The Christoffel symbols at the origin -- Why zero?

[GR191511]

"the christoffel symbols are all zero at the origin of a local inertial frame" Why must it be at the origin? If it is not?Thanks!

 

[Dale]

The Christoffel symbols are 0 throughout a global inertial frame. A local inertial frame only matches a global inertial frame at the origin.

 

[Ibix]

Hm. Agree that you can choose coordinates so that the Christoffel symbols vanish at an event in curved spacetime and you do so in such a way that they vanish at your event of interest. And I agree that they vanish everywhere in a global inertial frame. But is it necessary that the origin of the inertial frame is at the event of interest? I mean, I can't see why you'd want to choose a different location (I think it just adds terms to everything), but could you?

That said, the author has probably specified that the origin of coordinates in the inertial frame is at the event of interest, whether it's necessary or not.

 

[Dale]

But is it necessary that the origin of the inertial frame is at the event of interest? I mean, I can't see why you'd want to choose a different location (I think it just adds terms to everything), but could you?

Yes, you are certainly right about that.

 

[haushofer]

"the christoffel symbols are all zero at the origin of a local inertial frame" Why must it be at the origin? If it is not?Thanks!

It's not necessary, but often it's convenient. You can always perform a constant translation as general coordinate transformation which doesn't change your connection coefficients; just look at their transformation. So if they're zero at the origin of your local inertial frame, they also will be zero in your translated local inertial frame.

Actually, that a constant translation doesn't change the connection coefficients is just part of the fact that connection coefficients are tensors under the Poincaré group.

 

[PeterDonis]

connection coefficients are tensors

No, they aren't. If they were, it would be impossible to make them all vanish at a point by a coordinate transformation. You can't make a non-vanishing tensor vanish by a coordinate transformation.

 

[haushofer]

No, they aren't. If they were, it would be impossible to make them all vanish at a point by a coordinate transformation. You can't make a non-vanishing tensor vanish by a coordinate transformation.

Yes they are. They transform as tensors under the group of Poincaré transformations. That's an easy check to make; the inhomogenous term of the transformation disappears under such transformations (although these Poincare transformations are of course not the most general transformations to do this). So it is completely justified to say that "the connection coefficients transform as a tensor under the group of Poincare transformations". Just as e.g. Newton's second law is a tensor equation under the group of Galilei-transformations but not under accelerations or, more generally, general coordinate transformations (this is the whole idea of inertial observers being equivalent, but accelerating observers are not).

So whether something is a tensor depends on the group of coordinate transformations you specify. By "coordinate transformation" you (in your post I quote here above) implicitly mean "GENERAL coordinate transformation", but as far as I know, that implicit "GENERAL" is not part of the definition of a tensor. At least not how I used it during my graduate or PhD.

To clarify: I'd call e.g. the Klein-Gordon equation if one uses just partial derivatives instead of covariant ones (as you encounter it in everey QFT-book) also a tensor equation, because it transforms as a tensor under the Poincare group. I'd say it's pretty pedantic to point at the Klein-Gordon equation in a QFT-textbook and claim it's not a tensor equation "because it doesn't contain covariant derivatives". Everyone who is not autistic understands that in a QFT book the group of Poincare transformations (and not GCT's) is implied when you mention the word "tensor". That's why I think it's a bit silly you only partially quote my post, removing the addendum "under the Poincaré group".

 

[PeterDonis]

as far as I know, that implicit "GENERAL" is not part of the definition of a tensor

It is in GR, since in GR the laws of physics must be invariant under general coordinate transformations.

Even in the context of SR, where the laws only have to be invariant under Poincare transformations, it seems to me that your statement cannot be correct as it stands, since, for example, I can take a nonzero Christoffel symbol at a particular point in spherical polar coordinates and make it vanish by transforming to Cartesian coordinates. So your definition of "Poincare transformations" would have to only include Cartesian coordinates, which makes your claim trivial since in flat spacetime all Christoffel symbols vanish in such coordinates.

At least not how I used it during my graduate or PhD.

Can you give a reference that describes and justifies the definition of tensors you are using? I ask because it is not one I have seen in any textbook on relativity that I have read.

 

[PeterDonis]

I'd say it's pretty pedantic to point at the Klein-Gordon equation in a QFT-textbook and claim it's not a tensor equation "because it doesn't contain covariant derivatives".

