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Is Einstein's insight [1] that

All our well-substantiated space-time propositions amount to the determination of space-time coincidences [such as] encounters between two or more [...] material points.

and [2] his (so recognized)

[...] view, according to which the physically real consists exclusively in that which can be constructed on the basis of spacetime coincidences, spacetime points, for example, being regarded as intersections of world lines

applicable to propositions concerning spacetime curvature ?

How, for instance, is the proposition

"spacetime containing the worldline of material point A is curved"

constructed and expressed explicitly on the basis of spacetime coincidences (in which the "material point" identified as A or suitable other "material points" took part) ?

Edit in response to the answers and comments presently provided (Sept. 12th, 2013):

  • Trying to put my question more formally,

given that there is the set $S$ of any and all distinct "spacetimes" imaginable,
and that there is the function (or proposition)
$\kappa : S \rightarrow \{ true, false, undetermined \}$
which for any spacetime under consideration represents whether it is "curved", or "not curved", or not an eigenstate of "possessing any curvature" at all,

and further given a set of sufficiently many distinct names $W := \{ A, B, M, M', ... \}$,
and that there is the function
$coinc : S \rightarrow \text{ powerset}[ \text{ powerset}[ \, W \, ] \, ]$
which for any spacetime under consideration represents the set of (distinguishable) coincidences of (different, and distinctly named) "worldlines",

I'd like to know the explicit expression of the function (or proposition) $f : \text{ powerset}[ \text{ powerset}[ \, W \, ] \, ] \rightarrow \{ true, false, undetermined \}$,
for which
$\forall s \in S: f( coinc( s ) ) = \kappa( s )$.

Looking at the above quotes it may be expected that $f$ has been worked out and written down long ago already; therefore, please write it down in an answer here, or point me to the corresponding reference. However, here are

  • Considerations which answers would be acceptable otherwise:

either a proof that such a requested function $f$ doesn't exist at all; presumably by exhibiting two distinct spacetimes $s_a$ and $s_b$ for which $coinc( s_a ) = coinc( s_b )$ but $\kappa( s_a ) \ne \kappa( s_b )$. (But recall the hole argument discussion relating to the difficulty of distinguishing spacetimes at all.)

Finally, if such a requested function $f$ can neither be explicitly stated, nor refuted, then
define the notion "geodesic" (which has already been used/presumed in answers below) or at least the notion "null geodesic" explicitly in terms of $coinc( s )$.

user12262
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2 Answers2

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Nice question. In addition to coincidences, you also need the notion of geodesics. Here are a couple of simple examples.

Example #1: In a vacuum spacetime, say we have electrically test particles A and B. Their world-lines are geodesics. Suppose these two geodesics coincide twice. Then we are guaranteed that this spacetime is not flat.

Example #2: The Gravity Probe B experiment can be expressed in terms of coincidences. This experiment involved sending a gyroscope into orbit around the earth and detecting its precession. Suppose, in a simplified conceptual version of the experiment, that the axis of the gyroscope has a mark on it, and we also mark the point on the housing that is initially right next to it. After a while, we observe that the two marks no longer coincide. Then at some still later time, we observe that the two marks again coincide, because we've had one full cycle of precession.

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If you make a change of coordinates $x \to x'=x'(x)$, and if $x_0$ is a coincidence, then, in the new coordinates, $x'_0=x'(x_0)$ is a coincidence. Now, you can obtain the value of the new curvature tensor $R_{\alpha'\beta'\gamma'\delta'}$ from the old curvature tensor $R_{\alpha\beta\gamma\delta}$ by using transformation laws for tensors ($R_{\alpha'\beta'\gamma'\delta'} = \large \frac{\partial x^\alpha}{\partial x'^{\alpha'}}\frac{\partial x^\beta}{\partial x'^{\beta'}}\frac{\partial x^\gamma}{\partial x'^{\gamma'}}\frac{\partial x^\delta}{\partial x'^{\delta'}} R_{\alpha\beta\gamma\delta}$). If one of the components $R_{\alpha\beta\gamma\delta}(x_0)$ is different of zero, then there will exist one of the components $R_{\alpha'\beta'\gamma'\delta'}(x'_0)$ which will be different of zero.

