User talk:JRSpriggs

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Conversation on inflation?

Hi, I wonder if you remember that I told you a few years ago that I would seriously reconsider my belief in mainstream New Keynesian macroeconomic theory, if inflation exceeded 10% on an annualized basis in any two consecutive months. I believe the last 5 years has been a unique natural experiment and that it confirms the mainstream macro models that:

1. Price inflation is mainly determined by the relative 'slack' in the economy (measured by unemployment above normal, or by production below capacity) - it decelerates when there is positive slack, and accelerates when there is negative slack.
2. The quantity of money may be good as a long run guide, but it little to do with inflation in the short run, especially when the economy is near the zero nominal lower bound.
3. In a modern economy, it is very difficult for wages to fall (and hence deflation to occur), even in the face of depression levels of unemployment.

Given the inflation record of the last few years, I wonder if you would be willing to discuss your beliefs about monetary theory and 'mark to market' your beliefs? LK (talk) 03:05, 28 May 2014 (UTC)

Let me begin by pointing out that, if I recall correctly, you were the one who cut off most of our previous attempts to discuss economics. Although you were often the one to make the last argument, I only refrained from responding because your language made it clear that you did not want to hear my arguments and I did not feel comfortable continuing to argue (on your talk page) with someone under those conditions. Also, I felt that you often read things into what I said that were not there — especially assuming my total agreement with anyone who I mentioned favorably.

A "natural experiment" is not a scientific experiment and cannot serve as a reliable basis for testing hypotheses. In a scientific experiment, the experimenter controls the inputs and deliberately equalizes (or randomizes) the conditions (other than the variable being analyzed) between the control group and the experimental group. In this "natural experiment", there is no control group (indeed no experimental group other than the one case) and no equalizing of the other inputs. Thus any result could be (and probably is) caused by factors other than the variable being analyzed.

I agree that a large quantity of money does not necessarily result in price inflation; there are other considerations.

The ability of the economy to provide jobs has been impaired by taxes, regulations (especially labor law and obamacare), and government spending (raising the cost of raw materials and capital).

Bearing these in mind, if you still want to talk, go ahead. JRSpriggs (talk) 03:10, 29 May 2014 (UTC)

My memory is very bad nowadays, so I don't remember how our last convo ended. If I was brusque with you, I do apologize. I'm not sure if I agree with your assessment about natural experiments. Granted, it's not as good as a double blind random test, but it can be pretty good, and is often the best we have in the social sciences. Given that some events appear to consistently follow certain natural experiments, we can build up certain 'stylized facts' about the behavior of the economy. From this, we reject theories that contradict the stylized facts and try to construct theories that explain them. That's what economics is (at it's best). Science is not forming a set of beliefs, and then looking at the world through colored lenses, interpreting everything in a way to back up one's preconceptions (at least it shouldn't be like that). The last few years appear to have added (or solidified) a few stylized facts to our arsenal, namely 1, 2, and 3 above. LK (talk) 12:17, 30 May 2014 (UTC)

I was not asking for an apology, especially since some of the things I said may have been offensive to you. Rather, I was trying to remind you that in our conversations I have questioned some your ingrained beliefs in ways which may have been disturbing to you. You might want to look back at our previous conversations and consider that possibility before you decide to proceed.

The near impossibility of doing scientific experiments in social science does not mean that we should give more credence to unreliable methods.

I agree that it would be wrong to re-interpret every piece of evidence as necessary to construe it as confirmation of one's favorite hypotheses. However, no field of knowledge (such as economics) can be understood in isolation from the rest of our knowledge of reality. Economic theory must be reconciled with what we know about: physics, chemistry, biology, evolution, psychology, mathematics, business administration, law, politics, ethics, epistemology, metaphysics, etc..

What do you mean by a "stylized fact"? How is it different from an actual fact?

Regarding (1): Is this not just the theory of the Phillips curve which was discredited by the stagflation of the 1970s?

Regarding (2): This appears to be true.

