Galilean Spacetime (Neo-Newtonian Spacetime) P t = 2

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1 (Neo-Newtonian Spacetime) Figure V t t t = t* vt x P t = 2 t = 1 t = 0 O x x vt x I used figure V above to show you how to change the coordinates of point P in the coordinate system S to coordinates in S (and back again the other way). We use the Galileo transformation to change from one label of a point (its coordinates in S) to another label for the same point (its coordinates in S ).

2 Transformations viewed this way are called passive transformations. One sits at a point and, as it were, watches passively as the labels applied to that point are changed. But there is an alternative, more active, way to look at a transformation. We start with two coordinate systems, S and S, and we suppose that the transformation moves us in the space from from points with coordinates, say, (x,t) in S to points with the same coordinates in S. For example, if we are located at the point on the t-axis with coordinates (0,t*) in the diagram above, the (active) Galileo transformations (for two inertial frames moving with relative velocity v, of course) move us in the positive x-direction and parallel to the x-axis to the point (0,t*) on the t*- axis. In fact, every location along the t-axis is shifted over to the t*-axis. The action of the Galileo transformations begins to look like the beveling of the deck that Geroch talks about. The t-axis is, in effect, shifted to the t -axis. The line representing an object at rest at the origin is transformed into the line representing an object moving through the origin with velocity v in the original coordinate system (or, as an alternative way to say the same thing to the world 2

3 line of an object at rest in a CS moving with velocity v with respect to the original system). So now we can view transformations as shifting or altering structures--in this example, the slope of a line. The (geometric) slope of a line represents the (physical) velocity of a body. Under the Galilean transformations, then, velocity is not an invariant quantity. In particular, being at rest is not an invariant quantity under the Galilean transformations. In Aristotelian spacetime, the state of rest is singled out, with the vertical lines indicating it. Natural as that supposition was, this in an example of structure that is supposed to come with the spacetime, supposed to be intrinsic, that is not reflected in the coordinate transformations (transformations that reflect our physical understanding). If we suppose that only the invariants of transformation should be intrinsic features of spacetime, then rest (or, more generally, velocity of any magnitude) is not an intrinsic feature of spacetime. But as we showed before, acceleration is. What that means is that the Galilean transformations transform straight lines into other straight lines (lines that represent the histories of particles that 3

4 undergo no acceleration) and curved lines to curved lines (lines that represent the histories of particles undergoing acceleration). We are then left with a spacetime in which acceleration is absolute (that is, invariant), but velocity is not. We bring spacetime structure in line with coordinate transformation. Well, almost. The Galileo transformations preserve the origin, a point in spacetime that clearly can be chosen arbitrarily. It is not intrinsic geometric structure. What we have to do is to extend the Galileo transformations by adding an arbitrary constant to each identity--that is, by allowing translations as well as tilts or bevels. So the resulting transformations are actually: x = x - vt + cx, y = y + cy, z = z + cz, t = t + ct. When looked at as active transformations, they can send the origin to any point in the spacetime that we wish. Therefore it is no longer fixed or intrinsic structure. Notice that we widened or enlarged the set of allowable transformations by adding translations to 4

5 the original set. The result of expanding the class of allowable translations is to decrease the amount of invariant structure. And when we decrease the amount of structure in a clearly classical spacetime we throw new light on the impasse between Newton and Leibniz. We cut the link between absolute acceleration and absolute velocity that seems to have been in the minds of both. So where does that leave the dispute? One might see Leibniz as arguing for the following thesis: R1. All motion is the relative motion of bodies with respect to other bodies. Spacetime does not have, and cannot have, structures that support absolute quantities of motion. 1 (R1) presents or endorses a relational account of motion. Insofar as he was arguing for (R1) Leibniz was right that spacetime supports neither absolute position nor absolute velocity. But he was wrong to infer from these facts that it could not support absolute acceleration. 1 I am following John Earman in World Enough and Space-Time for (R1) and (R2). 5

6 On other occasions, one might understand Leibniz as arguing against the existence of substantival spacetime: R2. The spatiotemporal relations among bodies are direct. They do not derive from, are not parasitic upon relations between underlying spacetime points. But both Aristotelian and Galilean spacetimes are spacetimes, even if the latter lacks some of the geometric structure of the former. Newton and Leibniz could not resolve their differences this way, since the idea of Galilean spacetime arose (as far as I know) in the latter half of the 19th century. It was not an available conceptual option in the 17th. Any philosopher who encounters closely related theses like (R1) and (R2) will ask: What are the logical relations between them? Earman s discussion of this on page 13 of WEST is difficult. Perhaps the issues could be sharpened by altering (R1) and (R2). I find it hard to see how to improve (R1), but perhaps (R2) should become 6

7 (R2 ) There are objects that bear spatial and temporal relations to one another, but there is no substantival spacetime above or beyond those objects and their relations. It seems to me that (R2) would follow from (R2 ). (R2) is an ontological thesis, while (R1) asserts the relativity of motion. It also seems to me that (R1) would follow from (R2), but the converse would not, though it would be hard to see what motivation one would have for supposing there is such a thing as spacetime if (R1) were true. 7