I have for some time now been enjoying the Opinionated History of Mathematics Podcast. I do not always agree with his opinions, but his heresies are at least interesting and often enjoyable, even when I do not agree with them. In one episide, Rationalism 2.0: Kant's philosophy of geometry he spends significant time on the subject of absolute space, presented as a necessity of Newtonian physics. Apparently this was something Euler believed, so I shall not blame the podcaster for it, but I shall show below that Newtonian physics does not need it. Let's begin with the central argument claiming it is needed:
Think of the law of inertia. It says: If there is no outside force acting on a body, then the body keeps going in a straight line with the same speed. Forever. Like a metal ball rolling on a marble table, when there is no friction and no obstacles, it keeps going with the same velocity. Without external influence, the state of motion remains the same.
But note that this law talks about the state of motion of a body without reference to other bodies. The law of inertia presupposes that the body has some inherent velocity, a true velocity. That's the thing that stays the same in absence of interference.
The first paragraph of of what I've quoted (which starts part-way through one of his paragraphs, while the second ends part-way through the next) states the law of inertia just fine. It is with the second that I take issue. Its error lies in supposing that there must be an absolute velocity, tacitly also such a thing as being absolutely at rest, for the velocity of a body to remain unchanging.
However, if we can find even one frame of reference with respect to which
even one body subject to no outside influences continues moving at constant
velocity, then we can perfectly well infer that, in any frame of reference that
moves at constant velocity with respect to this first, that body shall also
continue moving at constant velocity. This arises because (whether you use
the add the difference in velocities of the frame
transformation of the
Galilean / Newtonian model or the somewhat trickier transformation of special
relativity) the velocity of the body in either frame is some straightforward
function of its velocity in the other frame and the relative velocity of the two
frames. In particular, you can restate the principle of inertia as saying that,
in any non-rotating frame of reference that sees even one body subject to no
outside influences moving at constant velocity, each body subject to no outside
influences moves at constant velocity. Any other non-rotating frame with
respect to which any one of these bodies moves at constant velocity differs from
the first only by mapping all bodies' velocities through some fixed
transformation that maps constants to constants, so that this other frame also
agrees that each body subject to no outside influence moves at constant
velocity.
Thus the principle of inertia can be perfectly well expressed in relativistic terms (indeed, that is pretty much the standard modern way of doing so), without any need for a recourse to absolute space.
In special relativity, transforming from a three-dimensional space
independent of time to a four-dimensional space-time, chosing
any time-like
trajectory as the time axis corresponds to chosing which
things are considered to be at rest
. In this context, chosing the
trajectory of any body subject to no outside influence as time axis (along with
non-rotating spatial axes) gives us a frame of reference with respect to which
each body subject to no outside influence moves in a straight line (that is, at
constant velocity). When we then let go of the co-ordinate we're using and just
accept this space-time as a vector space, seen in terms of the modern
conventional understanding of vector spaces (where one's coordinates are mere
description of a thing that exists independent of them), this space-time does in
fact emerge as absolute
in the sense that each body has a 4-velocity in
it that is independent of our description (i.e. choice of coordinates).
However, in this case, the very choice to think of the space independently of
any choice of coordinates in it constitutes abandoning the notion of absolute
velocity – that choice amounts to declaring the notion of velocity to be
invariant under change of representation.
The amusing foot-note to all of this is that the universe in
fact does turn out to have a privileged
state of motion, namely
that of the cosmic microwave background. When we look at that background, we
see a clear dipole variation in its spectrum: its spectrum as seen in one
direction is blue-shifted, while that in the opposite direction is red-shifted,
relative to the spectrum as seen in the directions sideways to these. That
Doppler shifting is entirely consistent with there being a frame of reference
– with respect to which we are moving – in which the background's
spectrum is (with a bit of local averaging, to smooth out its tiny variations)
the same in all directions. This still gives no prefered position, to be
considered as the fixed origin
or centre of a system of co-ordinates, but
it does give an state of motion with respect to which our universe is more
simply described than any other.
The other noteworthy quirk of this is that I specified non-rotating
as a requirement for my frames. This is needed, since I might chose a frame of
reference rotating about an axis that is the trajectory (with respect to that
frame) of some body subject to no outside influences; other bodies subject to no
outside influences but not moving along that same axis would then appear to move
in curved paths, not paths of constant velocity. This is not necessarily a
major obstacle, though, as it suffices that there exist at least one frame of
reference with respect to which each body subject to no outside influence does
move at constant velocity; then every frame of reference moving at constant
velocity with respect to this one, without rotating with respect to it, shall
also see each body subject to no external influence moving at constant
velocity.
Written by Eddy.