# Can the vanishing of the Riemann tensor be determined from causal relations?

Given a Lorentzian manifold and metric tensor, "$( M, g )$", the corresponding causal relations between its elements (events) may be derived; i.e. for every pair (in general) of distinct events in set $M$ an assignment is obtained whether it is timelike separated, or lightlike separated, or neither (spacelike separated).

In turn, I'd like to better understand whether causal separation relations, given abstractly as "$( M, s )$", allow to characterize the corresponding Lorentzian manifold/metric. As an exemplary and surely relevant characteristic (cmp. answer here) let's consider whether the Riemann curvature tensor vanishes, or not, at each event of the whole set $M$ (or perhaps suitable subsets of $M$).

Are there particular causal separation relations which would be indicative, or counter-indicative, of the Riemann curvature tensor vanishing at all events of set $M$ (or if this may simplify considerations: at all events of a chart of the manifold); or on some subset of $M$?

To put my question still more concretely, consider as possible illustration of "counter-indication":

(a)
Can any chart of a 3+1 dimensional Lorentzian manifold with everywhere vanishing Riemann curvature tensor (or, at least, a whole such manifold) contain [Edit in consideration of 1st comment (by twistor59): -- the Riemann curvature tensor vanish at least in one event of a 3+1 dimensional Lorentzian manifold if each of its charts contains -- ]

• fifteen events (conveniently organized as five triples):

$A, B, C$; $\,\,\,\, F, G, H$; $\,\,\,\, J, K, L$; $\,\,\,\, N, P, Q\,\,\,\,$, and $\,\,\,\, U, V, W$,

• where (to specify the causal separation relations among all corresponding one-hundred-and-five event pairs):

$s[ A, B ]$ and $s[ A, C ]$ and $s[ B, C ]$ are timelike,
$s[ F, G ]$ and $s[ F, H ]$ and $s[ G, H ]$ are timelike,
$s[ J, K ]$ and $s[ J, L ]$ and $s[ K, L ]$ are timelike,
$s[ N, P ]$ and $s[ N, Q ]$ and $s[ P, Q ]$ are timelike,
$s[ U, V ]$ and $s[ U, W ]$ and $s[ V, W ]$ are timelike,

$s[ A, G ]$ and $s[ G, C ]$ and $s[ A, K ]$ and $s[ K, C ]$ and
$s[ A, P ]$ and $s[ P, C ]$ and $s[ A, V ]$ and $s[ V, C ]$ are lightlike,

$s[ F, B ]$ and $s[ B, H ]$ and $s[ F, K ]$ and $s[ K, H ]$ and
$s[ F, P ]$ and $s[ P, H ]$ and $s[ F, V ]$ and $s[ V, H ]$ are lightlike,

$s[ J, B ]$ and $s[ B, L ]$ and $s[ J, G ]$ and $s[ G, L ]$ and
$s[ J, P ]$ and $s[ P, L ]$ and $s[ J, V ]$ and $s[ V, L ]$ are lightlike,

$s[ N, B ]$ and $s[ B, Q ]$ and $s[ N, G ]$ and $s[ G, Q ]$ and
$s[ N, K ]$ and $s[ K, Q ]$ and $s[ N, V ]$ and $s[ V, Q ]$ are lightlike,

$s[ U, B ]$ and $s[ B, W ]$ and $s[ U, G ]$ and $s[ G, W ]$ and
$s[ U, K ]$ and $s[ K, W ]$ and $s[ U, P ]$ and $s[ P, W ]$ are lightlike,

the separations of all ten pairs among the events $A, F, J, N, U$ are spacelike,
the separations of all ten pairs among the events $B, G, K, P, V$ are spacelike,
the separations of all ten pairs among the events $C, H, L, Q, W$ are spacelike, and finally

the separations of all twenty remaining event pairs are timelike
?

