# How are propositions concerning spacetime curvature constructed explicitly in terms of coincidences?

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 $coinc( s )$.

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.

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.

• "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 ? Commented Sep 9, 2013 at 19:20
• @user12262 : OK, so you might be interested by the example #1 given by Ben Crowell Commented Sep 10, 2013 at 6:23
• @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 : $\large \frac{D^2 v^i}{D \tau^2} = R^i_{klm}u^ku^lv^m$ Commented Sep 10, 2013 at 6:43
• "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.) Commented Sep 10, 2013 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?) Commented Sep 10, 2013 at 17:06