The gradient of a scalar field, which is what appears in the Klein-Gordon equation, is a tensor without any Christoffel symbols because the scalar field has no indexes to "correct" with Christoffel symbols in a covariant derivative. So for scalar fields the partial derivative is the covariant derivative.

In any case, your claim was not that the partial derivative of a scalar field is a tensor. It was that the Christoffel symbol is a tensor "under the Poincare group". As I pointed out in post #8, it seems to me that that claim can only be true if we restrict to Cartesian coordinates, which makes the claim true but trivial.

 

[strangerep]

I can take a nonzero Christoffel symbol at a particular point in spherical polar coordinates and make it vanish by transforming to Cartesian coordinates. So your definition of "Poincare transformations" would have to only include Cartesian coordinates, [...]

Hmm. But does the Jacobian transformation matrix "$M$" for Cartesian $\leftrightarrow$ Spherical Polar satisfy $M\,M^T = 1$? (Doesn't look like it to me. -- I could type it out here if you don't have it at hand.)

 

[PeterDonis]

does the Jacobian transformation matrix "$M$" for Cartesian $\leftrightarrow$ Spherical Polar satisfy $M\,M^T = 1$? (Doesn't look like it to me. -- I could type it out here if you don't have it at hand.)

What would it mean if it didn't?

 

[strangerep]

What would it mean if it didn't?

I was trying to see how going between Cartesian and SphPol could possibly be a Lorentz transformation.

 

[PeterDonis]

I was trying to see how going between Cartesian and SphPol could possibly be a Lorentz transformation.

I wasn't claiming that it was. Indeed, my point was, as I said in posts #8 and #9, that, as far as I can tell, "a tensor under the Poincare group" requires restricting to Cartesian coordinates, which makes the claim that the Christoffel symbols are a "tensor" under such transformations trivial, since they vanish. And if we reject that and allow anything other than Cartesian coordinates, then the claim is obviously false.

 

[strangerep]

Suppose we start off in spherical polar coords. Then a 3D rotation transformation is basically a translation in angle space, i.e., of the form $$r' = r ~,~~~~~ \theta' = \theta + a ~,~~~~~ \phi' = \phi + b ~,$$where $a,b$ are constants. The Jacobian matrix for this is $1$, so the 2nd derivative (which is what appears in the $\Gamma$ transformation formula) is 0. Unless I'm missing something, this means $\Gamma$ transforms like a tensor under 3D rotations if we're working in spherical polar coords.

 

[PeterDonis]

Unless I'm missing something, this means transforms like a tensor under 3D rotations if we're working in spherical polar coords.

You're missing the point.

Whether or not the transformation from spherical polar to Cartesian coordinates on 4D flat spacetime is a "Poincare transformation" or not, it is a valid general coordinate transformation, which makes formerly non-vanishing Christoffel symbols vanish. And in every GR textbook I have read, this is the sort of argument that is given to show why Christoffel symbols are not tensors.

The claim by @haushofer that I originally responded to was that Christoffel symbols are "tensors under the Poincare group". But what does this mean? The fact that non-vanishing Christoffel symbols in spherical polar coordinates can be made to vanish when transforming to Cartesian coordinates means that there are only two possibilities:

(1) The transformation from spherical polar to Cartesian coordinates on flat spacetime is a valid Poincare transformation. Your post makes an argument that it should be, but your argument only considers spatial rotations. The original transformation that was proposed in this thread was a translation. And boosts are also part of the Poincare group. So you would have to extend your analysis to those types of transformations to complete your argument.

But let's suppose your argument, suitably extended and completed, is valid. That means that Christoffel symbols are not tensors under Poincare transformations, since the transformation from spherical polar to Cartesian coordinates is a Poincare transformation and it makes previously non-vanishing Christoffel symbols vanish.

(2) The transformation from spherical polar to Cartesian coordinates on flat spacetime is not a valid Poincare transformation. Your argument, if correct, would contradict this, but let's suppose that, for whatever reason, your argument cannot be extended to translations or boosts.

That would mean that the only kind of coordinates that would be included in "Poincare transformations" would be Cartesian coordinates. And that would meant that the claim that Christoffel symbols are "tensors under Poincare transformations" would be true only because it was trivial--because in Cartesian coordinates in flat spacetime all Christoffel symbols vanish.

I see no other possibilities, and @haushofer has not responded to this argument. Can you?

