Is this an acceptable way to state the EP?

[nearlynothing]

I'm helping write some supplementary notes for an undergraduate GR class at my university.
I'm writing about the equivalence principle, do you find this way of stating matters acceptable?

The equivalence principle as stated by Einstein asserts that in a sufficiently small vicinity of
an event in space-time and in a freely falling frame, physics is the same as in a global inertial
frame in special relativity.

Now i want to know if i can state this in an equivalent way as follows:

Saying that physics is the same in a freely falling frame in a small enough vicinity of an event in space-time as in an inertial frame of special relativity amounts to saying that every mesurement one takes in such a frame
gives the same results, to first order, as the same measurements performed in a global inertial frame or SR.

Space-time thus has the same infinitesimal structure as the space-time of special relativity, up to
first order in displacements. This means that a reference frame can always be set up such that
at a single event in space-time, the metric has the same components as in an inertial frame of special
relativity, with the christoffel symbols vanishing there simultaneously, and that a freely falling frame is such a frame. A consequence of the above mentioned is that freely falling particles are represented by time-like geodesics in space-time.

Thanks in advance.

 

[PAllen]

Mostly that seems fine to me. A few points, though:

1) Your statement of the EP is the modern statement, that is often called (e.g. by Clifford Will) the Einstein equivalence principle, but it is not actually the way Einstein historically stated it. I would suggest leaving the statement the same, but dropping the claim that this is how Einstein stated it.

2) I don't buy the argument that the EP leads to the result that free fall particles follow timelike geodesics (at least not in any direct way). Any timelike path coincides to tangent with some geodesic at each point, and will be straight to first order around anyone point in a local inertial frame. Einstein needed to add the geodesic hypothesis as a separate axiom, and later derivations of it are based on the field equations (plus, they require energy conditions, but you could argue that those are related to physics being locally like SR). I guess you could argue that the field equations also follow from the requirement of local energy conservation in a free fall frame [but here you have to add some simplicity criterion, because this is not quite enough to uniquely determine the field equations], and thus (with a long chain of reasoning) bring everything back to EP. However, that seems tenuous to me.

 

[atyy]

I think it depends on what one means by the EP. Maybe the most tricky parts in what you write are the logical inferences like "thus".

1) Let's say one has no theory, and just plain English EP, can one derive GR or anything in GR? I think not, except by brilliant guesswork. After all, without a theory, we don't even know what a small region means.

2) Is some sense of the EP exact in GR, and in fact part of the definition of GR? Yes, as the principle of minimal coupling of for matter and gravity.

3) If one has GR, does the plain English EP hold locally approximately? Yes, it does. Now that we have the theory, we can set up Minkowski coordinates along a geodesic, and parameterize deviations as we go further from the geodesic. Also, we know that curvature is absolute and involves second derivatives of the metric, so we know that local means something like "up to first derivatives". This is local in the physics sense, not mathematically. For our naive physics intuition, second derivatives are more nonlocal than first derivatives because the naive approximation for the second derivative involves four positions, but the first derivative involves two positions. Mathematically, second derivatives are local, because the curvature exists at every point.

 

[nearlynothing]

Mostly that seems fine to me. A few points, though:

1) Your statement of the EP is the modern statement, that is often called (e.g. by Clifford Will) the Einstein equivalence principle, but it is not actually the way Einstein historically stated it. I would suggest leaving the statement the same, but dropping the claim that this is how Einstein stated it.

Good point. I should be more careful before making such a statement.

2) I don't buy the argument that the EP leads to the result that free fall particles follow timelike geodesics (at least not in any direct way). Any timelike path coincides to tangent with some geodesic at each point, and will be straight to first order around anyone point in a local inertial frame. Einstein needed to add the geodesic hypothesis as a separate axiom, and later derivations of it are based on the field equations (plus, they require energy conditions, but you could argue that those are related to physics being locally like SR). I guess you could argue that the field equations also follow from the requirement of local energy conservation in a free fall frame [but here you have to add some simplicity criterion, because this is not quite enough to uniquely determine the field equations], and thus (with a long chain of reasoning) bring everything back to EP. However, that seems tenuous to me.

I have to admit I never heard this objection before. Could you give me some reference where I can read more about it?
In Straumann's the argument is employed that since you can find such coordinates at an arbitrary point on the world-line of a freely falling particle
in which it is tangent to a geodesic, then the world-line is a geodesic.

