Covariant VS manifestly covariant

[superg33k]

What is the difference between covariant and manifestly covariant? And is this correct?

The equation for covariant differention:
$$\nabla_\lambda T^\mu=\frac{\partial{T^\mu}}{\partial{x^{\lambda}}}+{\sum}_{\rho}{\Gamma}^{\mu}_{\rho \lambda}T^{\rho}$$
And equation is manifestly coverint if I write it as:
$$\nabla_\lambda T^\mu = 0$$
Since its all covarient tensors. But if I write it as:
$$\frac{\partial{T^\mu}}{\partial{x^\lambda}}+{\sum}_{\rho}{\Gamma}^{\mu}_{\rho \lambda}T^{\rho} = 0$$
it is still a covarient equation, however it is not manifestly covarient becuase it isn't obvious because neither the reimann tensor nor partial derivatives are covarient

 

[Polyrhythmic]

"Manifestly" is just a word to emphasize that covariance is obvious or easy to see, it has no mathematical meaning.

 

[Sam Gralla]

As I said in the other thread, there is no universally agreed-upon meaning of these terms. Just don't worry about it

 

[superg33k]

From Wikipedia on Manifest covariance:

"In general relativity, an equation is said to be manifestly covariant if all expressions in the equation are tensors."

Which agree's with my statement. That said wikipedia has sent me wrong before. Physics forums too though. But at least here good resources get medals!

I doubt you'll find any references that state it's lack of meaning?

 

[dextercioby]

<Manifest> is just an adjective whose meaning is <explicit>. The really important word is <covariant> which means <expressed in terms of tensors/tensor components>.

 

[superg33k]

<Manifest> is just an adjective whose meaning is <explicit>. The really important word is <covariant> which means <expressed in terms of tensors/tensor components>.

I think your definition of manifest agree's with my understanding. If we know each component of an equation to be covariant/contravariant/invariant, then the equation will be explicitly covariant, i.e. its manifestly covariant.

However by (my understanding of) your definition of covariant, are you saying that a covaraint equation is one that is expressed in terms of covariant/contravariant/invariant tensors? Which is how I defined manifestly covaraint.

If this s the case it raises 2 points. Whats the difference between a covaraint equation and a manifestly covaraint equation? And the equation I stated to be a covaraint equation in the original post you are claiming is not, even though if it is true in one frame it would be in all.

(I notice in your post you said "expressed in terms of tensors", not covaraint/contravaint/invarant tensors. I just assumed this was what you meant as any equation can be written in tensors)

 

[Matterwave]

There's no real reason to worry too much about this, it's just terminology. The physics is in the equations themselves and not which specific word you choose to name them. It's like, we can argue whether the "Lorentz Contraction" should be called "Lorentz Fitzgerald Contraction" or whether we should call a covariant vector a dual vector or a one form, it's just a name.

 

[bobc2]

I can't ever read a post like this without my tensor math prof's booming comment on the first day of class ringing in my head: "Just remember--'Co' goes below."