Why the notion of covariance in GR is so important

[Kalimaa23]

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,

 

[DW]

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.

 

[Kalimaa23]

So as I understand it, it has little to do with the transformation behaviour of the 4-vectors involved? The forces and acceleration involved still have their components written in the uppercase, so they still represent contravariant components? Or am I seeing this wrong?

 

[DW]

Originally posted by Dimitri Terryn
So as I understand it, it has little to do with the transformation behaviour of the 4-vectors involved? The forces and acceleration involved still have their components written in the uppercase, so they still represent contravariant components? Or am I seeing this wrong?

Yes, both contravariant and covariant expressions of tensors are "frame covariant". I usually express tensors in contravariant element notation when possible, but one could just as well use abstract notation in which you don't even use elements. The expressions then aren't in either contra or co variant element notation but are still "frame covariant". This "frame covariance" just means that they have a particular isomorphism as they are mapped from one frame to another by the same transformation as the differential form of the coordinates.

 

[Kalimaa23]

Thank you. You have been most helpful.

 

[Arcon]

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.

Hi Dimitri - General relativity requires that the laws of physics are the same in all coordinate systems. This means that the equations are tensor equations. Tensors are geometric objects whose components transform tensorially from one coordinate system to another. General covariance, as defined by Einstein, is as follows

The general laws of nature are to be expressed by equations which hold good for all systems of coordinates, that is, are covariant with respect to any substitution whatsoever (generally covariant).

You ask

So as I understand it, it has little to do with the transformation behaviour of the 4-vectors involved?

That’s not quite true. 4-vectors have certain transformation properties which define them as geometric objects. The idea of general covariance incorporates this fact.

Arcon

 

[Kalimaa23]

D'oh, I knew that! Not any four numbers is a 4-vector, they have to agree with a behaviour under transformation.

My main question was about the use of the term "covariance", and DW cleared that up quite nicely.

 

[Arcon]

Originally posted by Dimitri Terryn
D'oh, I knew that! Not any four numbers is a 4-vector, they have to agree with a behaviour under transformation.

My main question was about the use of the term "covariance", and DW cleared that up quite nicely.

Okay. The examples DW gave pertained to force. Newton considered F = ma to be a law of physics. Today physicists consider F = ma a definition and not a law. An example of covariance is Maxwell's equations.

The problem with DW's response is that he seems to imply that if an equation is not written explicitly as a tensor equation then it isn't covariant. However covariance means that an law of physics must be expressable in tensor form. DW is speaking specifically with regards to what is called manifestly covariant form which means its a tensor equation.

 

[DW]

Originally posted by Arcon
The problem with DW's response is that he seems to imply that if an equation is not written explicitly as a tensor equation then it isn't covariant.

I didn't "seem" to say any such thing. You are just mad because you have a grudge against me and he appreciated my input.

By the way, instead of signing your post as pmb why don't you just post "as" pmb like you used to do in this forum instead of switching handles?

 

[Arcon]

I didn't "seem" to say any such thing.

The word seems is defined as to give the impression of being. And whether you think so or not that is the impression you gave. If you are unware of this then simply accept the criticism gracefully and more on. I.e. learn how to accept criticism. Your lack of accepting criticism was what got you kicked out of here before.

The last time you posted in this forum, before this most recent incarnation, you (i.e. "DavidW") decided to start a flame war because you have this sick vendatta against me for proving you wrong so often. In your earlier incarnation it was because I corrected you on the meaning of "scalar." However it seems once more that you're less interested in physics and more interesed in causing trouble.

Stop being so obsessed with me and stick to physics!

 

[DW]

Originally posted by Arcon
The term seems is defined as to give the impression of being. And whether you think so or not that is the impression you gave.

I "gave" no such impression. You just missunderstood having only superficially skimmed what was written. You do that a lot with just about everyones writtings! I explicitely said, "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."

Its not my fault if you didn't read all of what I wrote.

The last time you posted in this forum, before this most recent incarnation, you (i.e. "DavidW")

What on Earth are you talking about? I highly respect David for the depth of work he did on his online text from which I have learned a lot so I chose his initials as my Nick name. You don't actually think someone posting here as say Britney Spears would actually be Britney Spears do you?

If any such argument occurred between the two of you here based on these rude statements of yours I suspect it led to your dissmissal thus explaining why you no longer post "as" pmb.

 

[Kalimaa23]

Guys, cut it out. I had a question, I got an answer, this thread has served it's purpose. If you want to argue do it by PM or e-mail, but don't clutter the board.

I just wish the creator of the thread could lock it, but alas.

 

[Arcon]

Dimitri - If you're happy then so am I. I have no intention of responding to DW in the future so don't worry about it.

Arcon

 

[Phobos]

Originally posted by Dimitri Terryn
I just wish the creator of the thread could lock it, but alas.

Let me grant this holiday wish.