Trimok
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  • "you can obtain the value of the new curvature tensor from the old curvature tensor" -- Sure, but that's in no way addressing my question: How would "the old curvature tensor" (or at least: whether at least one its components differs from zero) be determined in the first place, explicitly on the basis of spacetime coincidences ? – user12262 Sep 09 '13 at 19:20
  • @user12262 : OK, so you might be interested by the example #1 given by Ben Crowell – Trimok Sep 10 '13 at 06:23
  • @user12262 : If you have a family of geodesics, where $v^i$ reprsents the separation between two neighbouring geodesics,$u^i$ represents the tangent to a geodesic, $\tau$ represents an affine parameter along the geodesic, you have the geodesic deviation equation : $\large \frac{D^2 v^i}{D \tau^2} = R^i_{klm}u^ku^lv^m$ – Trimok Sep 10 '13 at 06:43
  • "OK, so you might be interested by the example #1 given by Ben Crowell" -- Yes, I have been and continue to be interested in the answer submitted by Ben Crowell "20 hours ago". My comment to that answer, which I tried to submit roughly "19 hours ago", has apparently been lost or removed. However, since you (Trimok) are apparently using (now) some terminology which Ben Crowell had used in his example #1 as well, I may not have to repeat my comment and questions to Ben Crowell; but, for the moment, I'll rather ask you: (... to be continued.) – user12262 Sep 10 '13 at 17:05
  • "If you have a family of geodesics [...]" -- How do you define the notion "geodesic" (or at least, for starters, "null geodesic") in terms of terminology provided by the quotes stated in my question? Or do you mean that propositions concerning spacetime curvature require notions in addition or besides "coincidences"? (Similarly questionable are the various other notions you take for granted; such as "separation between geodesics", "affine parameter of a geodesic" and even "along a geodesic".) (And, btw., how about propositions concerning spacetime topology?) – user12262 Sep 10 '13 at 17:06
  • @user12262 : A geodesic is a path in space-time with a tangent vector $u^k$ verifying by $u^i \nabla_iu^k=0$, which could be written $\frac{\nabla u^k}{\nabla s}=0$, this means, that the variation of the tangent vector is null along the geodesic. Particles follow geodesics, and $u^k$ is the $4$-velocity of the particle. . The example shown by Ben Crowell, shows that, when several geodesics have 2 coincidences, it is a sign of curvature. – Trimok Sep 11 '13 at 07:30
  • @user12262 ....But it is not a local relation (From the first coincidence to the second coincidence, you are following a path, and at each point, you have a different curvature tensor). So I gave you an example (the geodesic deviation) which is a local relation between curvature and geodesics, but this relation is not directly linked to coincidences. – Trimok Sep 11 '13 at 07:30
  • Trimok: "So I gave you an example (the geodesic deviation) which [...] is not directly linked to coincidences." -- So you say; and, indeed, there's no explicit mentioning of coincidences in your presentation(s). But is this perhaps just a flaw of your choice of example, or form of presentation? Or: Do you really claim (know?) that it is not possible at all to define "geodesic deviation" (for instance) in terms of the "coincidence terminology" of the quotes in my question? --> – user12262 Sep 11 '13 at 18:51
  • A suggestion that "coincidence terminology" is at least very relevant can be found in MTW box 10.2 (along with box 16.4), where at least the operation "covariant differentiation, $\nabla$" is defined apparently with plenty of explicit references to coincidences, as far as in these drawings for instance the line containing ${\mathscr A}$, ${\mathscr N}$ and ${\mathscr P}$ as well as the line containing ${\mathscr X}$, ${\mathscr N}$ and ${\mathscr M}$ both represent "(worldlines of) material points". (Unfortunately, MTW also presumes "geodesic" and "affine parametrization" additionally.) – user12262 Sep 11 '13 at 18:52
  • @user12262 : I am only saying, that, in the example of Ben Crowell, 2 sucessive coincidences (so, for the same particles) are a non-local paradigm, and curvature is a local paradigm. That's all. A geodesic is a geodesic, even if no particle follows that geodesic. If you want to see the particles as source of the gravitation, this is is given by the Einstein equations via the stress-energy tensor. – Trimok Sep 12 '13 at 09:12
  • "in the example of Ben Crowell, 2 successive coincidences (so, for the same particles)" -- Considering that Einstein's conception of a "material point" supposedly endorses the possibility of it being closed (and thus "closed timelike") I'd be hesitant to call them "successive". Let's just call them "two distinct coincidences"; with some particles in common, and additionally distinct particles participating. So far, so good. But then there's the "geodesic" requirement (for at least some of the common particles) ... --> – user12262 Sep 12 '13 at 21:45
  • Trimok: "and curvature is a local paradigm." -- Hmm ... Is this (roughly) about covariant derivatives being evaluated event by event, and (in general, possibly) having different value event for event? Sure. Nevertheless: a derivative is by definition a limit of a series of ratios involving non-zero differences (cmp. again MTW box 10.2). The definition and evaluation of a covariant derivative therefore necessarily involves very many different events. "That's all. [...]" -- Please note the recent Edit to my question. – user12262 Sep 12 '13 at 21:46