## General Relativity – part 2
In the diagram above we have a large
mass M and
a small body moving
past it. The large mass could be the Sun or a
planet. And the small body could be a moon or grain of sand
for example. We’ll say the small body wants to get from A to
B and
needs to do so in a fixed amount of time. There are any
number of paths it could take and any number of variable speed changes
it could make along the way.
## The Schwarzschild MetricGeneral Relativity usually involves
lots of complicated tensor calculus and confusing symbols.
Because of this it is best to avoid delving into its specifics until a
proper understanding can be gained at a conceptual level.
Basically however, GR is built around something called Einstein’s field
equations
and these fully describe the effects of gravity and orbital
motion. The solution to the equations involves solving
multiple differential equations simultaneously and is very difficult
except in simple situations. The earliest and best known of
these is Karl Schwarzschild’s solution for a point mass. ---- (1) Where (alpha) α = 2GM/c ---- (2) Where v ---- (3) In most cases v ---- (4) The above should now start looking familiar. Notice that when we set M=0 (a zero gravity environment) we get ---- (5) This becomes the Lorentz
Transform! It tells us how much
slower ds
runs behind dt
when the spaceship containing clock ds
is
moving with velocity v.
Actually the Lorentz Transform is
expressed as the reciprocal of this because it describes dt/ds. ---- (6) And this is the equation for
gravitational time dilation as shown in the previous
chapter.
## What is velocity relative to?We’ll pause here for a closer look
at the situation. In earlier chapters we looked at Special
Relativity (SR) and noted one of its main problems is we don’t know
what velocity should be measured against. Velocity is
considered a relative quantity under both SR and classical
mechanics. Therefore in order to calculate velocity-based
time dilation it is necessary to know what velocity is relative
to. In the case of SR it is ambiguous.
## Where's the acceleration?Another oddity of the Schwarzschild
Metric is that it doesn’t mention acceleration.
## Light-bending and Perihelion advancesWe’ll now turn to a more
interesting
aspect of GR. This is where GR makes predictions different
from what classical mechanics would make. The first of these
is light-bending. ---- (7) Notice the v ---- (8) Without going through the
complicated math, what we have effectively
done is double the value of the gravitational constant. This
equates to doubling the ‘force’ on the light ray and we end up with
twice the bending. There’s obviously more to it than that but
that’s the basic idea. ---- (9) We want to make (9) a maximum, i.e.
make time dilation a
minimum. We already know this requires reducing overall
velocity (velocity = bad). What (9) tells us is that radial
velocity has a slightly greater impact on dilation than tangential
velocity (radial velocity = slightly badder). The grey line shows what Newtonian
gravity predicts. The
black line is what GR predicts. As the planet begins moving
away it slows down, making it closer to the Sun, and then speeds up as
it gets farther away. Here the blue dot indicates a point
that has moved closer. The result is the far-point of the
ellipse (the aphelion) is shifted along slightly. A similar
thing happens on the return trip in which the near point of the ellipse
(the perihelion) advances slightly with each orbit.
## Another Equivalence problemThe
fact that GR predicts a different amount of bending for light presents
another problem for the Strong Equivalence Principle. This
principle claims that if you were sealed inside an elevator you
wouldn’t know if it were sitting motionless on Earth or accelerating
through outer space at a rate of 9.8 m/s
## Vertical motionMost of the discussion so far has
been about orbital motion. That is, bodies moving around a
point mass at periodic intervals. But GR can be applied to
any type of motion such as vertically falling or rising objects. It’s interesting to note that GR
predicts the same thing. If
we label your hand as point A,
the top of the parabola as point B,
and
specify the time it takes to get there, GR also predicts the stone will
slow as it moves upward. Why it that?
## The stationary particleWe’ll now look at a situation that GR genuinely has problems with. Suppose you observed a particle to be situated some distance away from Earth. Your observation was made using high-speed film and the photograph gave no clue about which direction the particle was moving because there was no blur. Several hours later you took another photo and found the particle at the same location. Based on your understanding of gravity you would know that it is impossible for the particle to have been sitting there the whole time. So you would conclude that the particle must have either orbited the Earth and returned to the same point, or was earlier moving upward and had then fallen down to that same point. GR would also draw the same
conclusion and say the particle must have
taken one of those two paths. Problem is you wouldn’t know
which. In both cases, classical and GR, you’d be lacking
information.
## Questioning the motionThis leads us to some important
questions about what GR is describing. Under classical
mechanics a body can only respond to the forces that are acting upon it
at the time. It can’t base its motion on something that it
will encounter later on. Yet GR seems to describe a body
doing exactly that. Namely that a body determines a path to a
destination that it doesn’t yet know of, based on
conditions it
has not yet encountered. How is this possible?
## ConclusionsThe acceleration time dilation
aspects of General Relativity (GR) are internally self-contradictory
and thus could not be true. [1]
Schwarzschild’s original paper in
English http://arxiv.org/PS_cache/physics/pdf/9905/9905030v1.pdf |

Copyright © 2011
Bernard Burchell, all rights reserved.