 

[strangerep]

You're missing the point.

I think you might have misunderstood my posts above, and built a strawman as a result.
I'll make an attempt to clarify...

Whether or not the transformation from spherical polar to Cartesian coordinates on 4D flat spacetime is a "Poincare transformation" or not, it is a valid general coordinate transformation, which makes formerly non-vanishing Christoffel symbols vanish. And in every GR textbook I have read, this is the sort of argument that is given to show why Christoffel symbols are not tensors.

Yes, they are not tensors under the full group of general coordinate transformations.

The claim by @haushofer that I originally responded to was that Christoffel symbols are "tensors under the Poincare group". But what does this mean?

I'm surprised you need to ask that. Was it a rhetorical question?

The fact that non-vanishing Christoffel symbols in spherical polar coordinates can be made to vanish when transforming to Cartesian coordinates means that there are only two possibilities:

(1) The transformation from spherical polar to Cartesian coordinates on flat spacetime is a valid Poincare transformation. Your post makes an argument that it should be, [...]

No, that is not my argument. There has been a misunderstanding.

(2) The transformation from spherical polar to Cartesian coordinates on flat spacetime is not a valid Poincare transformation.

That is my understanding. Am I wrong about that?

 

[PeterDonis]

I think you might have misunderstood my posts above, and built a strawman as a result.

I think you have somehow missed the fact that this whole subthread was started by a response by me to a claim by @haushofer which I will quote again:

connection coefficients are tensors under the Poincaré group.

I am trying to make sense of that claim.

they are not tensors under the full group of general coordinate transformations.

Everyone in this discussion agrees with that. That's not the point at issue.

I'm surprised you need to ask that. Was it a rhetorical question?

No, it was a genuine question. I'm surprised you didn't realize that since all of my previous posts in this thread should have made it obvious.

I can only make sense of the claim in two ways, which I described in post #15. One way makes the claim false; the other makes it trivial. Since I don't think @haushofer meant his claim to be either false or trivial, I am trying to find another way to make sense of it, but I have not found one.

If your understanding, which you have now clarified, is that the transformation from spherical polar to Cartesian coordinates is not a valid Poincare transformation, then you are saying the only way to make sense of @haushofer's claim is in the second way I described in post #15, which makes it trivial. So apparently you also can't find any way to make sense of his claim that makes it not trivial. At least, none of your posts so far have shown one.

 

[PeterDonis]

I'm surprised you need to ask that. Was it a rhetorical question?

Since apparently what that claim means is obvious to you, what do you think it means?

 

[strangerep]

I think you have somehow missed the fact that this whole subthread was started by a response by me to a claim by @haushofer which I will quote again:
[...]

No, I did realize that the whole time. I was just having an extremely hard time understanding how or why or what you didn't understand about haushofer's claim. It happens.

I'll respond to your question about what I "think it means" tomorrow (my time), if haushofer or someone else doesn't clarify it first.

 

[martinbn]

I find this terminology very unusual. "This is a tensor under a given group of transformations, or this transforms like a tensor under so and so transformations." Are there any textbooks that use this terminology? I am ready to bet a lot that there are no maths textbooks that use such terminology.

 

[martinbn]

No, I did realize that the whole time. I was just having an extremely hard time understanding how or why or what you didn't understand about haushofer's claim. It happens.

I think he understands haushofer's calim, he just disagrees with the use of terminology.

 

[renormalize]

Actually, that a constant translation doesn't change the connection coefficients is just part of the fact that connection coefficients are tensors under the Poincaré group.

I believe haushofer refers here to the following argument:

