What is the significance of the Christoffel symbol as a tensor?

[pmb_phy]

On page 34 of General Relativity, Robert Wald, the author refers to the Christoffel symbol as a tensor of rank (1,2). That whole derivation that led to that is quite involved so I don't quite understand how Wald can say that the Christoffel symbols are tensors since normally one does not refer to them as tensors, in fact its commonly understood that they are not tensors.

So what's the deal??

Pete

 

[HallsofIvy]

I have no idea! The Christoffel symbols are NOT tensors! Could you give the complete quote from Wald?

 

[robphy]

Here's a quote from Wald. I have highlighted what I think are the important parts.

From Wald, p. 33.
Thus, we have shown that \tilde\nabla_a - \nabla_a defines a map of dual vectors at p (as opposed to dual vector fields defined in a neighborhood of p) to tensors of type (0, 2) at p. By property (1), this map is linear. Consequently \tilde\nabla_a - \nabla_a defines a tensor of type (1,2) at p, which we will denote as C^c{}_{ab}. Thus, we have shown that given any two derivative operators \tilde\nabla_a and \nabla_a there exists a tensor field C^c{}_{ab} such that
\nabla_a \omega_b= \tilde\nabla_a \omega_b - C^c{}_{ab}\omega_c (3.1.7)

...snip...

Continuing in a similar manner, we can derive the general formula for the action of \nabla_a on an arbitrary tensor field in terms of \tilde\nabla_a and C^c{}_{ab}. For T \in {\cal T}(k,l) we find
\nabla_a T^{b_1 \cdots b_k}{}_{c_1 \cdots c_l}=<br /> \tilde\nabla_a T^{b_1 \cdots b_k}{}_{c_1 \cdots c_l} +<br /> \sum_i C^{b_i}{}_{ad} T^{b_1 \cdots d \cdots b_k}{}_{c_1 \cdots c_l}<br /> -\sum_j C^{d}{}_{ac_j} T^{b_1 \cdots b_k}{}_{c_1 \cdots d \cdots c_l}<br /> (3.1.14)
Thus, the difference between the two derivative operators \textcolor{red}{\nabla_a} and \textcolor{red}{\tilde\nabla_a} is completely characterized by the tensor field \textcolor{red}{C^c{}_{ab}}.

From Wald, p. 34.
The most important application of equation (3.1.14) arises from the case where \tilde\nabla_a is an ordinary derivative operator \partial_a. In this case, the tensor field C^c{}_{ab} is denoted \Gamma^c{}_{ab} and called a Christoffel symbol. Thus, for example, we write
\nabla_a t^b = \partial_a t^b + \Gamma^b{}_{ac} t^c (3.1.15)
Since we know how to compute the ordinary derivative associated with a given coordinate system, equation (3.1.15) (and, more generally, eq. [3.1.14] with \partial_a and \Gamma^b{}_{ac} replacing \tilde\nabla_a and C^b{}_{ac}) tells us how to compute the derivative \nabla_a if we know \Gamma^b{}_{ac}. Note that, as defined here, a Christoffel symbol is a tensor field associated with the derivative operator \nabla_a and the coordinate system used to define \textcolor{red}{\partial_a}. However, if we change coordinates, we also change our ordinary derivative operator from \partial_a to \partial&#039;_a and thus we change our tensor \Gamma^c{}_{ab}, to a new tensor \Gamma&#039;^c{}_{ab}. Hence the coordinate components of \textcolor{red}{\Gamma^c{}_{ab}}, in the unprimed coordinates will not be related to the components of \textcolor{red}{\Gamma&#039;^c{}_{ab}} in the primed coordinates by the tensor transformation law, equation (2.3.8), since we change tensors as well as coordinates.

 

[mathwonk]

Hi Pete,

Input from the peanut gallery.

Well that quote is rather confusing to me, not least because i do not know what christoffel symbols are.

However, some people call a "tensor" anything that acts on families of vectors and covectors, sending them to other such things, with two defining properties:
1) the action is (multi) linear.
2) the value of the action depends only on the pointwise values of the arguments and not on their value in a nbhd.

From this perspective, even without knowing what a christoffel symbol is, one can deduce that Wald thinks his object is a tensor because of the first two sentences of your first quote from him:

"Thus, we have shown that *** defines a map of dual vectors at p (as opposed to dual vector fields defined in a neighborhood of p) to tensors of type (0, 2) at p. By property (1), this map is linear. Consequently *** defines a tensor of type (1,2) at p,"


So maybe the confusion is that when he changes coordinates, in order to have a tensor, he would have to transform both derivative operators by the tensor law, whereas he is merely taking one of the coordinate derivative operators as his delta a, rather than taking whatever he gets when he transforms the primed one.