Regarding (3): What do you mean by "modern economy" that would have any bearing on the flexibility of wage rates? If deflation is unlikely to occur, then why has the Federal Reserve been striving so desperately to avoid it?

JRSpriggs (talk) 03:40, 31 May 2014 (UTC)

Economists use the term "stylized fact" to refer to something that (almost) everyone agrees happens most of the time in a certain part of the economy, and that needs to be part of a good theory about that particular part/aspect of the economy.

• Regarding (1): Not at all! What was discredited was the naive assumption (which few actually believed) that we could 'choose' to stably remain a particular point on a stable Philips curve. What came out of it is the theory of short and long run Philips curves. There obviously is a Philips curve in the short run, but persistently being on one side of the Philips curve (e.g. unemployment persistently lower than the natural rate) will cause the short run Philips curve to shift over time.
• Regarding (3): Non-modern (especially agrarian) economies apparently could experience relatively high rates of deflation. Post WWII developed economies (with complex supply chains, long term contracts, unions, etc) apparently cannot. Instead, they fall into economic depression – associated with very low inflation or slight deflation. It is this situation that the Fed is trying to avoid.

Best, LK (talk) 03:02, 10 June 2014 (UTC)

Regarding (1): If you want the Phillips curve to be a scientific theory, you need to specify how it "shifts". That is, you need a formula for the change in the curve's parameters. If you have such, please provide a link to it.

What really bothers me about this is that you are treating the economy as if it were some kind of recalcitrant engine which needs to be tweaked to make it work correctly. You are ignoring the fact that it consists of living people. Those people have their own goals which they are trying to pursue to the extent possible given the contradictory demands of reality and the government bureaucracy. Essentially, the policies you are advocating are a kind of manipulation or deception by which you try to get people to do something which they would not knowingly and freely do.

I do not believe that there is ever any "slack" in the economy. If the owner of a factor of production chooses not to use it, he does so because it is not profitable to do so. Probably because some co-factors of production are missing or prices/regulations have shifted in a way that ruined his plans for it. Or perhaps he is holding onto it against a possible future contingency (a scarcity of that factor). If you trick him into using it (by causing inflation, lending money at unsustainably low interest rates), then it will not end well. You may get a temporary burst of economic activity, but the result will be waste.

Regarding (3): So government and the unions it protects have created a situation where instead of correcting itself quickly via rapid deflation, economic activity is choked-off when deflation would be appropriate. JRSpriggs (talk) 05:24, 12 June 2014 (UTC)

Hi, sorry for the long delay, I've been distracted by other things. Regarding the question of how the Philips curve shifts, you should know that nowadays, almost everything in mainstream economics is rigorously mathematically modeled. For an intro to New Keynesian inflation theory (with graphs but little maths), I suggest Macroeconomics[1], by Hubbard and O'Brien, Chapter 16. For the state of the art, see Interest and Prices: Foundations of a Theory of Monetary Policy[2], by Michael Woodford (warning, probably only readable to people with PhDs in economics). For something for free, here's "Keynesian Macroeconomics without the LM Curve"[3], by David Romer, about how to teach New Keynesian economics to undergraduates. Here's a more mathematical version of the same[4]. Here's an even more mathematical version about teaching the basic model[5].

About slack in the economy, it has often been observed that crowds and mobs can be irrational. E.g. Stock market bubbles; people trampling each other to death during panics; people at a stadium concert all standing, when they could all have a equally good view sitting down. Why not the same for the macroeconomy? The failure to coordinate actions can cause individually rational agents to be collectively irrational. Milton Friedman had an interesting thought experiment on this. Rationality implies that money and prices should be transparent to the real economy — it should be equivalent if bread costs $1 and hourly wages are $2, or if bread costs $5 and hourly wages are $10. An hour of work buys 2 bread. Hence monetary policy should have no effect. Equivalently, it shouldn't matter if we label the hour when the sun is highest as 11:00, 12:00, or 13:00, people rationally should behave the same, changing activities depending on how bright it is. Hence, daylight savings time should have no effect. However, experience proves otherwise — monetary policy and daylight savings time change behavior. Why? Coordination failure. Given long enough, people will adapt no matter what the nominal numbers are, but in the short run, changing nominal numbers have a real effect.

