5.3 The Global Break-Down of Special Relativity (General Relativity)
Description
This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by others.
5.3 The Global Break-Down of Special Relativity (General Relativity)
Now that we have tried to argue that we can continue using special
relativity even when gravity is involved (by appropriately defining a new
inertial frame), we are now in a position to explain where the argument
breaks down.
Consider Diagram 5-6. There we see a ship which is much wider than the
ships we have shown thus far. It is in free fall towards the surface of the
Earth, and there are two observers shown, one at either side of the ship.
Now, according to our argument, both observers are said to be in inertial
frames of reference because they are both in free-fall. However, as they
each fall towards the center of the Earth, because they are at great
distances from one another, they accelerate in different directions as
shown. If one observer looks at the other, he will see that other observer
accelerating towards him. But if they are both supposed to be inertial
observers, then how can they also each be accelerating in the frame of the
other?
Diagram 5-6
+--------------------------------------------------------------+
| |
| O O |
| \ / |
| O \| |/ O |
|/|\ ""`Acceleration Acceleration '"" /|\|
| | toward center toward center | |
|/ \ \ of Earth of Earth / / \|
| \| |/ |
| ""` '"" |
+--------------------------------------------------------------+
_______==========_______
___===""""" """""===___
The Earth's Surface
Long Ship Falling in Earth's Gravitation
Also, consider Diagram 5-7 in which there is a ship which is much
taller than the ships we have been considering. Here, two observers are
again shown, one at the bottom of the ship and one at the top. Because the
one near the bottom is much closer to the surface of the Earth, he is
accelerating at a greater rate than the other observer. Again, these two
observers are both supposed to be inertial observers, yet each is
accelerating in the other observer's frame. Further, as the observer on the
top continues to accelerate downward, he will eventually be where the
observer at the bottom is now. Thus, as time passes, he will fall into a
stronger gravitational field, and he will be in a "different" inertial frame
than he his now.
Diagram 5-7
+------+
| O |
| |
| O |
|/|\ |
| | | |
smaller\|/ |/ \ |
accel | | G |
| | r |
| | a |
| | v |
| | i |
| | t |
| | y |
| | \|/
| O |
| |
| | O |
| |/|\ |
larger | | | |
accel \|/ |/ \ |
| |
+------+
==========
Earth's
Surface
Tall Ship Falling in Earth's Gravitation
What does all this say? Well, we have shown that for small distances
and over small amounts of time, a free falling frame has all the properties
we want in an inertial frame when gravity is present. However, in each of
the last two cases above, we have observers who are all free-falling and
thus (by our new definition of an inertial frame in the presence of gravity)
are all supposed to be in inertial frames. Yet, if we draw a space-time
diagram for one of the observers, and extend it so that the other observer
can be drawn on the diagram, that other observer will be accelerating on the
space-time diagram. Therefore, a space-time diagram which well describes an
inertial frame for all of space-time in special relativity can only well
describe an inertial frame of reference over a small distance in space and
time when a general gravitational field is involved.
This is analogous to the situation in which a flat map can well
describe a small, local piece of the curved surface of the Earth (such as a
city). However, globally, as you extend the map, it no longer describes the
curved surface of the Earth.
We therefore find that when gravity is involved, we can still define an
inertial frame of reference LOCALLY (meaning local in both space and time),
but globally, there is no way to define a single, rigid frame of reference
which describes an inertial frame of reference everywhere in space-time.
Therefore, globally we cannot use special relativity to describe space-time
in the presence of a general gravitational field. We must therefore re-think
relativity in the presence of gravity.
What we will find is that gravity is actually caused by a curvature of
space-time, and like the map trying in vain to describe the curved surface
of the Earth, special relativity cannot describe the curved space-time
caused by gravity. It is general relativity which describes curved
space-time, and for us to fully appreciate it, we will need to discuss some
basic ideas used to describe such a geometry.
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