So his definition of the christoffel symbols is not coordinate invariant, i.e. he says right there: (as you pointed out in red)

"as defined here, a christoffel symbol depends on a choice of coordinate system"
(roughly quoted).

But if it depends on the coordinate system it is not a true tensor, in the sense that it does not transform like one.

I.e. maybe the problem is something like this: maybe there IS a tensor asociated to a DIFFERENCE of two derivative operators, but when he tries to FIX one of the operators as differentiation in a coordinate direction, he steps out of bounds as far as defining a true tensor.

?
just a guess.


roy

 

[robphy]

mathwonk,

Pete is pmb_phy, who asked the first question in this thread.
I am robphy, who transcribed the passage from Wald.

Although we have the same last name , we're not related .

 

[mathwonk]

thank you! My apologies. When Halls of Ivy asked Pete for the quote and then the quote appeared I did not look at who supplied it.

What did you think of my guess as to the problem he posed?

 

[pmb_phy]

Here's a quote from Wald. I have highlighted what I think are the important parts.

Thanks rob. I guess that's the part that doesn't make sense to me. How can a tensor be associated with, in part, a coordinate system? I don't see that it can from the definition given by Wald for "tensor".

Schutz has an entire section on this topic, which I had forgotten about. The section starts on page 143 and is entitled The tensorial nature of G^a_bu (where G^a_bu are the Christoffel symbols. Schutz states

Since \bold e_{\alpha} is a vector, \nabla \bold e_{\alpha} is a tensor whose components are \Gamma^{\mu}_{\alpha \beta}. Here \alpha is fixed and \mu and \beta are component indices: changing \alpha changes \nabla \bold e_{\alpha}, while changing \mu or \beta changes only the component under discussion. So it is possible to regard \mu and \beta as component indices and \alpha as giving the particular tensor referred to. etc

(sorry, unable to sit longer to type more).

If there is a sense where you can call the Christoffel symbols "tensors" then would it be reasonable to say that there is a sense in which you can call the gravitational force a tensor as well?

Pete

 

[mathwonk]

how silly of me. I just repeated what Rob said.

 

[PFanalog57]

In the book "A first course in general relativity" by Bernard F. Schutz, on page 143, chapter 5, section 5.5, the tensorial nature of the Christoffel symbols are explained as the components of a set of tensors but there is no single tensor whose components are C^mu_ab.

And the combination V^b_,a + V^mu C^b_mu a is explained as a component of a single tensor.

If I understand correctly, the Christoffel symbols are tensors in the respect that they're linear maps from tangent vectors and dual vectors to the reals. Christoffel symbols don't transform like tensors though.

 

[sal]

Thanks rob. I guess that's the part that doesn't make sense to me. How can a tensor be associated with, in part, a coordinate system?

So maybe the confusion is that when he changes coordinates, in order to have a tensor, he would have to transform both derivative operators by the tensor law, whereas he is merely taking one of the coordinate derivative operators as his delta a, rather than taking whatever he gets when he transforms the primed one.

So his definition of the christoffel symbols is not coordinate invariant, i.e. he says right there

Not exactly... There's a subtle but important point which has, perhaps, not been given sufficient emphasis here.

The tensor which Wald actually defines (and proves to be a tensor) is the thing he calls C^a_bc. It describes the relationship between two abstract derivative operators. It depends on the choice of operators, but no coordinate system is referred to in its definition.

Let me say that again: Wald defines derivative operators without reference to a coordinate system. His C^a_bc tensor relates a pair of these operators; each such pair of operators defines a tensor. Given a particular pair of operators, the tensor which relates them is valid in any coordinate system.

He later narrows the discussion to derivative operators which are associated with particular coordinate systems, and the (unique) derivative operator which is associated with the metric. And in that context, he defines

\Gamma^a_{bc} = C^a_{bc}

to be the tensor which maps between the covariant derivative and the derivative operator for the selected coordinate system.

His C^a_bc thingies are obviously tensors. Whether you choose to call the Gamma version, which is associated with the coordinate system, a tensor or not seems like a matter of taste. He's shown that it actually depends on the derivative operator, not the CS, and with that knowledge in hand, I'd say what you decide to call it is just a semantic issue.

Incidentally, the Christoffel symbols are mislabelled anyway. The Greek letter Gamma is a "g" which has no business here. The Greek letter corresponding to "C", which is what we ought to use, is chi, which looks just like "X". But perhaps that would be too confusing.

 

[robphy]

Here's an interpretation I have held for while (since I never had the chance to sort it out myself... but this might be my opportunity!)... if it's wrong, please correct me.