In any spacetime, flat or curved, the general change of coordinates $x^{\sigma}\rightarrow x^{\prime\sigma}(x)$ transforms the Christoffel connection $\Gamma_{\alpha\mu}^{\sigma}$ according to: $$\Gamma_{\alpha\mu}^{\prime\sigma}=\frac{\partial x^{\prime\sigma}}{\partial x^{\beta}}\frac{\partial x^{\rho}}{\partial x^{\prime\mu}}\frac{\partial x^{\tau}}{\partial x^{\prime\alpha}}\Gamma_{\tau\rho}^{\beta}+\frac{\partial x^{\prime\sigma}}{\partial x^{\beta}}\frac{\partial^{2}x^{\beta}}{\partial x^{\prime\alpha}\partial x^{\prime\mu}}\\=\frac{\partial x^{\prime\sigma}}{\partial x^{\beta}}\frac{\partial x^{\rho}}{\partial x^{\prime\mu}}\frac{\partial x^{\tau}}{\partial x^{\prime\alpha}}\Gamma_{\tau\rho}^{\beta}-\frac{\partial x^{\tau}}{\partial x^{\prime\alpha}}\frac{\partial^{2}x^{\prime\sigma}}{\partial x^{\tau}\partial x^{\beta}}\frac{\partial x^{\beta}}{\partial x^{\prime\mu}}$$ It's clear from the last two terms that $\Gamma$ transforms purely as a tensor under the subgroup of linear coordinate transformations, i.e., those satisfying $$\frac{\partial^{2}x^{\prime\sigma}}{\partial x^{\tau}\partial x^{\beta}}=0$$ The solution of this equation is $$x^{\prime\sigma}(x)=M{}^{\sigma}{}_{\tau}\:x^{\tau}+\xi^{\sigma}$$ where $M$ is an arbitrary invertible constant matrix and $\xi$ is any constant vector. In particular, choosing the matrix $M\in SO(1,3)$ demonstrates that $\Gamma$ is always a tensor under the action of the Poincaré group for any 4D Riemannian spacetime.

But is there any utility in these Poincaré transformations, especially in a curved spacetime? They might prove useful in rectangular-coordinates $(t,x,y,z)$ for something like gravitational plane waves, where $M$ and $\xi$ could be chosen to boost the momentum vector, rotate polarization directions and move the origin, thereby inducing a tensor-transformation of the connection coefficients. But what about $(t,r,\theta,\phi)$ in Schwarzschild spacetime? The same choice of $M$ and $\xi$ would scramble those coordinates in a completely nonsensical way. And first converting that spacetime to rectangular coordinates, Poincaré transforming, and then converting back would be an algebraic nightmare. So I remain skeptical.

 

[haushofer]

It is in GR, since in GR the laws of physics must be invariant under general coordinate transformations.

Even in the context of SR, where the laws only have to be invariant under Poincare transformations, it seems to me that your statement cannot be correct as it stands, since, for example, I can take a nonzero Christoffel symbol at a particular point in spherical polar coordinates and make it vanish by transforming to Cartesian coordinates. So your definition of "Poincare transformations" would have to only include Cartesian coordinates, which makes your claim trivial since in flat spacetime all Christoffel symbols vanish in such coordinates.

[1] Well, yes, but in GR one often gauge-fixes the coordinates and restrict oneself to a subgroup of gct's. One can then perfectly ask the question whether a field transforms as a tensor under this subgroup. My own research was a lot about using the Newton-Cartan formalism, where you constantly gauge fix coordinates and see how the resulting transformation laws change. We used this terminology all the time.

[2] Well, yes, that's a good point, but I wouldn't call a transformation from Cartesian to polar coordinates a Poincare transformation. Of course, the Christoffel symbols don't transform tensorially under such a transformation. A Poincare transformation is a Lorentz transformation and/or a spacetime translation. So if you've picked Cartesian coordinates, the christoffel symbol is zero; every Poincare transformation keeps it zero. If you've picked spherical coordinates, the christoffel symbol is non-zero, and stays that way under Poincare transformation.

[3] I have to check, but I just use the philosophy "if it quacks like a duck and walks like a duck, let's call it a duck". So if something transforms tensorially (i.e. multilinearly) under a subgroup of gct's, let's call it a tensor under this subgroup.

Renormalize indeed shows what I mean.

 

[haushofer]

I find this terminology very unusual. "This is a tensor under a given group of transformations, or this transforms like a tensor under so and so transformations." Are there any textbooks that use this terminology? I am ready to bet a lot that there are no maths textbooks that use such terminology.

I'm not sure. But even then, someone like Wald calls the Christoffel symbol "a tensor field associated with the covariant derivative" after eqn. 3.1.15. Just to say that "what textbooks say" is not always a good argument; I think many of us would say Wald's statement there can be confusing.

