User talk:Differential 0celo7

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Please come back

I see that you have only edited Wikipedia once since our disagreement. I hope that you were not discouraged by my opposition to the section you added to the article Lagrangian. If you were discouraged by me, I apologize.

Upon re-reading your message to me, I see some points to which I should have responded earlier.

• You said that action (and thus the Lagrangian) is unphysical. I do not think that we know that. And the fact that action is quantized in quantum mechanics suggests otherwise.
• I very much prefer the coordinate+index notation. I think that the coordinate-free and index-free notation used by some in differential geometry conceals what is really going on. I have to think much harder to understand it.
• You said that I objected "that the integral is not in the form ∫f(x)dx.". I do not remember making any such objection.
• The action integral has coordinate time as its variable of integration (or one of several variables of integration) in all other cases in this article. So it should be so in this case also for consistency and also to allow adding Lagrangians of components to get Lagrangians of systems (not all components are likely to have the same proper time).
• You said "your Lagrangian is not invariant under diffeomorphisms.". But it is invariant, since

$\int -mc^{2}{\frac {d\tau }{dt}}dt=\int -mc^{2}d\tau =\int -mc^{2}{\frac {d\tau }{d\lambda }}d\lambda \,.$

• You said "The square root makes this very hard to quantize.". Why? And even if it is difficult, that does not change the fact that it is the correct Lagrangian.
• As I pointed out before, your Lagrangian fails to handle massless particles also.
• Your comment about an auxilliary bosonic field went completely over my head. Please explain that in more detail.
• Whether one interprets your action integral as

$\int {\frac {1}{2}}g_{\mu \nu }{\frac {dx^{\mu }}{d\tau }}{\frac {dx^{\mu }}{d\tau }}d\tau ={\frac {-c^{2}}{2}}\int d\tau$

or as

$\int {\frac {1}{2}}g_{\mu \nu }{\frac {dx^{\mu }}{dt}}{\frac {dx^{\mu }}{dt}}dt={\frac {-c^{2}}{2}}\int {\left({\frac {d\tau }{dt}}\right)}^{2}dt\,,$

it makes no sense to do it this way. JRSpriggs (talk) 17:40, 7 January 2015 (UTC)

I was discouraged, because this argument is rather arbitrary and I'm not that attached to my section. I honestly do not remember the original argument. Could you please restate precisely what you find wrong with the section I added? I feel like we're arguing about random things at this point.

But today I have some time, so I'll formulate a response.

First I'll respond to your question about using $\sigma =\xi ^{0}$ as a parameter for light. Let p be a point on a manifold M. We know that in SR, globally, we have

${\frac {d^{2}x^{\alpha }}{d\tau ^{2}}}=0$

Mathematically, the equivalence principle demands that there is a coordinate system such that

$\left.{\frac {d^{2}\xi ^{\mu }}{d\tau ^{2}}}\right|_{p}=0$

What I mean by this is that ξ is only locally inertial. There should in general exist a smooth extension to the rest of the manifold, but the SR equation will not hold anywhere else.

I would like to state, once and for all, that your Lagrangian is correct. So is mine. The Lagrangian is not unique. The equations of motion are. I know how to reproduce the geodesic equation from the square root Lagrangian. This calculation is performed in both of Zee's books, Becker, Becker & Schwarz, Wald, Weinberg, and Straumann. Probably in Cahill too. Perhaps even in Polchinski! I'm not arguing against the square root! The problem I had was a rather semantical one. By parameterizing by time and deriving the equations of motion, you got (F=0)

${\frac {dp_{\sigma }}{dt}}=\Gamma _{\alpha \sigma }^{\rho }p_{\rho }{\frac {dx^{\alpha }}{dt}},\quad p^{\mu }=m{\frac {dx^{\mu }}{d\tau }}$

which is totally the geodesic equation! However a geodesic is, in its purest form, a curve along which the tangent is kept autoparallel. Here is an instance where coordinate-free notation shines. (I can also give an example where it is crap: It is not possible to contract tensors in coordinate-free.) Let t be the parameter along the geodesic (not coordinate time, I'm using mathematical notation here, not phyical) and $u=d/dt$ be the tangent, then

$\nabla _{u}u=0$

This is the definition of a curve whose tangent vector is autoparallel. We of course also have the length functional

