Did Einstein Cheat (Perihelion motion of Mercury)

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by rush</i>
<br />I have to know who is correct in this issue:<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">The article in question made no attempt to be scientifically correct or even to check its "facts". I certainly never suggested that Einstein cheated. But he did know the correct result for Mercury's perihelion in advance (as everyone did). And there was a previous version of general relativity released in 1913 that had to be subsequently withdrawn and amended. You can find that history in Misner, Thorne and Wheeler's "Gravitation".

The article itself is on a par with tabloid journalism. Personally, I have no time for nor interest in such trash. But at least one reader (not someone I know) did take the author to task, which the journal published in a later issue: salon.com/people/letters/2001/07/23/hughes/index.html

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">[tvf in the article]: "the choice of coefficients of potential phi in the space-time metric is arbitrary. Einstein knew the unmodeled perihelion motion of Mercury, and therefore confined his attention to metrics that predicted this quantity correctly."

[Carlip in the article]: "No, it makes no sense at all. Van Flandern seems to have invented a free parameter where none exists. There is one free parameter, but it's just Newton's gravitational constant, G, and is fixed completely by the requirement that the theory reduce to Newtonian gravity in the weak-field, low-velocity limit. Once you've fixed that, everything else is completely determined." According to Carlip, "Van Flandern seems to be under the impression that there are a bunch of adjustable parameters in general relativity that can be fiddled with. This is certainly not true."

[rush]: Who is correct about this issue?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">There are, of course, an infinite number of possible metrics, distinguished by their metric coefficients. General relativity is a unique theory, which proposes that one particular member of that infinite set is the metric that represents the real world. I have no idea what question was put to Carlip or if he was quoted out of context. But his reply is true only in the trivial sense, that metric coefficients are not adjustable parameters <i>within GR</i>, because any other coefficients would represent a different theory, not GR.

So if you are following the issue, Carlip's remark is beside the point I made, which was about how Einstein arrived at the particular choice of coefficients used in GR from an infinite set of possibilities. And one of several factors he used in narrowing his choice was agreement with the known perihelion advance of Mercury. It would have been foolish to ignore this information and propose a theory that predicted the wrong perihelion advance, and Einstein was no fool.

Moreover, metric coefficients have integer multipliers. If one changes a coefficient to a different integer multiplyer, that changes the predicted perihelion advance by a relatively large amount. A "free parameter" is something (like that gravitational constant) that can be adjusted through some continuous range and be made to match an observed result by setting its value at some particular spot in that range. Metric coefficients are not of that character and are therefore definitely not "free parameters" in that sense. But they can take on an infinite number of different, discrete values.

The bottom line is that each of the things Carlip said about GR are true, but irrelevant to what I said. I suspect sloppy or malicious journalism rather than any intent by Carlip to be deceptive.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">After all, what was EXACTLY the equation used by Einstein to calculate this quantity?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">What quantity? The perihelion advance? The Einstein formula for that is 3C/p per revolution, where C is a constant related to the gravitational constant and p is the semi-latus-rectum of the orbit. The "3" comes from the metric coefficients.

So here is how it works. There are an infinite number of ways to write metric equations so as to produce a net of 3 for the perihelion coefficient. Einstein's way produced three terms. The first contributed a coefficient of +4. The second term contributed another +1. And the last term contributed -2. The net is +3. But from this you can get a sense of there being lots of ways to get to +3 as the bottom line.

My own analysis of this matter appeared in an article titled "The perihelion advance formula" in Meta Research Bulletin 8:10-15 (1999). If you don't mind some heavy math, an article by Brunstein on pp. 54-59 of that same volume shows the metric equations in full generality, and how to get to Einstein's choice from there. -|Tom|-

Yes, the quantity I was referring to was the perihelion advance of Mercury. Thank you Tom.

reply to anonymous email:<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">[tvf]: Einstein's way produced three terms. The first contributed a coefficient of +4. The second term contributed another +1. And the last term contributed -2. The net is +3.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">&gt; [anon]: Could you post a followup with some elaboration of this? I'm familiar with several different derivations of Mercury's extra precession from Einstein's field equations, but none of them arrives at the multiplier 3 as the sum 4+1-2, so I'm curious to know more about the derivation you are referring to.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">Are you familiar with GR equations of motion? Here are some places where you can see these:

** Damour, T. and Deruelle, N. (1985). "General relativistic celestial mechanics of binary systems I. The post-Newtonian motion." Ann.Inst.HenriPoincare 43, 107-132.