As I said, I'm perfectly fine with people who think"my terminology" is unusal; it's just how we used it during my PhD in my particular field of research. I think it's a helpful way to think about it, because it naturally arises whenever one restricts himself to certain subgroups of GCT's and checks how tensors or other fields transform under this restriction. E.g., for me the expression "all inertial observers are equivalent" is similar to "Newton's second law only transforms tensorially under the restricted group of Galilei-transformations." But if other people find that confusing, be my guest :)

 

[Cepatiti]

I was taught in GR that a tensor ought to be a "geometrical object". Hence in particular, it cannot be annihilated by a mere coordinate transformation.

"if it quacks like a duck and walks like a duck, let's call it a duck"

What would you say about pseudovectors, then? They are not geometrical objects, but still transform like vectors under certain transformations. Would you call these vectors, too?

 

[haushofer]

I was taught in GR that a tensor ought to be a "geometrical object". Hence in particular, it cannot be annihilated by a mere coordinate transformation.
What would you say about pseudovectors, then? They are not geometrical objects, but still transform like vectors under certain transformations. Would you call these vectors, too?

I'm not sure what you mean exactly by "geometrical object". If it can be considered to be a multilinear mapping from a dual space to the reals and if the components transform multilinearly according to the very definition under a group of transformations, I'd call it a vector (tensor) under these transformations.

Look, I'm not saying something silly here, as far as I know. I just say that if one talks about tensors, one should specify the group of transformations one considers. Often this happens implicitly, as in GR or quantum field theory.

 

[PeterDonis]

in GR one often gauge-fixes the coordinates and restrict oneself to a subgroup of gct's. One can then perfectly ask the question whether a field transforms as a tensor under this subgroup.

Given what you mean by this, yes, one can. But if that subgroup is the subgroup of Poincare transformations between Cartesian coordinate systems, then the claim that the Christoffel symbols are a tensor under this subgroup is trivial, since you're "transforming" a tensor that vanishes. It appears that that is indeed what you meant.

I wouldn't call a transformation from Cartesian to polar coordinates a Poincare transformation.

Yes, and that just reinforces what I said above, since the "Poincare transformations" you speak of are restricted to transformations between Cartesian coordinate charts.

if something transforms tensorially (i.e. multilinearly) under a subgroup of gct's, let's call it a tensor under this subgroup.

Even if it vanishes, apparently.

But actually, now that you have confirmed what you meant, there is more. I'll address that in a follow-up post.

 

[haushofer]

(2) The transformation from spherical polar to Cartesian coordinates on flat spacetime is not a valid Poincare transformation. Your argument, if correct, would contradict this, but let's suppose that, for whatever reason, your argument cannot be extended to translations or boosts.

That would mean that the only kind of coordinates that would be included in "Poincare transformations" would be Cartesian coordinates. And that would meant that the claim that Christoffel symbols are "tensors under Poincare transformations" would be true only because it was trivial--because in Cartesian coordinates in flat spacetime all Christoffel symbols vanish.

Your conclusion does not follow. How I see it is that Poincaré transformations connect different inertial coordinate systems (you could call such a coordinate system "an observer"). Within such a coordinate system (or: "observer", if you like), you can choose Cartesian or cylindrical or spherical or whatever spatial coordinates you want. But such a transformation is not a Poincare transformation. If I interpret such a transformation with a mental picture in my mind, I just see the same observer pondering how to label her events in space. If that observer picture confuses you, just leave it out; that's not the point.

So once you've chosen your spatial coordinates (Cartesian, spherical,...), you can calculate your christoffel symbol. If you then apply a Poincaré transformation (Lorentz transformation, rotation or spacetime translation), this symbol you've calculated transforms tensorially (because the inhomogeneous term in the transformation drops out). If you want the symbol to become zero again, you have to step out of the group of Poincaré transformations. One way is to choose spherical coordinates (a mere spatial coordinate transformation); another way is to start accelerating (in which you involve time in your coordinate transformations).

 

[haushofer]

Given what you mean by this, yes, one can. But if that subgroup is the subgroup of Poincare transformations between Cartesian coordinate systems, then the claim that the Christoffel symbols are a tensor under this subgroup is trivial, since you're "transforming" a tensor that vanishes. It appears that that is indeed what you meant.Yes, and that just reinforces what I said above, since the "Poincare transformations" you speak of are restricted to transformations between Cartesian coordinate charts.Even if it vanishes, apparently.

But actually, now that you have confirmed what you meant, there is more. I'll address that in a follow-up post.