$L=\int {\sqrt {\langle u,u\rangle }}dt$

Suppose we choose to parameterize with the length of the curve. Then the square root must be unity. This gives us a nice constraint on the curve (in coordinates)

$g_{ij}{\frac {dx^{i}}{dl}}{\frac {dx^{j}}{dl}}=1$

where l is the arc length parameter. You can write out the length functional in coordinates, do variation, and derive an equation

${\frac {d^{2}x^{i}}{dt^{2}}}+\Gamma _{\;jk}^{i}{\frac {dx^{j}}{dt}}{\frac {dx^{k}}{dt}}=0$

where

$\Gamma _{\;jk}^{i}={\frac {1}{2}}g^{il}(\partial _{j}g_{kl}+\partial _{k}g_{jl}-\partial _{l}g_{jk})$

We call this the Christoffel symbol. But how do we know that it is the Christoffel symbol? Well, there is also a three-indexed object in the covariant derivative. This term can be calculated by a standard argument: the covariant derivative of the metric is zero. We write out the covariant derivative in three different ways and add a specific combination and also suppose that the bottom two indices are symmetric. Then we find the usual definition. As it turns out, that is a lot supposing. In fact, the supposing is the definition of a Riemannian manifold. Thus, if and only if the manifold is Riemannian, are geodesics the curves of the shortest arc length. When we write out the definition of a geodesic starting from the coordinate free notation in index notation, the first term will give the coordinate with two derivatives with respect to the parameter. In your momentum equation, you have one derivative with respect to the proper time and one with respect to coordinate time. You have two parameters! Of course the chain rule eliminates the coordinate time parameter. It's not wrong the way you wrote it, but conceptually confusing and nonstandard. This is why I don't like parameterizing with coordinate time.

When I said that objection with the integral, I was being hypothetical. Not many mathematically rigorous folk like to see integrals with differentials under a square root.

Your action is invariant, of course, but your Lagrangian is not. This is because the combination (going back to physics notation)

$g_{\mu \nu }{\frac {dx^{\mu }}{dt}}{\frac {dx^{\nu }}{dt}}$

is not a scalar. t has a nontrivial transformation. It is generally highly desirable that Lagrangians are scalars.

I'm not sure what you mean in your comment about systems. Could you please elaborate?

I don't know how much quantum field theory you know. The quantum mechanics of a particle is governed by a path integral over all possible paths. For an action S and particle path X we write the transition amplitude as

$\int DX\,\exp(iS[X])$

So let us put in the action for some arbitrary affine parameterization:

$\int DX\,\exp \left(i\int {\sqrt {-g_{\mu \nu }{\frac {dx^{\mu }}{d\lambda }}{\frac {dx^{\nu }}{d\lambda }}}}\,d\lambda \right)$

This integral is quite horrible. If we use the action I derived earlier using the auxiliary field (more on that later), the path integral has a standard quadratic form.

I'm not sure why quantization makes the action physical. Once again, I don't know how much field theory you know, but it is generally possible to add a divergence or constant to a Lagrangian. The equations of motion are not affected and these terms produce constant factors in path integrals. We only concern ourselves with ratios of path integrals, so those terms do not matter. Please explain further if you find my comment unsatisfactory.

An auxiliary field is defined as a field without kinetic energy. This means that auxiliary fields can be integrated out of the path integral. The overall factor is canceled because we are only concerned with ratios. There are a wealth of examples from condensed matter physics (Hubbard-Stratonovitch transformation), quantum field theory (certain treatments of Yang-Mills), supersymmetry (the most basic superymmetric action) and string theory (the derivation of the Polyakov action contains an auxiliary metric).

I have to ask this. You say you've been studying this for a while, but since you disagree with my action, this leads me to believe you have not read Wald or Carroll. I dare say both are standard textbooks. Have you read them?

Differential 0celo7 (talk) 21:51, 7 January 2015 (UTC)

It is not clear to me why you think that having a scalar Lagrangian is so desirable. I think that it is more in keeping with the spirit of relativity to allow arbitrary smooth transformations of the variable of integration — spacetime is then parameterized by x⁰, x¹, x², x³ and the particle's trajectory is parameterized by a fifth coordinate t. Then one can later choose t to be the same as: coordinate time x⁰ or the proper-time τ or the longitude (in spherical coordinates) x³ or whatever is convenient. If we do it this way, the Lagrangian is covariant with respect to t. Requiring the action to be invariant with respect to changes in all five coordinates ensures that the theory be completely explicit and transparent. This is the main advantage of general covariance in relativity — we avoid hiding our assumptions and simultaneously ensure that a different observer will get a comparable result.