Conversely, consider as possible illustration of "indication":

(b)
Is there a 3+1 dimensional Lorentzian manifold with everywhere vanishing Riemann curvature tensor (or, at least, one of its charts) which doesn't [Edit in consideration of 1st comment (by twistor59): -- nowhere vanishing Riemann curvature tensor such that all of its charts -- ] contain

• twenty-four events, conveniently organized as

four triples ($A, B, C$; $\,\,\,\, F, G, H$; $\,\,\,\, J, K, L$; $\,\,\,\, N, P, Q$) and

six pairs ($D, E$; $\,\,\,\, S, T$; $\,\,\,\, U, V$; $\,\,\,\, W, X$; $\,\,\,\, Y, Z$; $\,\,\,\, {\it\unicode{xA3}}, {\it\unicode{x20AC}\,}$),

• where (again explicitly, please bear with me$\, \!^*$):

the sixty-six separations among the twelve events belonging to the four triples are exactly as in question part (a),

each of the six pairs is timelike separated,

the separations of all fifteen pairs among the events $D, S, U, W, Y, {\it\unicode{xA3}}$ are spacelike,
the separations of all fifteen pairs among the events $E, T, V, X, Z, {\it\unicode{x20AC}\,}$ are spacelike,

$s[ D, {\it\unicode{x20AC}\,} ]$ and $s[ S, Z ]$ and $s[ U, X ]$ are spacelike,
$s[ E, {\it\unicode{xA3}} ]$ and $s[ T, Y ]$ and $s[ V, W ]$ are spacelike,

$s[ A, {\it\unicode{xA3}} ]$ and $s[ A, Y ]$ and $s[ A, W ]$ are spacelike,
$s[ A, {\it\unicode{x20AC}\,} ]$ and $s[ A, Z ]$ and $s[ A, X ]$ are timelike,
$s[ A, E ]$ and $s[ A, T ]$ and $s[ A, V ]$ are timelike,

$s[ C, {\it\unicode{x20AC}\,} ]$ and $s[ C, Z ]$ and $s[ C, X ]$ are spacelike,
$s[ C, {\it\unicode{xA3}} ]$ and $s[ C, Y ]$ and $s[ C, W ]$ are timelike,
$s[ C, D ]$ and $s[ C, S ]$ and $s[ C, U ]$ are timelike,

$s[ F, {\it\unicode{xA3}} ]$ and $s[ F, D ]$ and $s[ F, S ]$ are spacelike,
$s[ F, {\it\unicode{x20AC}\,} ]$ and $s[ F, E ]$ and $s[ F, T ]$ are timelike,
$s[ F, V ]$ and $s[ F, X ]$ and $s[ F, Z ]$ are timelike,

$s[ H, {\it\unicode{x20AC}\,} ]$ and $s[ H, E ]$ and $s[ H, T ]$ are spacelike,
$s[ H, {\it\unicode{xA3}} ]$ and $s[ H, D ]$ and $s[ H, S ]$ are timelike,
$s[ H, U ]$ and $s[ H, W ]$ and $s[ H, Y ]$ are timelike,

$s[ J, D ]$ and $s[ J, U ]$ and $s[ J, Y ]$ are spacelike,
$s[ J, E ]$ and $s[ J, V ]$ and $s[ J, Z ]$ are timelike,
$s[ J, T ]$ and $s[ J, X ]$ and $s[ J, {\it\unicode{x20AC}\,} ]$ are timelike,

$s[ L, E ]$ and $s[ L, V ]$ and $s[ L, Z ]$ are spacelike,
$s[ L, D ]$ and $s[ L, U ]$ and $s[ L, Y ]$ are timelike,
$s[ L, S ]$ and $s[ L, W ]$ and $s[ L, {\it\unicode{xA3}} ]$ are timelike,

$s[ N, D ]$ and $s[ N, S ]$ and $s[ N, W ]$ are spacelike,
$s[ N, E ]$ and $s[ N, T ]$ and $s[ N, X ]$ are timelike,
$s[ N, V ]$ and $s[ N, Z ]$ and $s[ N, {\it\unicode{x20AC}\,} ]$ are timelike,