For simplicity, take an affine space ("a vector space, but forget the origin").
Given a point x on that space, its location is represented by a tuple of numbers that depends on the choice of origin.
Given two points x and y on that space, the displacement from x to y (i.e., y-x) is a vector, independent of any choice of origin.

Is something like this happening with the Christoffel symbols and the tensor C^c_ab?

(If so, is this why the Christoffel symbols are sometimes called affine connections? Or does "affine" refer to the "affine parameter" in the geodesic equation? Based on http://planetmath.org/encyclopedia/Connection.html#foot196 , I might have to go find the references to Cartan.)

 

[pmb_phy]

The tensor which Wald actually defines (and proves to be a tensor) is the thing he calls C^a_bc. It describes the relationship between two abstract derivative operators. It depends on the choice of operators, but no coordinate system is referred to in its definition.

Let me say that again: Wald defines derivative operators without reference to a coordinate system. His C^a_bc tensor relates a pair of these operators; each such pair of operators defines a tensor. Given a particular pair of operators, the tensor which relates them is valid in any coordinate system.

Correct me if I'm wrong but myu rad of Wald is that the coordinate system is not explicit in the definition - it is implicit. I.e. Wald writes on page 34

Note that, as defined here, A Christoffel symbol is a tensor field associated with the derivative operator nabla_a and the coordinate system used to define partial_a.

That seems to mean that a coordinate system is tied to the tensor, not by definition, but as a result of the definition.

I think I understand this "Christoffel symbol as tensor" thing. If you recall, there are several things which have the name "tensor" and which are not the same thing. For example, a Lorentz tensor is not a tensor in the normal sense of the term since the Lorentz tensor is only a tensor under a Lorentz transformation. Thus the tensor is tied to Lorentz transformations and not generalized coordinate transformations.

Perhaps this is yet another use of the term "tensor". By the way, the Christoffel symbols are Lorentz tensors. In fact they are tensors under any linear transformation. But they are not general tensors.

Pete

 

[sal]

The tensor which Wald actually defines (and proves to be a tensor) is the thing he calls C^a_bc. It describes the relationship between two abstract derivative operators. It depends on the choice of operators, but no coordinate system is referred to in its definition.

Correct me if I'm wrong but [my reading] of Wald is that the coordinate system is not explicit in the definition - it is implicit. I.e. Wald writes on page 34

Note that, as defined here, A Christoffel symbol is a tensor field associated with the derivative operator nabla_a and the coordinate system used to define partial_a.

That seems to mean that a coordinate system is tied to the tensor, not by definition, but as a result of the definition.

I can't completely agree with this interpretation.

It's the C^a_bc object which Wald clearly states is a tensor, and there is no coordinate system, either explicit or implicit, in its definition. CONSIDER: It relates two abstract derivative operators, and in general, a particular derivative operator may not be related to any coordinate system! In curved space, the derivative operator associated with the metric is identical with the coordinate derivative of any particular coordinate system at at most one point. (If that were not true, you could find globally flat coordinates.) Every possible "curved" metric has a (different!) derivative operator associated with it, so there are an uncountable number of derivative operators, all different, which are associated with no coordinate system. Each pair of these non-coordinate derivative operators determines a C^a_bc tensor.

Again, the C^a_bc objects are tensors, by anybody's definition, and as I just pointed out, they are not typically associated with any particular coordinate system.

When Wald goes on to define [Gamma]^a_bc as a Christoffel symbol associated with a particular derivative operator which is in turn the coordinate derivative for a particular CS, then he has, indeed, bound it to a particular CS. At that point it's no longer clear whether it's better to just drop the name "tensor" to avoid confusing everybody.

I think I understand this "Christoffel symbol as tensor" thing. If you recall, there are several things which have the name "tensor" and which are not the same thing. For example, a Lorentz tensor is not a tensor in the normal sense of the term since the Lorentz tensor is only a tensor under a Lorentz transformation.
Thus the tensor is tied to Lorentz transformations and not generalized coordinate transformations.

Perhaps this is yet another use of the term "tensor". By the way, the Christoffel symbols are Lorentz tensors. In fact they are tensors under any linear transformation. But they are not general tensors.

Again, I can't fully agree.

Wald is using the word "tensor" without qualification, unlike its use in "Lorentz tensor", where it is qualified. But when he defines the Christoffel symbols, which are bound to a coordinate system, he seems a bit hesitant about calling them "tensors". They refer to objects which are tensors, under any transformation, but it's not useful to call them that because they refer to a different tensor in every (non-linear) frame.

I think I'm repeating myself, so I should stop.