The problem of formulating the geodesic equation for massless particles is solved by eliminating τ from the equation as done at Geodesics in general relativity#Equivalent mathematical expression using coordinate time as parameter. Once the proper-time is gone, the geodesic equation can be extended to include photons. For a free-falling particle, we get that the ordinary acceleration is

$a^{\beta }=-\Gamma _{\alpha \mu }^{\beta }v^{\alpha }v^{\mu }+v^{\beta }\Gamma _{\alpha \mu }^{0}v^{\alpha }v^{\mu }\,.$

If you use

$\int \exp \left({\frac {i}{\hbar }}\int mg_{\mu \nu }{\frac {dx^{\mu }}{d\tau }}{\frac {dx^{\mu }}{d\tau }}{\frac {d\tau }{dt}}dt\right)DX$

for your path integral, do you not still have to deal with the square-root implicit in dτ/dt? JRSpriggs (talk) 12:46, 10 January 2015 (UTC)

Spacetime is parameterized locally by coordinates. We can only use a set of coordinates in a chart that they belong to. This is something that is obscured by component notation. The statement

$\nabla _{u}u=0$

is manifestly coordinate independent and therefore chart independent. It holds globally. But the statement

${\frac {d^{2}x^{\mu }}{d\tau ^{2}}}+\Gamma _{\;\;\nu \rho }^{\mu }{\frac {dx^{\nu }}{d\tau }}{\frac {dx^{\rho }}{d\tau }}=0$

only holds in the chart of $\{x^{\mu }\}$.

For a specific example of what I mean, we specialize to Schwarzschild spacetime with Schwarzschild coordinates $(t,r,\theta ,\phi )$. For this example, t, r and various Γs are singular at the Schwarzschild radius. The proper time, however, is perfectly well defined everywhere. So it makes sense to parameterize with an affine parameter (a parameter linearly related). And of course the coordinate invariant equation $\nabla _{u}u=0$ holds at the horizon just fine.

Also, the proper time is not a coordinate, but rather a parameter. Just like the regular time in nonrelativistic quantum mechanics. This is evidenced by the fact that we do not need to specify a chart when taking proper time derivatives.

Using coordinate time really bugs the mathematician in me. The definition of a geodesic is a curve which parallel transports its own tangent vector. The parallel transport operator is the affine connection along the tangent, $\nabla _{u}$. Thus for u to be parallel transported we must have $\nabla _{u}u=0$. I wrote a short proof showing that the only way this equation can hold in this form is if the parameterization is affine. Proof. When we expand the coordinate free equations in a chart, there will always be a term that has two derivatives with respect to the affine parameter on it. I do not know why you want to replace one of those derivatives with a coordinate time one. And turning all derivatives into coordinate time ones just makes the geodesic equation look ugly. The spirit of relativity is manifest covariance, which is not there with coordinate time parameterization.

Please explain what you mean by "covariant with respect to t" and provide a proof.

I flipped open MTW and did a bit of skimming. (Never going to read this whole thing.) Eq. 13.17 looks like the geodesic equation with affine parameterization!

Differential 0celo7 (talk) 19:53, 10 January 2015 (UTC)

I should clarify that I think that the postulates (or action integral) of a physical theory should be manifestly invariant. But I have no objection to using non-tensor quantities in derived equations. In fact, ideally the theory should be cast into the form of classical physics to make it more useable.

The trajectory of a particle may be regarded as a one-dimensional manifold embedded into the four-dimensional manifold of spacetime. That one-dimensional manifold has its own coordinate charts, and with respect to them the Lagrangian of the particle is a covariant vector.

$L={\frac {dS}{dt}}={\frac {dS}{dt^{\prime }}}\,{\frac {dt^{\prime }}{dt}}$

by the chain rule. JRSpriggs (talk) 01:39, 13 January 2015 (UTC)

Part 2 (separated for editing convenience)

I believe we have lost sight of the original argument. Please restate your objection to my post. If I recall correctly, you did not like my Lagrangian. Please provide a source which says Wald, Carroll, Zee, BBS and Polchinski are incorrect, because those texts use that action.