** Einstein, A., Infeld, L., and Hoffmann, B. (1938). "The gravitational equations and the problem of motion." Ann.Math. 39, 65-100.

** Misner, C.W., Thorne, K.S., and Wheeler, J.A. (1973). Gravitation. W.H. Freeman and Co., San Francisco.

** Robertson, H.P. and Noonan, T.W. (1938). Relativity and Cosmology. W.B. Saunders Co., Philadelphia.

Many of the terms in these equations are insignificant for perihelion advance purposes. But three terms are not, and these three are responsible for the +4, +1, -2 contributions to the basic form. The basic form itself comes straight from the properties of elliptical motion and the standard perturbation equations used in celestial mechanics. The only freedom one has to get away from this basic form is by integer multiples that arise from the metric coefficients in solutions to the field equations.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">[anon]: By the way, you also said "a free parameter can be adjusted through some continuous range - metric coefficients are not of that character - but they can take on different discrete values." Don't understand what that means. Metric coefficients aren't limited to a set of discrete values. In general relativity the metric coefficients vary continuously from place to place.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">I was referring to the integer multipliers of potential/c^2 that appear in, e.g., the Schwarzschild solution. How to describe these integers depends on whether one is looking at the exponential or the linearized form of the metric. The only point of relevance here is that they are *integers*, and are therefore discrete parameters without the possibility of continuous variation. Best wishes. -|Tom|-

Second reply to anonymous:

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">[anon]: Thanks for posted reply.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">You're welcome, but this method of communicating is onerous, and others reading this can't see your original messages. Why not register and post anonymously to the Message Board? To register, you don't have to provide ID, only a real email address and the ability to follow directions. (This is for the purpose of excluding spambots, whose attacks are now up to 5-8/day.)

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">What you say doesn't seem right. The profound issue of "the equations of motion in general relativity" tackled by Einstein and Hoffmann in the 1938 paper you mentioned has nothing to do with deriving Mercury's excess perihelion motion.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">??? To my knowledge, there is no way to get perihelion motion or even basic elliptical motion directly from the Schwarzschild solution to the field equations. It is derivable only from equations of motion, which of course arise by taking gradients or partial derivatives of field equation solutions. But until we convert to expressions for Euclidean 3-space accelerations (the equations of motion), we cannot get orbital motion, compare to observations, or derive perihelion motion.

Anyway, what is "profound" about equations of motion? They represent garden-variety 3-space dynamics, no?

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">Now, your first reference (Damour) does indeed address post-Newtonian celestial mechanics, which of course is also discussed in the standard texts you listed as your remaining two references, but these references would suggest that you're alluding to something like the parameterized post-Newtonian (PPN) formalism, which however is inconsistent with your other remarks<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">??? I said nothing about PPN, and the references all derive GR equations of motion. I have no idea why you took this PPN detour.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">In sum, I find nothing in your references that supports what you said. So, regretably, the meanings of your "discrete integer metric coefficients" and your "4+1-2" remain veiled in mystery (for me).<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">Do you see the integer coefficient "4" in the term representing the source mass potential divided by c^2? See p. 1095 of MTW, where c = 1, or the equivalent in any of the other references. That "4" (rather than some other integer multiplier) arises from the integer coefficient that multiplies potential / c^2 in the Schwarzschild solution.

The Schwarzschild metric begins ds^2 = c^2 dt^2 (1-2 GM/rc^2) - ... The factor of "2" multiplying GM/rc^2 (where GM/r is potential) is the metric coefficient I am referencing.

A one-parameter family of solutions to Einstein's field equations for the Schwarzschild problem would substitute the following for the factor in parentheses: (1+2GM/nrc^2)^-n, where n is any integer. Because these are all solutions, Einstein had to find a way to specify n. Each different value of n gives a different perihelion motion. But since n occurs in integer steps, the possible perihelion motions also occur in integer steps. Einstein's choice gave 43"/cy. The next closest choices would have given either 29"/cy or 57"/cy. One cannot get solutions that yield, say, 35"/cy.

I hate ascii equations, but I hope this is close enough to the language you speak to convey the point. -|Tom|-