Lastly, I hope you recognize that your (3) is akin to a statement of religious belief. LK (talk) 11:39, 27 June 2014 (UTC)

The paper by Carlin and Soskice is more understandable than most papers on economics, but it contains some simple mathematical errors. They lost a factor of two when they differentiated the cost function of the central bank (although this does not matter since the derivative is then set to zero). More importantly, they ignored the continuing effects (future costs) of uncorrected deviations of the rate of inflation from its target. They arbitrarily assume at one point that some of their constant coefficients are equal to 1. And they never specify the units of the variables and constants in their equations. Also, like most papers, it is much too wordy and yet manages to leave important matters to the guess-work of the reader.

Any attempt by business firms to coordinate their activities to be more rational risks running afoul of government regulations and especially the anti-trust laws. Also it would make their tax returns into an even bigger nightmare (how do they account for the tax consequences of the contingent costs and benefits in their contracts?).

My recognition of the effects of minimum wage laws and the contracts which businesses are compelled to negotiate with various unions is not religion, it is a fact. JRSpriggs (talk) 04:26, 28 June 2014 (UTC)

I probably shouldn't have pointed you at that paper, it's a suggested reformulation, not the standard model. The two previous are better representations of the standard New Keynesian model. It's also an internal working document and not peer reviewed. If you are sure that you have correctly identified errors, you should write the authors, I'm sure they'll appreciate the chance to fix it. BTW, Krugman just posted an interesting blog entry on the history of macro, it's a good read[6].

Your statement that "government and the unions it protects have created a situation where instead of correcting itself quickly via rapid deflation, economic activity is choked-off when deflation would be appropriate" is not a "fact'" - at least not one that 99% of macroeconomists would agree with. In fact more economists would probably agree with these "facts": "Because it is impossible to get rid of market imperfections, an enlightened bureaucrat can always find a way to improve on a market outcome." (See Theory of the second best). "Modest increases in the minimum wage have negligible or slightly positive effect on employment." (See Card & Kruger 1995). LK (talk) 11:17, 2 July 2014 (UTC)

Here we get to what I see as the basic philosophical difference between us. You have this belief that an "enlightened bureaucrat" will be making the government's decisions. That is magical-thinking. Rationality is not something that comes automatically or free of charge. In fact, it is very rare outside of matters within a person's painful and hard-earned experience. That is why the people making decisions should always be those who are directly affected by the results. Hence my belief in freedom and capitalism. JRSpriggs (talk) 10:06, 3 July 2014 (UTC)

I hope you do recognize that both your views and the ones you attribute to me are essentially beliefs, and little different from catechisms like "Jesus Christ is my personal savior". BTW, the way I see the difference in our views is like this. "Motor vehicles exist, they cause pollution and hurt/kill a lot of people every year, however, they are fundamental to our modern lifestyles. I'm in favour of restricting the amount of motor vehicles, using them only where necessary, and improving the way they work and how they fit into our societies. You believe that they are fundamentally evil and favour getting rid of them altogether." Substitute government for motor vehicles in the above, and that's how I see our disagreement. LK (talk) 04:26, 7 July 2014 (UTC)

1. I have good reasons for what I believe, it is not like a religion.

2. Argument by analogy (to automobiles or religion) is not a valid form of argument.

3. Automobiles provide a very large benefit — carrying people and goods from one place to another where they may be more valuable. On the other hand, government has only one "benefit" that is not available in the private sector, it can provide overwhelming force to ensure that it gets its way. But force is inherently destructive, so it is not appropriate to use it except in situations where the target is a net evil which cannot be eradicated by any lesser method.