 

[atyy]

In Straumann's the argument is employed that since you can find such coordinates at an arbitrary point on the world-line of a freely falling particle in which it is tangent to a geodesic, then the world-line is a geodesic.

I too was wondering about PAllen's comment, and I thought his objection might be that while your statement is true for geodesics, it requires an additional assumption to say that a free falling test particle follows a geodesic. It's clear that one can add the notion of a test particle that follows a geodesic, but for a real particle with energy, the particle's energy should itself contribute to spacetime curvature, and it seems tricky to say that a particle moves exactly on a geodesic once backreaction is taken into account. I think it can be shown that when backreaction is taken into account, one only has geodesic motion to some very good approximation.

 

[nearlynothing]

I think it depends on what one means by the EP. Maybe the most tricky parts in what you write are the logical inferences like "thus".

1) Let's say one has no theory, and just plain English EP, can one derive GR or anything in GR? I think not, except by brilliant guesswork. After all, without a theory, we don't even know what a small region means.

2) Is some sense of the EP exact in GR, and in fact part of the definition of GR? Yes, as the principle of minimal coupling of for matter and gravity.

3) If one has GR, does the plain English EP hold locally approximately? Yes, it does. Now that we have the theory, we can set up Minkowski coordinates along a geodesic, and parameterize deviations as we go further from the geodesic. Also, we know that curvature is absolute and involves second derivatives of the metric, so we know that local means something like "up to first derivatives". This is local in the physics sense, not mathematically. For our naive physics intuition, second derivatives are more nonlocal than first derivatives because the naive approximation for the second derivative involves four positions, but the first derivative involves two positions. Mathematically, second derivatives are local, because the curvature exists at every point.

I think I see you point, I can't logically infer something because the theory is not complete. Would it be better to phrase it differently? Along the lines of a plausibility argument.

Edit: Not complete if all you have is the EP

 

[atyy]

I think I see you point, I can't logically infer something because the theory is not complete. Would it be better to phrase it differently? Along the lines of a plausibility argument.

I think a plausibility argument is fine. For example, if one doesn't have the full field equations, just the vacuum equations, if we add the assumption of geodesic motion from an EP-like plausibility argument, we can get the perihelion precession.

On the other hand if one is stating full GR, then the exact version of the EP as minimal coupling is nice to know.

Then given full GR, that Fermi normal coordinates are Minkowski along a geodesic is a form of the EP as an approximate local principle.

So we have 3 EPs: hand-wavy brilliant EP, exact EP, and local approximation EP. It's as if the hand-wavy EP can be made precise in two ways.

 

[nearlynothing]

So we have 3 EPs: hand-wavy brilliant EP, exact EP, and local approximation EP. It's as if the hand-wavy EP can be made precise in two ways.

This certainly helps. Thanks :)

 

[atyy]

BTW, the terminology for "minimal coupling" does vary. I said the EP as minimal coupling is exact, whereas http://arxiv.org/abs/1310.7426 says that minimal coupling can be wrong. The problem is that minimal coupling can be right or sometimes wrong depending on how one defines the "basic laws of physics". Minimal coupling is right if the "basic laws" are formulated using a Lagrangian, and the Lagrangian for a matter field is a function of the field and its first derivatives only (eg. Hawking and Ellis). On the other hand, section 24.7 of http://www.pma.caltech.edu/Courses/ph136/yr2011/ shows some cases in which minimal coupling does fail for other notions of "basic laws".

 

[PAllen]

I have to admit I never heard this objection before. Could you give me some reference where I can read more about it?
In Straumann's the argument is employed that since you can find such coordinates at an arbitrary point on the world-line of a freely falling particle
in which it is tangent to a geodesic, then the world-line is a geodesic.

I've never heard you can make this argument (I've never seen Straumann's book). If it could be made, why did Einstein need a separate hypothesis, and why have there been great efforts derive geodesic motion with the fewest asssumptions?

As to what you state here, any timelike world line is tangent to some timelike geodesic at every point (different ones at different points). To get anywhere, you need a further assumption that it must remain tangent to the same geodesic, which is assuming your conclusion.

 

[atyy]

I guess the EP argument that free-falling test particles follow geodesics is as follows.

