Malament's Theorem and Allen Janis's Construction
Maybe you are already familiar with David Malament's famous 1977 paper "Causal Theories of Time and the Conventionality of Simulataneity" (Nous 11). If you do not know, see Janis's article or Norton's exposition*.
Malament's Theorem shows that, given several "unproblematic" conditions, the relation of standard simulataneity (which assumes that the time a light-signal takes when going to, and coming back from, another event is the same, so that a distant simultaneous event can be identified by taking a half of the round-trip time of the light signal) is the only binary relation definable in terms of the light cone structure (causal connectibility, signified by ƒÈ) and the inertial worldline O of an observer.
Those conditions are: (i) Such a binary relation is invariant under all O causal automorphism, (ii) it is an equivalence relation, (iii) there exists two events p and q, one of which is on O while the other is not, and the two satisfy such a binary relation, and (iv) the relation is not a trivial universal relation (holding between any two events).
Here, "O causal automorphism" means a one-one mapping such that it maps the worldline O onto itself. The word looks difficult, but the idea is very easy. That is, you may translate O along itself, you may make a spatial rotation around O itself, you may expand the scale of spatial axis and time axis as you like, or you may change O upside down (reflection about orthogonal hypersurface). In other words, condition (i) demands various symmetries of causal structure of spacetime and O.
And, Malament's major point of using this theorem is that, granting causal theories of time such as Reichanbach's or Gruenbaum's, another assertion associated with these theories, i.e. the conventionality of simultaneity (you need not adopt the standard simultaneity; or in a roudtrip of a light signal, the time of reflection may be conventionally chosen), cannot be maintained.
Gruenbaum and others (such as Redhead and Janis) argue against this. Gruenbaum's newst paper employs Janis's construction (original was made in 1983), in particular, in order to refute Malament's claim that his theorem eliminates the latitude for conventionality (Gruenbaum further argues that conditions (i) and (ii) are not unproblematic). See the following figures. Figure 1 shows two inertial trajectories and their respective hypersurfaces orthogonal to them.
Figure 1
Figure 2
Allen Janis (a physicist, long associated with the Center for Philosophy of Science, Univ. of Pittburgh) pointed out the construction of Figure 2, which may be used against Malament's argument that the simulaneity in the special relativity is uniquely determined by the causal structure in it. This is quoted in Gruenbaum's recent paper "David Malament and the Conventionality of Simulataneity: a reply."
Suppose O is the inertial trajectory of an observer. This observer can choose (as in Figure 2), for his/her time axis, the inertial trajectory A of any other observer uniformly moving relative to him/her. Then, according to Malement's theorem, hypersurfaces of the standard simultaneity for A are uniquely determined orthogonal to A. But the observer in O, since he/she has chosen A as his/her time axis, must regard such hypersurfaces as his/her hypersurfaces of simultaneity. And these hypersurfaces determine a non-standard definition of simultaneity vis-a-vis O itself! Thus Janis argues that, granting Malament's theorem, there remains the same latitude as that of Reichenbach's or Gruenbaum's conventionalism.
*Norton, John. D., "Philosophy of Space and Time", in Introduction to the Philosophy of Science (Salmon et al.), Prentice-Hall, 1992.
Last modified Dec. 1, 2004. (c) Soshichi Uchii