I'm surprised someone with a mathematics background would want to rip apart nice tensor equations and make them look like classical equations. You are definitely in the minority there. Since that is nonstandard, I guess it's pointless to argue with you on other points where you deviate from the norm.

Differential 0celo7 (talk) 01:59, 13 January 2015 (UTC)

You have never justified the section you added as it is written. You have given other arguments to support the geodesic equation, but I never said that the geodesic equation was false. Your section offers a Lagrangian which is a constant, and (as I said in my very first comment on this) that means that you can only derive an identity 0=0 from doing the variation. You "applied" the Euler-Lagrange equation, but you never showed that it is properly applied in this situation where your variable of integration τ is path-dependent. Your claim that your method handles massless particles (and mine does not) fell apart upon examination.

Your repeated arguments from authority are invalid since argument from authority is a well-known logical fallacy. Many textbooks and other supposedly reliable sources contain very serious errors. And such errors are often propagated from one source to another. I am just trying to prevent them from propagating into Wikipedia. JRSpriggs (talk) 08:28, 14 January 2015 (UTC)

If it is such a serious error, one would think that it would have been discovered by now. And this "error" is present in the mathematical literature as well.

As I have said many times, the Lagrangian is not a priori constant. This arises from the geodesic equation itself. My support of the geodesic equation was to prove this, but I guess I did not say it enough times.

You might want to check out the geodesic article. I looked at it when I wrote my section. Specifically this section. I wanted to see if you would bother to check and edit that article. I guess not. So they can use the energy functional there, but I can't here? What gives? (Also see their justification using the Holder inequality.)

Differential 0celo7 (talk) 12:29, 14 January 2015 (UTC)

• The article Geodesic is not on my watchlist and I had not read that section before. There are over a million articles in Wikipedia and I can only handle about 250 on my watchlist. So I could not assume responsibility for the accuracy of every article in Wikipedia even if I was free to edit them without worrying about getting into edit-wars with other editors.
• Notice that the section you linked is about Riemannian manifolds, not pseudo-Riemannian manifolds like our spacetime. The "length" (duration of proper time) of a time-like geodesic is maximized in our spacetime unlike Riemannian manifolds where the length of a geodesic is minimized. Thus the Cauchy–Schwarz inequality, which provides an upper bound, is not helpful in our case.
• Even if the Cauchy–Schwarz inequality were applicable, minimizing an upper bound is not the same as minimizing the thing being bounded.
• The argument in the section proves too much because it works for any parameter t whether it is affine (a linear function of proper-time) or not. So if it were a valid argument it would lead to a contradiction since the resulting "geodesic equations" would be mutually inconsistent.
• In any case, you did not make this argument based on the Cauchy–Schwarz inequality in your section to justify your Lagrangian. Nor does this have any bearing on the problems I mentioned in my last post which have to do with how you got from that Lagrangian to the geodesic equation.
• Your Lagrangian certainly is a constant. The proper-time is related to the metric by

$-c^{2}(d\tau )^{2}=g_{\mu \nu }dx^{\mu }dx^{\nu }\,.$

Dividing through by 2 dτ² gives

$-{\frac {1}{2}}c^{2}={\frac {1}{2}}g_{\mu \nu }{\frac {dx^{\mu }}{d\tau }}{\frac {dx^{\nu }}{d\tau }}$

which is your Lagrangian. JRSpriggs (talk) 16:53, 14 January 2015 (UTC)

After further thought, I realize that it is possible to rescue your derivation with some minor changes (which nonetheless make it more complicated than my derivation).

Let u be a monotone function of t (but not of x^α and its derivatives) to be specified later.