4. History has shown that governments are almost invariably captured (or originally established) by corrupt people who use them for evil purposes. JRSpriggs (talk) 10:49, 10 July 2014 (UTC)

BLP

I suggest that this edit is in contravention of Wikipedia:Biographies of living persons policy, which "applies to all material about living persons anywhere on Wikipedia". You might like to revert it. Deltahedron (talk) 17:36, 10 July 2014 (UTC)

Thanks for that. Deltahedron (talk) 06:24, 11 July 2014 (UTC)

Thanks for tensor edits!

Thanks (as usual) for the edits, in this case to help correct my additions to tensor.  :)

So I don't clog the talk page with dumb questions: does the distinction you are making mean that the co/contra-variant aspects of the tensor cannot be separately assessed (in a meaningful way)? I was thinking dimensions b/c of the contribution to the dimensionality of the final matrix (and in a sense the dimensionality of the tensor as a whole), but I wish to understand where the flaw is in my perception. TricksterWolf (talk) 19:17, 12 August 2014 (UTC)

One does not normally talk about the dimensions of the array, perhaps because of the possibility of confusion with the dimensionality of the underlying space.

Each dimension of the array is associated with one index. The index ranges over a finite number of values — each value corresponding to one of the dimensions of the underlying space. So a tensor with type (1, 1) over a ten-dimensional underlying vector space would consist of a 10×10 matrix having one hundred components. Each row is associated with a distinct element in the basis of the underlying vector space. Each column is associated with an element of the basis of the covector space dual to the vector space.

If the underlying vector space has d dimensions, then a tensor of type (n, m) will have d^n+m components each of which is a real number (or a real valued function of location on the manifold, if we are talking about a tensor field). Is that clear? JRSpriggs (talk) 23:55, 12 August 2014 (UTC)

ISIS & the ISI

Hello, there. Although I wasn't before, I am now concerned about the length of the ISIS page and Gazkthul as you will have seen has suggested doing with the ISI group what he did with ISIS's earlier predecessors, giving it its own article and leaving a resume in "History" in the ISIS page, which would reduce the size of the "History" section considerably. He will need to know whether he has consensus to do this and I wonder if I could ask you to give your response at the end of the thread here, pro or con. I cannot see why the 2014 timeline is duplicated on this page when it already has its own article! Reducing the ISI section and removing the 2014 timeline would drastically reduce the article's size! Thanks for telling the IP how to go about becoming an editor; the page could do with some more input. BTW, there has still been no response to my request at the WP:RSN about the Israel citations! It was even archived at one point, but I retrieved it and put it back in the list. --P123ct1 (talk) 12:08, 23 September 2014 (UTC)

deleted talk stuff on natural numbers

Hello JR it is nice to make your acquaintance.

Yeah of course I'm not out to delete other contributions. What you don't see is that there was a second half of that conversation already deleted by MjolnirPants. Rather than deleting it directly he said he was moving it to another section, put the pointer for the move, but the text never appeared in the other section. It could have been an accident. I don't mind. He now seems to be excited about my observation that there is a conflict here between formal math definition and what school book writers would like to see. He has opened a talk section on that. I think it will be productive.

Wikipedia does allow for talk page "refactoring", and I do think that section should go according to those guidelines. https://en.wikipedia.org/wiki/Wikipedia:Refactoring_talk_pages

There was an effort to prevent zero from being mentioned as a natural when I arrived at the site. The first sentence was a circular definition and called naturals {1,2, 3... } etc. The point about not linking to other pages is correct. Every time I mentioned zero, my edit was deleted. When I added pointers to the set-theoretic natural number, my edit ws deleted. When I gave the first of Peano's axioms, zero is a natural, it was deleted - despite being on the page about the Peano axioms. When I pointed out that zero is an additive identity needed in arithmetic, my edit was deleted. When I mention Von Neuman's definition my edit was deleted. When I copied another editors comment that "The convention used by set theorists .." it was deleted as was the other editors remark. It is back now. When I fixed the circular definition on whole numbers, that was deleted... These deletes were often when ignoring my talk page entries. Majorpants ignored six talk page entries and the confirmation of two editors when he deleted stuff.

