instead of going through all those rediculous itterations of the law you just state a tensor equation. For example start with a four vector force
F^\lambda = \frac{DP^\lambda}{d\tau} an invariant for mass m and a four vector acceleration
A^\lambda = \frac{DU^\lambda}{d\tau} and say
F^\lambda = mA^\lambda
is your tensor equation law and automatically every frame observer will agree that it describes the physics according to every frame as long as it describes it according to any single one (with a hypothetical complete accuracy). Tensors are beautiful!
To say "covariant form of Maxwell's equations" is kind of a strange way to fraise it because the everyday form is actually special relativistically covariant again in the literal sense of the word. What that form is not is "generally covariant" nor generally invariant in form. An accelerated frame observer will disagee that Maxwell's equations in old form describe the physics as he observes it. The generally covariant expressions for the electromagnetic field are the electromagnetic and electromagnetic duel tensors. The tensor equation given by equations 7.1.5 or 7.1.8 at
http://www.geocities.com/zcphysicsms/chap7.htm#BM84
which you heard referred to as "covariant form of Maxwell's equations" have a frame invariant form. If even one observer finds that these equations describes physics (with a hypothetical complete accuracy) than every observer for every frame must agree whether inertial or accelerated whether in the depths of space or in considering a strong varying gravitational field. This expresses a general law.