Let the 'energy function' be

$E={\frac {1}{2}}g_{\mu \nu }{\frac {dx^{\mu }}{du}}{\frac {dx^{\nu }}{du}}{\frac {du}{dt}}=-{\frac {1}{2}}c^{2}\left({\frac {d\tau }{du}}\right)^{2}{\frac {du}{dt}}\,.$

We find the maximum (of the absolute value) of the integral of E as if E were a Lagrangian

$0=\delta \int _{a}^{b}Edt$

which after some work yields

${\frac {d^{2}x^{\alpha }}{{du}^{2}}}=-\Gamma _{\beta \gamma }^{\alpha }{\frac {dx^{\beta }}{du}}{\frac {dx^{\gamma }}{du}}\,.$

According to the Cauchy–Schwarz inequality, we have

$\left(\int _{a}^{b}-mc^{2}{\frac {d\tau }{du}}\,{\frac {du}{dt}}dt\right)^{2}\leq \int _{a}^{b}-2m^{2}c^{2}{\frac {du}{dt}}dt\,\times \int _{a}^{b}Edt$

with equality when the integrands on the right side are proportional to each other. (Notice that the first integral on the right side is independent of the trajectory.) If we let u=τ along the trajectory which maximizes E, then we get that proportionality. So the square of the action

$\int _{a}^{b}Ldt=\int _{a}^{b}-mc^{2}{\frac {d\tau }{dt}}dt=\int _{a}^{b}-mc^{2}{\frac {d\tau }{du}}\,{\frac {du}{dt}}dt$

will reach a maximum when E reaches a maximum, i.e. when

${\frac {d^{2}x^{\alpha }}{{d\tau }^{2}}}=-\Gamma _{\beta \gamma }^{\alpha }{\frac {dx^{\beta }}{d\tau }}{\frac {dx^{\gamma }}{d\tau }}\,.$

I am sorry that I did not figure this out earlier. JRSpriggs (talk) 06:38, 16 January 2015 (UTC)

Still there is some circularity since which trajectory is the geodesic and the definition of proper-time along the geodesic each depend on the other. JRSpriggs (talk) 06:50, 16 January 2015 (UTC)

I haven't read your recent posts yet, but I will. In the meantime, check out this P.S.E. post. Thoughts?

Differential 0celo7 (talk) 01:15, 17 January 2015 (UTC)

I noticed that the four equations

${\frac {d^{2}x^{\alpha }}{{du}^{2}}}=-\Gamma _{\beta \gamma }^{\alpha }{\frac {dx^{\beta }}{du}}{\frac {dx^{\gamma }}{du}}$

are equivalent to the three equations (k≠0)

$a^{k}=-\Gamma _{\alpha \mu }^{k}v^{\alpha }v^{\mu }+v^{k}\Gamma _{\alpha \mu }^{0}v^{\alpha }v^{\mu }$

together with the equation

$u=\int \exp \left(\int \Gamma _{\alpha \mu }^{0}v^{\alpha }v^{\mu }dt\right)dt\,.$

So perhaps we can avoid the circular reasoning with u and also avoid the Cauchy–Schwarz inequality. JRSpriggs (talk) 03:31, 20 January 2015 (UTC)

Circular reasoning?

Differential 0celo7 (talk) 03:42, 20 January 2015 (UTC)

I recently changed Lagrangian#General relativistic test particle to include a derivation of a variant of the geodesic equation. Please check it and make any improvements which you think are needed.

Regarding "circular reasoning": when I wrote my 'fix' of your derivation (above here), I thought at first that the geodesic formula would depend on the choice of u and vice versa. Thus one could start with a version of u, get a trajectory, recalculate u, get another trajectory, recalculate u again, etc.. Question was, would this converge? Now however, I see that the choice of u is forced upon us by the time-component of the geodesic equations. Any other u (modulo a linear transformation) would result in a system of equations which are mutually contradictory, i.e. have no solution. JRSpriggs (talk) 15:34, 21 January 2015 (UTC)

Isn't Wikipedia supposed to be a source of mainstream physics? There is nothing wrong with your addition per se, but it sure isn't the mainstream way of writing the equation.

As to your 'fix'... I'm not sure what needed fixing.

What I'd really like is for you to state explicitly the reason why this debate started and for us to bring in a third party. It's obvious neither one of us is planning to budge. Differential 0celo7 (talk) 21:23, 21 January 2015 (UTC)

Thank you for taking the time to check my contribution to the article.

As for an explanation of my problem with your version, I have already explained it several times. So all I can say is that it was not convincing to me. Perhaps my standard of proof is higher than yours. Or perhaps my background lacks some supporting ideas that yours has. In any case, I feel that further attempts to explain it would not be effective. So I will sign off and wish you success in your other endeavors at Wikipedia and elsewhere. JRSpriggs (talk) 05:58, 22 January 2015 (UTC)