So yeah, I wish people would have more respect about deletes. You didn't ask me about my edit before you reverted it either. I have a talk page.

Now I'm on the fence about "refactoring" and talking out that section about the war on zero because people don't understand it without more context, and it is too much work to put the links to the deletions mentioned above in order to support the point, but most of all it is not on the topic of natural numbers but on the topic of editor etiquette. Besides what good does it do? Furthermore it isn't fair that my text stays but the other text goes.

So I hope you understand why I might refactor that out. — Preceding unsigned comment added by Thomas Walker Lynch (talk • contribs) 18:40, 8 October 2014 (UTC)

graph cuts

hi you seem to be a math/stats guy. I was trying to make sense of Cut (graph theory) and was struggling, and tagged it for being too technical, and was drilled on per Talk:Cut_(graph_theory)#Technical. Maybe you might be willing to add some discussion to that article to make it more understandable to your average college educated person at least? thanks for considering... Jytdog (talk) 01:22, 24 October 2014 (UTC)

Sorry. Graph theory is not an aspect of mathematics to which I have paid much attention. So anything I said on that subject would be no better than what a layman might say. JRSpriggs (talk) 12:26, 24 October 2014 (UTC)

thanks for having a look and replying! Jytdog (talk) 12:34, 24 October 2014 (UTC)

Loving relation in FOL-article

Hi! Thank you for looking after my tinkering there. I only by chance stepped into this article and you were so kind to repair my rubbish. Just now I noticed this and tried to improve by marginals again. I am fully OK with substituting formulas for formulae, but like to remark that I deliberately chose someone loves someone over everyone loves someone, because I feel less a priori quantification in the former formulation. This is not to say that I want to revert it, but just give a reason for me doing so. Sorry, if I bothered you. Purgy (talk) 08:00, 11 December 2014 (UTC)

I just felt that the examples in first sentence of First-order logic#Loving relation should agree with one another to avoid confusing the readers. If you want to change both of them, while retaining their agreement, then go ahead. JRSpriggs (talk) 15:44, 11 December 2014 (UTC)

If I ever get the feeling I could do some substantial improvement to this article, I'll reconsider this idea. :) Thanks! Purgy (talk) 17:59, 11 December 2014 (UTC)

Lagrangian of a free particle in GR

OK, here I will give a detailed discussion. First, some notes on Lagrangian and actions, as well as the geodesic equation:

• The Lagrangian is unphysical.
• The action is unphysical.
• Lagrangians and actions are equivalent if they generate the same equations of motion.
• δS=0 implies the Euler-Lagrange equations, not the other way around.
• Overall factors in the Lagrangian are irrelevant. This includes signs and constants.
• The geodesic equation does not rest on an Euler-Lagrange style argument. It can be derived by other methods.
• The mass shell condition is a consequence of the geodesic equation.

The first three are basic tenets of the Lagrangian prescription. See any textbook on QFT and you will see Lagrangians and actions mangled and beaten into shape to extract physical information. Now, because of the linearity of integration and functional differentiation, if the Lagrangian is multiple of something else, then that multiple cancels in δS=0, leaving the equations of motion unaffected.

Some comments on your "talk" post:

• How does one derive the geodesic equation from your Lagrangian? That Lagrangian IS correct, in a sense, as I will show later. However, it is not in a form that I can see leading to the properly parametrized geodesic equation. (A multiple of τ is the correct parameterization.)
• The units really don't matter. I could be using units in which m=c=1. This does not affect anything.
• The Newtonian argument breaks down when we consider massless particles. For such particles your action is zero! By your reasoning, this leads to trivial equations of motion.