EP: In free fall, gravity should be canceled out*.
Fermi normal coordinates**: Fermi normal coordinates are Minkowski along the entire spacetime trajectory if the trajectory is geodesic. Thus even in curved spacetime which has gravity, we can get flat spacetime coordinates along some trajectory if it is geodesic. So in some sense, geodesics cancel the gravity of curved spacetime. This cancellation is local, because the coordinates are Minkowski only exactly on the geodesic, and it is approximate because spacetime is of course still curved in the strict mathematical sense on the geodesic.

Putting both arguments together we get that free falling test particles move on geodesics. I think Einstein, and later successful derivations of the approximate geodesic motion of small bodies (Geroch, Jang, Ehlers, Gralla, Wald) were not satisfied with this argument, because in the full theory there is no such thing as a test particle, which has mass but no energy that warps spacetime.

*Newton's statement of the EP, as stated in Rovelli's quantum gravity book (footnote 19): If bodies, moved among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will continue to move among themselves after the same manner as if they had not been urged by those forces.

*Wikipedia only defines Fermi normal coordinates for a geodesic, but http://arxiv.org/abs/1102.0529 defines Fermi normal coordinates along a timelike curve that is not necessarily geodesic.

 

[PAllen]

I am well aware of FN coordinates, and their use for non-geodesic worldlines (there are even extensions to rotating as well as non-inertial frames, but some prefer to give a different name to coordinates built from a local frame that is rotating as well subject to proper acceleration; Fermi-Walker transport is used to define the reference against which local frame rotation is defined). I am also well aware that FN coordinates produce Minkowski metric [edit: and zero connection coefficients] all along a world line if and only if the world line is a geodesic. What I've never seen (in any of the GR textbooks I own) was treating these facts as implying geodesic motion from the EP. I do find your argument plausible, but it is conceptually and mathematically more sophisticated than common deductions from the EP. Note, the machinery to define and prove these facts did not exist in the early history of GR, so this argument was beyond anything Einstein conceivably used up through the 1920s.

[edit: Note the in line correction above: FN coordinates produce Minkowski metric all along any world line. What characterizes a geodesic is that the connection coefficients vanish as well, all along the world line. This extends the 'locally flat' feature to first derivatives of the metric. This justifies the statement that geodesic motion leads to absence of 'inertial forces' locally.]

 

[PAllen]

I guess a coordinate free way to make such an argument is that a world line experiences no proper acceleration if and only if it is a goedesic. This requires covariant derivative to define proper acceleration, the statement the proper acceleration so defined corresponds to what accelerometers measure, and a formal definition of geodesic. But now, the connections to EP is even more tenuous.

 

[Nugatory]

1) Your statement of the EP is the modern statement, that is often called (e.g. by Clifford Will) the Einstein equivalence principle, but it is not actually the way Einstein historically stated it. I would suggest leaving the statement the same, but dropping the claim that this is how Einstein stated it.

Or leave the statement the same, but also note that that's not how Einstein historically stated it? There's something to be said for reminding students that they're getting the benefit of a century of 20/20 hindsight, with all the false starts and wrong turns removed. There's no need to delve into exactly what Einstein did say unless someone asks.

 

[PAllen]

Looking at things in local coordinates also generates the following observation. I noted that any world line is tangent to some geodesic everywhere (different ones at different points). This can be related to the ability to define Riemann Normal coordinates at any point on any world line such that:

1) The metric is Minkowski at the origin
2) The connection coefficients vanish at the origin
3) The velocity of the world line is zero at the origin

Thus, this implements an MCIF (momentarily co-moviing inertial frame). However, unless the world line is geodesic, there will coordinate acceleration (= proper acceleration at this point, in this construction). I think maybe this argument better implements an EP argument: only a geodesic has zero acceleration in all MCIF's.

Thus, you could say: The EP requires that a free fall trajectory have no acceleration in any of its MCIFs. This, with differential geometry, requires that the trajectory be a geodesic.

 

[PAllen]

Finally, the way I've always seen the geodesic hypoethesis introduced, with no reference to EP, is by heuristic generalization of the law of inertia. The simplest generalization of inertial bodies move in straight lines (assumed in the context of SR spacetime), to curved spacetime, is the geodesic hypothesis: inertial motion is motion on spacetime geodesics.

 

[atyy]

What I've never seen (in any of the GR textbooks I own) was treating these facts as implying geodesic motion from the EP. I do find your argument plausible, but it is conceptually and mathematically more sophisticated than common deductions from the EP. Note, the machinery to define and prove these facts did not exist in the early history of GR, so this argument was beyond anything Einstein conceivably used up through the 1920s.

