The Illusion of Time's Arrow - Kurt Nalty

The Illusion of Time’s Arrow
Kurt Nalty
January 19, 2007
Abstract
Parameterization of physics using time in a four dimensional spacetime
is capricious, and leads to infinities in mathematics. The four dimensional
pathlength s is a better choice. Such a description allows resolution of
paradoxes such as electron pre-acceleration during radiation, and helps
provide a plausible basis for quantum mechanical phenomonea, such as
the uncertainty principle and virtual particles.
Drift Velocity and the Time as a River Analogy
Before dealing with four dimensional spacetime, I wish to look at coordinates
and drift motion in regular experience.
Imagine a small boy on a boat, dragging a fishing bob in a circular path.
When the boat is stationary, there is no debate that the x and y components
of motion of the fishing bob go both positive and negative. Displacement is
clearly bipolar. However, as the boat begins to move faster and faster (say in
the x direction), while the y displacement remains bipolar, the drift velocity in
the x direction can easily overwhelm the oscillator component, leading to the
situation that x always increases, although the rate of increase may vary.
14
12
10
8
6
4
2
0
-2
-15 -10 -5 0 5 10 15
Circular Motion without Drift
Circular Motion with Small Drift
Circular Motion with More Drift
Circular Motion with Large Drift
Circular Motion with Very Large Drift
1

Our experience with space time is analogous to the boy on a fast flowing
river. Our ’velocity’ has components of motion, where the time component of
velocity is so high that our usual experience has time always increasing.
Describe the coordinates of a point in spacetime by ˜r = (t, x, y, z). The
change in this coordinate is d˜r = (dt, dx, dy, dz). The magnitude of the change
(+ sign choice) is ds = |d˜r| = √ dt 2 + dx 2 + dy 2 + dz 2 .
When we look at the four dimensional unit (tangent) vector, we have
ũ = d˜r
ds
(dt, dx, dy, dz)
= √
dt2 + dx 2 + dy 2 + dz 2
=
=
(1, dx/dt, dy/dt, dz/dt)
√
1 + ( )
dx 2
( ) 2
dt + dy
(
dt + dz
dt
1, ⃗v
√
1 + v
2
) 2
It is now customary in physics to measure time in meters using the speed of
light as a conversion factor. (Using this convention, the speed of light becomes
1.)
From COBE data (find references and most accurate number), we know that
earth has a relative velocity of 371 km/sec to the cosmic radiation background.
While this number seems impressive, if we express this velocity as a fraction
of the speed of light, we find that our speed is only 0.00124c. If we measure
time and space in the same units (for example, meters), the four dimensional
cartesian unit tangent will have components of
(
)
1
ũ = √ , ⃗v
√ = (0.999999235, 0.00124) (1)
1 + v
2 1 + v
2
Simply put, our velocity in the time direction is just barely under the speed
of light. It is this high velocity in the time direction that gives time it’s special
status, from our point of view of choice of coordinates.
Our situation with respect to time is analogous to that of a particle directly
approaching the event horizon of a black hole in conventional spacetime. As the
particle falls, the radial speed increases such that just before the event horizon,
the radial speed is just about the speed of light, and the transverse speed components
are insignificant. For this particle, there is clearly a difference between
the radial and non-radial directions. This particle may prefer to parameterize
its events using radial position as an always increasing parameter, and could do
so fairly accurately.
Poor Choice of Parameterization Leads to Infinities
Return to the example of circular motion imposed upon a drift velocity. For the
case where the motion is parameterized by the x component, when the x velocity
changes direction, tangential infinities will occur. For a contrived example, let
y = sin(t) and x = v ∗ t + cos(t).
2

dx = (v − sin(t))dt
dy = cos(t)dt
dy
dx = cos(t)
v − sin(t)
When v