First I will show that the geodesic equation is not strictly a consequence of δS=0. For this, I will adapt the discussion found in Weinberg's Gravitation and Cosmology. Let ξ be a locally inertial coordinate system (free-falling) and τ be the proper time along the worldline (the time on a clock as measured by the particle in its rest frame). According to the Principle of Equivalence, there is such a coordinate system in which the equation of motion for SR holds:

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

In natural units, the proper time in this frame is

$d\tau ^{2}=-\eta _{\alpha \beta }d\xi ^{\alpha }d\xi ^{\beta }$

Now suppose we use any other coordinate system x, which may be a cartesian system in a laboratory, curvilinear, accelerated, or whatever. The freely falling coordinates ξ are functions of the x, thus the first equation becomes

$0={\frac {d}{d\tau }}\left({\frac {\partial \xi ^{\alpha }}{\partial x^{\mu }}}{\frac {dx^{\mu }}{d\tau }}\right)={\frac {\partial \xi ^{\alpha }}{\partial x^{\mu }}}{\frac {d^{2}x^{\mu }}{d\tau ^{2}}}+{\frac {\partial ^{2}\xi ^{\alpha }}{\partial x^{\mu }\partial x^{\nu }}}{\frac {dx^{\mu }}{d\tau }}{\frac {dx^{\nu }}{d\tau }}$

A line of algebra and the chain rule leads to the geodesic equaton

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

where

$\Gamma _{\;\;\mu \nu }^{\lambda }={\frac {\partial x^{\lambda }}{\partial \xi ^{\alpha }}}{\frac {\partial ^{2}\xi ^{\alpha }}{\partial x^{\mu }\partial x^{\nu }}}$

is the affine connection. The proper time may also be written in an arbitrary coordinate system:

$d\tau ^{2}=-\eta _{\alpha \beta }{\frac {\partial \xi ^{\alpha }}{\partial x^{\mu }}}dx^{\mu }{\frac {\partial \xi ^{\beta }}{\partial x^{\nu }}}dx^{\nu }=-g_{\mu \nu }dx^{\mu }dx^{\nu }$

where g is the metric tensor

$g_{\mu \nu }=\eta _{\alpha \beta }{\frac {\partial \xi ^{\alpha }}{\partial x^{\mu }}}{\frac {\partial \xi ^{\beta }}{\partial x^{\nu }}}$

Denote by $\partial _{\mu }$ the operator $\partial /\partial x^{\mu }$. You can have the pleasure of verifying that, given the previous equations

$\partial _{\lambda }g_{\mu \nu }=\Gamma _{\;\;\lambda \mu }^{\rho }g_{\rho \nu }+\Gamma _{\;\;\lambda \nu }^{\rho }g_{\rho \mu }$

Add to this equation the same equation with μ and λ interchanged and subtract the same equation with ν and λ interchanged. Remembering that the affine connection as defined above is guaranteed to be symmetric, this leads to the usual Christoffel symbols

$\Gamma _{\;\;\mu \nu }^{\lambda }={\frac {1}{2}}g^{\lambda \rho }(\partial _{\mu }g_{\nu \rho }+\partial _{\nu }g_{\mu \rho }-\partial _{\rho }g_{\mu \nu })$

So it is easy to see that the geodesic equation does not rest upon any particular variational principle of any particular Lagrangian, because you don't need the action principle to derive it.

From here on I will use some coordinate free notation mixed in with the usual Ricci notation. Let γ(τ) be the worline of the particle, parameterized by the proper time τ. Let (M, g) be a 4-dimensional pseudo-Riemannian manifold with a Lorentz metric g and inner product $\langle .,.\rangle =g(.,.)$. Let $\nabla$ be the Levi-Civita connection on M. Let

${\dot {\gamma }}={\frac {d}{d\tau }}$

be the tangent vector to the curve. In this notation, the geodesic equation is

$\nabla _{\dot {\gamma }}{\dot {\gamma }}=0$

i.e. the tangent vector is autoparallel along the geodesic. Next we use the Ricci identity. See any text on differential geometry for a proof. For any three vectors X, Y and Z

$X\langle Y,Z\rangle =\langle \nabla _{X}Y,Z\rangle +\langle Y,\nabla _{X}Z\rangle$