Yes, I was just trying to figure out the version of the EP in the OP. Originally, I had reservations because I thought it was assuming that free fall motion is geodesic, but now I see it used the EP as a local, approximate principle to require that free fall motion is Minkowski with vanishing connection coefficients on the world line, and then deduced geodesics as satisfying the requirement

[edit: Note the in line correction above: FN coordinates produce Minkowski metric all along any world line. What characterizes a geodesic is that the connection coefficients vanish as well, all along the world line. This extends the 'locally flat' feature to first derivatives of the metric. This justifies the statement that geodesic motion leads to absence of 'inertial forces' locally.]

Good catch, I missed that point in my post.

Thus, you could say: The EP requires that a free fall trajectory have no acceleration in any MCIF. This, with differential geometry, requires that the trajectory be a geodesic.

That seems to be what the OP was arguing. It's nice because then it also captures that accelerated motion fakes gravity by making the connection coefficients non-zero. Of course acceleration cannot make the derivatives of the connection coefficients non-zero, which is what only true gravity does.

Finally, the way I've always seen the geodesic hypoethesis introduced, with no reference to EP, is by heuristic generalization of the law of inertia. The simplest generalization of inertial bodies move in straight lines (assumed in the context of SR spacetime), to curved spacetime, is the geodesic hypothesis: inertial motion is motion on spacetime geodesics.

Can we get this by writing the special relativistic 4-force definition in arbitrary coordinates, apply the version of the EP that says "comma goes to semicolon", then set the 4-force to zero to get the geodesic equation? (As you know, I don't believe in 4-force , so don't know this off the top of my head.)

 

[PAllen]

Can we get this by writing the special relativistic 4-force definition in arbitrary coordinates, apply the version of the EP that says "comma goes to semicolon", then set the 4-force to zero to get the geodesic equation? (As you know, I don't believe in 4-force , so don't know this off the top of my head.)

Yes that does work, but that version of EP is, shall we say, not very phenomenological.

 

[nearlynothing]

Yes, I was just trying to figure out the version of the EP in the OP. Originally, I had reservations because I thought it was assuming that free fall motion is geodesic, but now I see it used the EP as a local, approximate principle to require that free fall motion is Minkowski with vanishing connection coefficients on the world line, and then deduced geodesics as satisfying the requirement

That seems to be what the OP was arguing. It's nice because then it also captures that accelerated motion fakes gravity by making the connection coefficients non-zero. Of course acceleration cannot make the derivatives of the connection coefficients non-zero, which is what only true gravity does.

Yes, this is what i was arguing. Considering complete equivalence between gravitational fields and fictitious forces, a gravitational field would come from non-vanishing Christoffel symbols. Going to a local inertial frame you make the Christoffel symbols vanish, and noting that free fall is motion with zero proper-acceleration will then mean that the worldline of a freely falling particle is a geodesic.

 

[WannabeNewton]

Saying that physics is the same in a freely falling frame in a small enough vicinity of an event in space-time as in an inertial frame of special relativity amounts to saying that every mesurement one takes in such a frame
gives the same results, to first order, as the same measurements performed in a global inertial frame or SR.

Not all measurements will satisfy this. If the measurements are sensitive to tidal gravity then they will only agree with analogous SR measurements to zeroth order. The distinction to make is the measurements can depend at most on first derivatives of the metric. For example I can decompose the Riemann tensor into its electric and magnetic parts in the local Lorentz frame of some observer and from it get the measurements of differential precession and tidal deformations relative to this observer and you cannot make these vanish to any non-trivial order in the momentarily comoving local inertial frame so it will not reduce at first order to the SR measurements.

A consequence of the above mentioned is that freely falling particles are represented by time-like geodesics in space-time.

This is certainly true albeit a posteriori which is fine; in other words, keep in mind it's not a consequence but rather an inference. You cannot prove that freely falling particles are represented by timelike geodesics using just the EP but you can certainly infer it using the simple observation that freely falling particles move inertially in the momentarily comoving local inertial frame. A truly careful proof requires a much more formidable calculational framework c.f. Gralla and Wald (2008). A less careful proof simply follows from evaluating $\nabla_{\mu}T^{\mu\nu} = 0$ under the multipole expansion of $T^{\mu\nu}$ around the worldtube of an object in the limit as the worldtube shrinks to a worldline (to monopole order) c.f. Geroch and Jang (1975) or Papapetrou (1951).