[1] I never said it's not trivial. Within the choice of cartesian coordinates you effectively transform a "zero tensor" if you apply poincare transformations on the christoffel symbol, just like a rotation of the zero vector doesn't magically make your tensor become non-zero.

[2] Or spherical coordinates. Or whatever fancy spatial coordinate system you want. Maybe I missed something, but since when do we call the transformation from cartesian to spherical coordinates possibly a poincare transformation?

[3] Ok. :)

 

[PeterDonis]

is it necessary that the origin of the inertial frame is at the event of interest? I mean, I can't see why you'd want to choose a different location (I think it just adds terms to everything), but could you?

Careful. The posts in this thread have been conflating two different things.

Strictly speaking, a "local inertial frame" is a frame on the tangent space at a point, and it describes, not displacements, but tangent vectors at that point. So it doesn't even make sense to ask which "location" in the frame corresponds to the point you picked; none of them do since the frame is not describing "locations", it's describing vectors. To the extent you can label any "point" in the frame as corresponding to the chosen point in the spacetime, it would have to be the origin, since that is the "base" of all the vectors.

If you are talking about placing a particular event at a particular point in "local coordinates", the "local coordinates" would be Riemann normal coordinates, which are not coordinates on the tangent space, they are coordinates on the spacetime. Those coordinates are constructed such that the event you choose is at the origin and the Christoffel symbols vanish at that point (and only at that point). I have never seen a construction that places the chosen event anywhere except at the origin, and I'm not sure that such a construction would be as simple as has been implied.

You can always perform a constant translation as general coordinate transformation which doesn't change your connection coefficients

This is true in an inertial chart on flat spacetime in Cartesian coordinates. But as the above makes clear, that is not what we are talking about in this thread. We are talking about some kind of transformation between Riemann normal coordinate charts that would move a chosen event where the Christoffel symbols vanish from the origin to some other non-origin coordinates. Whatever that transformation is, it's not a Poincare transformation.

 

[haushofer]

Just another example: within the Newtonian limit of General Relativity, one has gauge-fixed the coordinate system in such a way to obtain Newton's second law with the gravitational force in the following form:

$$\ddot{x}^i + \Gamma^i_{00} = 0 \ \ \ \ \ (1)$$

With the group of Galilei transformations, this equation transforms as a tensor (or: vector equation). Physically, this means no Galilei transformation will introduce a fictitious force. But if you e.g. apply a time-dependent rotation,

$$x^{'i} = R^i{}_j (t)x^j \ \ \ with \ \ \ R^i{}_j (t)\ \ \in SO(3) \ \ \forall t$$

then eqn.(1) does not transform as a tensor anymore, but inhomogenously. The inhomogeneous terms are the fictitious forces. I don't see the problem to speak about eqn.(1) this way, and why it would be so bad to call eqn.(1) a tensor equation under the group of Galilei transformations.

 

[PeterDonis]

Poincaré transformations connect different inertial coordinate systems

Yes. But in a curved spacetime, these inertial coordinate systems are on the tangent space at a particular point, not on the spacetime itself. See post #30. So "translations" in the usual sense of that term make no sense. A "translation" in the context of this thread would be, as I said in post #30, a transformation between Riemann normal charts on a local patch of the spacetime, which is not a Poincare transformation.

 

[PeterDonis]

once you've chosen your spatial coordinates (Cartesian, spherical,...), you can calculate your christoffel symbol. If you then apply a Poincaré transformation (Lorentz transformation, rotation or spacetime translation), this symbol you've calculated transforms tensorially (because the inhomogeneous term in the transformation drops out).

Is this true for translations in spherical coordinates? I don't think such a transformation is linear.

 

[haushofer]

Yes. But in a curved spacetime, these inertial coordinate systems are on the tangent space at a particular point, not on the spacetime itself. See post #30. So "translations" in the usual sense of that term make no sense. A "translation" in the context of this thread would be, as I said in post #30, a transformation between Riemann normal charts on a local patch of the spacetime, which is not a Poincare transformation.

I agree.

 

[Dale]

So once you've chosen your spatial coordinates (Cartesian, spherical,...), you can calculate your christoffel symbol. If you then apply a Poincaré transformation (Lorentz transformation, rotation or spacetime translation), this symbol you've calculated transforms tensorially (because the inhomogeneous term in the transformation drops out).

Hmm, is that true? I have never tried that, and I am not even sure exactly how to do it, but I am rather skeptical about it.