So let $X=Y=Z={\dot {\gamma }}$. Then

${\frac {d}{d\tau }}\langle {\dot {\gamma }},{\dot {\gamma }}\rangle ={\dot {\gamma }}\langle {\dot {\gamma }},{\dot {\gamma }}\rangle =2\langle \nabla _{\dot {\gamma }}{\dot {\gamma }},{\dot {\gamma }}\rangle =0$

where the last equality follows from the geodesic equation. We integrate the very first term along the worldline P. Thus

$\langle {\dot {\gamma }},{\dot {\gamma }}\rangle ={\text{const}}$

The convention is to take this constant to be -1. This comes about as follows. Suppose a particle in SR travels along a worldline P. Then the change in proper time is given by the line integral

$\Delta \tau =\int _{P}d\tau =\int _{P}{\sqrt {-\eta _{\alpha \beta }d\xi ^{\alpha }d\xi ^{\beta }}}$

Now suppose we turn on the gravitational field. Then by the Equivalence Principle (I will drop the P from now on)

$\Delta \tau =\int {\sqrt {-g_{\mu \nu }dx^{\mu }dx^{\nu }}}$

You object that the integral is not in the form ∫f(x)dx. So we choose to parameterize using the proper time and write

$\Delta \tau =\int {\sqrt {-\langle {\dot {\gamma }},{\dot {\gamma }}\rangle }}\,d\tau$

Comparing with above, we conclude that the choice

$\langle {\dot {\gamma }},{\dot {\gamma }}\rangle =-1$

is consistent. But what if we choose some other parameter λ? As of this moment we have no restrictions on what can be a parameter. Thus

$\Delta \tau =\int {\sqrt {-\langle {\bar {\gamma }},{\bar {\gamma }}\rangle }}\,d\lambda$

where

${\bar {\gamma }}={\frac {d}{d\lambda }}$

Now I want to bring the action principle into the fold. I think that we can both agree that in SR, if we care about overall signs and factors (c=1 still remains though), the free action for a point particle of mass m is

$S=-m\int {\sqrt {1-{\vec {v}}^{2}}}\,dt=-m\int d\tau$

The GR generalization is

$S=-m\int d\tau =-m\int {\sqrt {-\langle {\bar {\gamma }},{\bar {\gamma }}\rangle }}\,d\lambda$

The principle of least action is of course δS=0. Now I hope you see that the factor -m does not do anything at all! Thus the variational principle is simply

$\delta \int {\sqrt {-\langle {\bar {\gamma }},{\bar {\gamma }}\rangle }}\,d\lambda =0$

I also hope you see that $\langle {\bar {\gamma }},{\bar {\gamma }}\rangle$ is in no way restricted or fixed and can thus S can be varied. In this article this variation is performed. I will not do it here. The result is

${\frac {d^{2}x^{\lambda }}{d\lambda ^{2}}}+\Gamma _{\;\;\mu \nu }^{\lambda }{\frac {dx^{\mu }}{d\lambda }}{\frac {dx^{\nu }}{d\lambda }}=0\quad {\text{or}}\quad \nabla _{\bar {\gamma }}{\bar {\gamma }}=0$

Once again we can easily show that

$\langle {\bar {\gamma }},{\bar {\gamma }}\rangle ={\text{const}}$

from which it follows that $\lambda \propto \tau$. This follows only from the equations of motion and is not obvious before. As a constraint derived from the equations of motion, it cannot be applied to the Lagrangian! This would be like doing QFT on the mass shell only. (In fact, as I will show, the mass shell condition is a consequence of the geodesic equation.)

I said that the Lagrangian you presented was not incorrect. This is because you have defined the action as

$S=\int L\,dt$

whereas I have defined it as

$S=\int L\,d\tau$

This is a simple change of parametrization. However, your Lagrangian is not invariant under diffeomorphisms. The coordinate t has a nontrivial transformation rule as opposed to λ, which allows for invariant parameterization. Some food for thought: try deriving the geodesic equation from your Lagrangian. Maybe you can surprise me and get it.