 

[pmb_phy]

Wald is using the word "tensor" without qualification, unlike its use in "Lorentz tensor", where it is qualified. But when he defines the Christoffel symbols, which are bound to a coordinate system, he seems a bit hesitant about calling them "tensors". They refer to objects which are tensors, under any transformation, but it's not useful to call them that because they refer to a different tensor in every (non-linear) frame.

But its a tensor which is attached to a coordinate system and general tensors do not have this property.

Pete

 

[mathwonk]

If interested, here is a reference I think is very scholarly on Christoffel symbols. Volume 2, of Differential Geometry, by Michael Spivak, chapter 4D, the Birth of the Riemann curvature tensor.

I have not studied it, but as has appeared in these posts, apparently the Christoffel symbols are objects defined by taking derivatives, are not strictly tensors, but a certain difference of them is so.

The interest of Spivak's treatment is that he actually went back and read Riemann's original treatment and even provides a translation of Riemann's "habilitationshrifft" in which they first appear.

In particular, these expressions are apparently due not to Christoffel but to Riemann originally. Christoffel did work on them after Riemann, Even the gamma notation used now for the connection is not Christoffels original notation, which Riemann remarks on.

 

[sal]

If interested, here is a reference I think is very scholarly on Christoffel symbols. Volume 2, of Differential Geometry, by Michael Spivak, chapter 4D, the Birth of the Riemann curvature tensor.

I'm still trying to get through Volume 1 of that monster... and volumes 2 through 5 glare at me and make me feel guilty every time I walk into the spare room...

In particular, these expressions are apparently due not to Christoffel but to Riemann originally. Christoffel did work on them after Riemann, Even the gamma notation used now for the connection is not Christoffels original notation, which Riemann remarks on.

I always assumed that the gammas were chosen to stand in for Latin C's. Of course, they're the wrong letter -- they happen to be third in the Greek alphabet, and we tend to think of gamma as being like C, but it's really G; the greek X is C...

 

[mathwonk]

So what? obviously volume 1 is not a prerequisite for understanding volume 2, since Riemann did not know any of those modern definitions in volume 1, and volume 2 is his stuff. volume 1 is just to scare off the faint of heart, or the obsessive consecutive reader.

 

[mathwonk]

Well more accurately, I guess it functions to discourage people with jobs. I'm one of those lucky people who gets paid to go to the library and read books I find interesting. Of course the downside is I still can't afford the last three volumes of Spivak!

But really Sal, I think volume 1 is Mike's version of a very through treatment of modern differentiable manifolds and deRham theory, and then volume 2 is the actual classical differential geometry. So I bet you can start reading right at the beginning of chapter 4D, of vol 2, knowing only partial derivatives, and get the gist of it.

best regards,

roy

 

[sal]

But really Sal, I think volume 1 is Mike's version of a very through treatment of modern differentiable manifolds and deRham theory, and then volume 2 is the actual classical differential geometry. So I bet you can start reading right at the beginning of chapter 4D, of vol 2, knowing only partial derivatives, and get the gist of it.

Thanks for the hint, and the encouragement. I'll definitely look at it, when I get out from under the sand pile I'm stuck in at work just now.

I find Spivak's style very engaging but I often find his explanations less than enlightening, and so I stalled in Volume 1 ... and I'm kind of compulsive about reading things in order, it's true. But after spending my free time for the past year softening my brain with tensor calculus I suspect I'll find the text less daunting when I pick it up again, anyway (or at least I hope I will).

 

[humanino]

In the book "A first course in general relativity" by Bernard F. Schutz, on page 143, chapter 5, section 5.5, the tensorial nature of the Christoffel symbols are explained as the components of a set of tensors but there is no single tensor whose components are C^mu_ab.

And the combination V^b_,a + V^mu C^b_mu a is explained as a component of a single tensor.

If I understand correctly, the Christoffel symbols are tensors in the respect that they're linear maps from tangent vectors and dual vectors to the reals. Christoffel symbols don't transform like tensors though.

I think the paradox has been faded away since this post and clarified later.

I would like to point that , for us physicists, these symbols are just what we need to add to the partial derivative to get a covariant derivative. If they were tensors, put them in the LHS with covariant derivative. This imply that the RHS, which is the partial derivative, should be covariant too, and why bother at all with correcting it to make it covriant !? This is indeed confusing.

Wald's book is an excellent one, but requires much attention, especially pay attention to greek and latin indices. The formal indice formalism of Penrose is very powerful.

 

[mathwonk]

I just learned by reading spivak, that riemann invented the curvature tensor before tensors were invented. i always like stories ike that.

the point being it does not matter what a "tensor" is, what matters is knowing what you are doing.