$s[ Q, E ]$ and $s[ Q, T ]$ and $s[ Q, X ]$ are spacelike,
$s[ Q, D ]$ and $s[ Q, S ]$ and $s[ Q, W ]$ are timelike,
$s[ Q, U ]$ and $s[ Q, Y ]$ and $s[ Q, {\it\unicode{xA3}} ]$ are timelike, and finally

the separations of all ninety-six remaining event pairs are lightlike
?

(*: The two sets of causal separation relations stated explicitly in question part (a) and part (b) are of course not arbitrary, but have motivations that are somewhat outside the immediate scope of my question -- considering Lorentzian manifolds -- itself. It may nevertheless be helpful, if not overly suggestive, to attribute the relations of part (a) to "five participants, each finding coincident pings from the four others", and the relations of part (b) to "ten participants -- four as vertices of a regular tetrahedron and six as middles between these vertices -- pinging among each other".)

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This sounds like a very complicated set of conditions - have you tried to see if you can satisfy your (a) and (b) counterexamples by doing the usual trick of taking Minkowski space and identifying various points, lines etc to change the causal structure? –  twistor59 May 30 '13 at 6:26
@twistor59: "This sounds like a very complicated set of conditions" -- still the simplest I could think of, to expect them being relevant. (The footnote may hint at why I thought so.) "have you tried [...] the usual trick of taking Minkowski space and identifying [...] the causal structure?" -- (1): thanks for pointing out that this "usual trick" applies to both question parts (a) and (b), as presently stated (may I rephrase (b), perhaps? ...) Is the applicable "causal structure of Minkowski space" even spelt out anywhere explicitly enough; even by J.W.Schutz, or A.A.Robb? ... And (2): –  user12262 May 30 '13 at 19:31
... (2): yes, I tried; case (a) seriously for about a week, until I had an idea how to solve it just a few days ago; while case (b), as presently stated, is sort of trivial once it is clear that I'm asking about Minkowski space (but I'm also interested already quite a while, and unable to quite solve so far, the question of whether the structure of case (b) would then identify "mutual rest" of participants). Finally (3): I failed to realize that this "usual trick" is directly applicable because the answer here (see above) doesn't mention it either. –  user12262 May 30 '13 at 19:32
p.s. Reading the first comment again I now also notice that I had significantly crippled the suggested "usual trick" in quoting. Sorry, FWIW; I still took occasion to edit part (b) of my question, to be perhaps less easily answered, and perhaps even be addressable by applying the suggested "usual trick" as intended ... –  user12262 May 30 '13 at 20:14

The casual structure of spacetime is invariant under conformal transformations, transformations where the metric is $g$ is changed to a new metric $\tilde{g} =\Omega^2g$[1]. The Riemann tensor as a whole is not invariant under conformal transformation, and therefore it cannot be a good measure of causal structure.

The Weyl tensor, however, is invariant under conformal transformations and therefore it would be a measure of causal structure. So If you took some appropriately causally closed domain containing your events and demonstrated that there was a chart where the Weyl tensor vanishes everywhere, then you would know that the causal relations between those events would have to look like Minkowski space. The Weyl tensor is a component of the Riemann tensor, so the vanishing of the Riemann tensor would be sufficient but not at all necessary.

I believe the topic that would deal with these questions would be the conformal geometry of pseudo-Riemmanian spaces.

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That's very helpful (thanks; up-vote). I suppose that "being a measure of causal structure" is at some level interchangable with "being derivable from causal structure". (Though I'm very much for adhering to the latter approach, eventually.) Therefore: could you please expand/detail the step how to "demonstrate that [or whether] there was a chart where the Weyl tensor vanishes everywhere"; given some particular "causal structure"? –  user12262 May 31 '13 at 22:52