The current discussion is adapted from Becker, Becker & Schwarz String Theory and M-Theory. The action -m ∫dτ has some problems with it:

• In the path integral sense, it is a disaster. The square root makes this very hard to quantize.
• It is completely unsuited to handling massless particles for obvious reasons.

These problems can be circumvented by introducing an auxiliary (bosonic) field α(τ) in the modified action

$S={\frac {1}{2}}\int d\lambda \,\left({\frac {\langle {\bar {\gamma }},{\bar {\gamma }}\rangle }{\alpha }}-m^{2}\alpha \right)$

First we determine the equations of motion for α(τ):

${\frac {\delta S}{\delta \alpha }}=-{\frac {1}{2}}\left({\frac {\langle {\bar {\gamma }},{\bar {\gamma }}\rangle }{\alpha ^{2}}}+m^{2}\right)=0\longrightarrow \langle {\bar {\gamma }},{\bar {\gamma }}\rangle =-m^{2}\alpha ^{2}$

Plugging this back into S, we see that we recover the original action. Now, it may be shown that this modified action is parameterization invariant if α has certain transformation properties. (BBS exercise 2.3) This leads to a sort of gauge invariance. The gauge α=1 is nice. In this gauge

$\langle {\bar {\gamma }},{\bar {\gamma }}\rangle =-m^{2}$

is the familiar mass shell condition. This means ${\bar {\gamma }}$ is the momentum vector. The additive constant in the Lagrangian can be just thrown away because it contributes nothing to the equations for γ. Thus the action is effectively

$S={\frac {1}{2}}\int \langle {\bar {\gamma }},{\bar {\gamma }}\rangle \,d\lambda$

Now this makes perfect sense for massless particles. For massive particles, we have τ=mλ. So, up to an overall factor (which we just throw away, the 1/2 is kept for historical reasons)

$S={\frac {1}{2}}\int \langle {\dot {\gamma }},{\dot {\gamma }}\rangle \,d\tau$

As I showed in the post you deleted, this leads to the geodesic equation

$\nabla _{\dot {\gamma }}{\dot {\gamma }}=0$

The really nice thing about this action is that it is computationally simple to work with.

Thus, the Lagrangian of general relativity can be taken to be

$L={\frac {1}{2}}g_{\mu \nu }{\dot {x}}^{\mu }{\dot {x}}^{\nu }$

Because this leads to the correct equations of motion, it is an acceptable Lagrangian.

I hope this clears up any confusion.

Differential 0celo7 (talk) 22:43, 18 December 2014 (UTC)

It would have been better to hold this discussion at Talk:Lagrangian#What is the correct Lagrangian for an isolated test particle in general relativity? so that other interested people could find it and participate.

Much of what you say is true and already well known to me. However, I disagree on some important points.

I disagree about ignoring constant factors because we may wish to add Lagrangians for various parts of a system together to get the Lagrangian for the total system and this will fail unless their proportions are correct. Also you need the constant factors to get the correct stress-energy tensor from the Lagrangian.

Your Lagrangian was either pulled out of the air and justified by its results (which I think should be avoided when possible) or it is an attempt to generalize ${\frac {mv^{2}}{2}}$ in an inappropriate way.

I do not disagree with the geodesic equation, but only with your method of deriving it.

I have shown how to derive a variation of the geodesic equation (using momentum) from my Lagrangian at User:JRSpriggs/Force in general relativity#Derivation from Lagrangian.

I have not yet seen a fully satisfactory derivation of the equation of motion for massless particles except by taking a limit of the equation for massive particles as the mass approaches zero. However, the only particles which might be massless are the photon and the graviton (if it exists).

Using the proper-time τ as a parameter for the particle's path is inappropriate because the value of τ is path-dependent. In particular, the limits of integration might have to be changed as you do the variation which is highly undesirable. JRSpriggs (talk) 12:27, 19 December 2014 (UTC)

I'll move the discussion to the Lagrangian talk page.

Differential 0celo7 (talk) 19:55, 19 December 2014 (UTC)

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