The Hitchhiker’s Guide to Relative Space
Image by Nick Olsouzidis
James Lythgoe
As you read these words the page, screen or crumpled napkin from which you are reading them is travelling at the speed of the earth’s rotation, plus the speed of the earth’s orbit of the sun, plus the rotational speed, relative to its position in the radius, of the Milky Way, plus the speed of the Milky Way’s movement from the centre of the Big Bang to you. Or at any rate it would be travelling at that speed if space were absolute; as it is you can rest secure in the knowledge that not only is it moving at each of these cumulative speeds and many others it is also staying still, and all this at the same time. Welcome to relative space.
The essential feature of what I am calling ‘relative space’ is that space itself is nothing over and above the distance between objects. ‘Objects’ here is used in the sense in which a part of an object is also an object (so that every object is in principle divisible into an infinite number of infinitely small objects). So, then, if the universe consisted of one (macroscopic) object then the entirety of space would be that lone object’s dimensions; if it consisted of two (macroscopic) objects then the entirety of space would be their dimensions and the distance between them. Furthermore relative space is infinite, not because it has no boundaries, but because it has the potential to increase infinitely as the most distant objects move away from each other. As a consequence speed, definable as distance divided by time, becomes relative to perspectives. To illustrate this, imagine the two-object universe again; let’s call the objects ‘A’ and ‘B’. Now, if A moves towards B then its speed will be the difference between the distance between A and B before the move divided by the time taken for the move. On the other hand, A and B are not moving just in case they maintain a constant distance from each other (so that each of A’s infinitely divisible parts maintain a constant distance from B’s infinitely divisible parts). Now if we introduce a third object (I’ll imaginatively title it ‘C’), we can imagine that A and B maintain a constant distance from one another whilst moving away from C. From the perspective of C, A and B are moving away at the same speed; but from the perspective of B, C is moving away and A is staying still.
This notion of space is to be contrasted with what I will call ‘absolute space’. On the absolute model, space is defined from without, as if the universe lay between a set of infinite three-dimensional axes. On this view all objects are locatable as a portion of the continuum of each axis, and hence distance is the change along at least one axis. This means that in a one-object universe the one object can move whilst maintaining a constant size, and does so just in case its position along at least one axis changes. Obviously this is not possible in relative space. On this model, for any given period of time an object is travelling at a speed equal to the distance it moves along at least one axis divided by the time taken, and hence your copy of Dialectic/computer screen/napkin is travelling at the incredible total speed mentioned above, and any sentence which asserts that it is travelling at any other speed is made false. Conversely, on the relative space account your preferred choice of reading media is travelling at that speed; but it is also equally true that it is travelling at the speed of the earth’s rotation plus its orbit (from the perspective of the Sun) and that it is staying still (from your perspective or that of anything travelling at the same speed).
This should also be contrasted with a subjective account of absolute space where an object seems to be in motion or at rest depending on the perspective of another object. This subjective account is crucially different from the relative model, firstly, in that it makes the same speed assertions false as the objective account, but makes sentences of the (tacit or explicit) form ‘A seems to be travelling at speed s from the perspective of B’ true. Secondly, as the predicate in relative-account assertions is speed, the perspectives in those assertions can be stipulative; whereas the subjectivist seeming-speed assertions become counterfactual conditional statements unless there is an actual conscious being answering to the perspective in question.
The relative picture of space, though it seems initially odd and unappealing, seems to be the only way that we can make sense of the notion of space at all. The absolute view seems to make too much of our every day ’speed’ talk false. Furthermore, the absolute model makes the unintelligible claim that space is both measurable above and beyond the distance between objects and indeterminate (infinite in the sense of having no boundaries). Moreover, the subjective account’s attempts to remedy this by making speed claims actually tacit seeming-speed conditionals seem unsatisfactory because they fail to say anything about speed itself.
James Lythgoe is a third year philosophy undergraduate at the University of York.
February 25, 2009 at 12:24 pm
Your account of absolute space is an interesting one because there is no reason why the absolutist would define space in such a way that it is equal to “the speed of the earth’s rotation, plus the speed of the earth’s orbit of the sun, plus the rotational speed, relative to its position in the radius, of the Milky Way, plus the speed of the Milky Way’s movement from the centre of the Big Bang to you.” Conversely the absolutist defines his space based on inertial effects that result from rotation and wouldn’t necessarily say that “the big bang” defines a stationary point in the absolute framework. Even Newton himself admits that there is no way of determining absolute velocity and I certainly don’t think that he, along with virtually any serious absolutist, would try to define absolute velocity in the way you have in your article. He holds that absolute space is detectable only when there is a rate of change of velocity (acceleration). A good example might be Newton’s account of the two globes.
The absolutist only defines position and velocity indirectly insofar as acceleration is rate of change of velocity and velocity is rate of change of position (it ought to be noted that both acceleration and velocity are both directional quantities and position is scalar). If we are trying to define an absolutist conception of position and velocity based on an absolutist conception of acceleration this can not be done.
If a=dv/dt then, by integration of the expression using calculus, we obtain v=at+u (assuming for simplicity that ‘a’ is constant and that ‘u’ is equal to some initial velocity). In the expression ‘u’ is undefined so there is no way of moving from an absolute judgment concerning space to an absolute judgment concerning velocity. Given that the absolutist is only claiming that we can know that space is absolute by looking at the inertial effects of acceleration we can not transfer this judgment to velocity (and by the same argument position) since the ‘u’ in the expression is arbitrary due to the limits of the integration of ‘a’ being undefined.
I think you’re right in claiming that talk of velocity is meaningless without reference to some other point or object. It seems that, on this account, the absolutist is simply a relativist who picks one privileged reference frame from which to define velocity. What I would argue is that this conception of absolute space is not one that any philosopher really subscribes to without basing his argument somehow on inertial frames and acceleration. Unless you take into account such arguments then you can’t legitimately dismiss the absolutist.