5.2 The "New Inertial Frame" (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.
5.2 The "New Inertial Frame" (General Relativity)
Before starting this section, I want to mention something to the
reader: in the end, when gravity is concerned, we will not be able to find a
single inertial frame of reference which will correctly explain the geometry
of all situations. This will be the actual death-blow to special relativity.
In this section, it will start to look as if the situation is hopeful, and
that by defining a proper inertial frame, SR will be saved. However, in the
next section, we will see where this all falls apart, and I want the reader
to realize this from the beginning.
Now, in the previous section we showed that a space-time diagram drawn
for an inertial frame of reference doesn't explain the way things really are
for a frame of reference sitting stationary on the Earth's surface. If such
a frame cannot be called an inertial frame because of some effect of
gravity, then perhaps there is another way to define an inertial frame of
reference in the presence of gravity.
First, let's consider the properties of a frame which we know to be an
inertial frame without gravity. Consider a space ship sitting far from any
source of gravity. Here we will assume that the ship isn't
accelerating--it's just sitting there in the middle of space. Diagram 5-2
shows such a space ship at different times. Also shown is an observer and a
ball, both of which start out stationary in this frame of reference. Both
the observer and the ball are weightless along with the ship, and as time
goes on neither move with-respect-to the sides of the ship. This is
obviously what we would consider to be an ideal inertial frame of reference.
Diagram 5-2
--------------------- time -------------------->
1 2 3 4
+------+ +------+ +------+ +------+
| | | | | | | |
| O | | O | | O | | O |
| | | | | | | |
| O | | O | | O | | O |
|/|\ | |/|\ | |/|\ | |/|\ |
| | | | | | | | | | | |
|/ \ | |/ \ | |/ \ | |/ \ |
| | | | | | | |
| | | | | | | |
| | | | | | | |
| | | | | | | |
+------+ +------+ +------+ +------+
Ship Floating in Space
Next, consider the same ship, but let it be sitting stationary on the
Earth. Diagram 5-3 shows such a ship at different times, and again there is
an observer and a ball shown as well. Obviously, the observer and the ball
in this case cannot remain stationary with respect to the ship--rather they
must fall in the Earth's gravity and accelerate towards the Earth's surface.
Note that because of the way gravity works, the observer and the ball and
anything else in the ship will accelerate downward at the same rate
regardless of their mass (as long as they are at relatively the same height
above the Earth's surface, and neglecting air resistance). This
distinguishes gravity from all other forces in nature. With the other three
forces (electromagnetism, the strong nuclear force, and the weak nuclear
force) the motion of an object in the presence of the force depends on the
composition of the object. For example, electromagnetism doesn't act on
neutral particles, but does act on charged ones. However, when we consider
gravity, the path taken by an object which is released with a given velocity
in a gravitational field does not depend on the composition of the
object--not even its mass. So, both the ball and the observer in Diagram 5-3
accelerate at the same, constant rate towards the bottom of the ship. In
step 3 on that diagram, the observer hits the bottom of the ship, and in
step 4 the ball reaches the bottom as well. Obviously this situation isn't
like the inertial frame of reference we described above, and the observer in
these two situations could easily tell the difference between the two cases.
Diagram 5-3
--------------------- time -------------------->
1 2 3 4
+------+ +------+ +------+ +------+
| | | | | | | | |
| O | G | | | | | | |
| | r | | O | | | | |
| O | a | | | | | | |
|/|\ | v | | O | | | | |
| | | i | |/|\ | | O | | |
|/ \ | t | | | | | | | |
| | y | |/ \ | | O | | O |
| | \|/ | | |/|\ | |/|\ |
| | | | | | | | | |
| | | | |/ \ | |/ \ O |
+------+ +------+ +------+ +------+
========== ========== ========== ==========
Earth's Earth's Earth's Earth's
Surface Surface Surface Surface
Ship Sitting on the Earth's Surface
Further, consider the same ship again, this time letting it accelerate
at a constant rate in the middle of space. Diagram 5-4 shows such a ship at
different times (again with an observer and a ball). Note that in the
diagram, the observer and the ball start out at a constant speed (in steps
1, 2, and 3, both move one interval up during each step of time). However,
the acceleration of the ship causes it to move further between steps 2 and 3
than it did between steps 1 and 2, and so on. Therefore, at step 3 the
bottom of the ship meets with the observer, and the observer begins to be
pushed by the ship, accelerating along with the it from then on. This would
cause the observer to feel the force of the ship against him, "holding" him
against the floor. In the final step, the ball meets with the bottom of the
ship, and it too accelerates from then on because the ship is pushing
against it. This case thus looks very much like the case just above where
the ship was sitting on the Earth's surface--in both cases objects in the
ship will seem to accelerate at the same, constant rate towards the bottom
of the ship (regardless of their mass) and once there they will feel a force
against them as they sit on the floor of the ship. The observer in each of
these cases would find it hard to tell which of the two situations he was
really in.
Diagram 5-4
--------------------- time -------------------->
4
+------+
| |
| |
| |
3 | |
| |
+------+ | |
1 2 | | | |
| | | O |
accel | | |/|\ |
^ +------+ | | | | |
| | | | | |/ \ O |
+------+ | | | O | +------+
| | | O | | | \/ \/
| O | | | | O |
| | | O | |/|\ |
| O | |/|\ | | | |
|/|\ | | | | |/ \ |
| | | |/ \ | +------+
|/ \ | | | \/ \/
| | | |
| | | |
| | +------+
| | \/ \/
+------+
\/ \/
Ship Accelerating in Space
Given all three examples above, it seems obvious that a frame sitting
stationary on the Earth is much more like an accelerating frame than it is
like an inertial frame. Seeing that, it now seems perfectly reasonable for
us to find that an experiment performed on the surface of the Earth can't be
explained by a diagram drawn for an inertial frame.
But, can we now find a frame of reference in the presence of gravity
which DOES look like an inertial frame? Well, look back to Diagram 5-4
(where the ship is accelerating in space) and notice the state of the ball
and the observer during the first part of that illustration. Even though the
ship in that case is not an inertial frame because it is accelerating, the
observer and the ball don't begin to accelerate until the bottom of the ship
reaches them and begins to push them. Thus, until that point, the ball and
the observer are not accelerating. They are shown moving at a constant
velocity. Thus, until the bottom of the ship reaches them, the observer and
the ball are inertial observers. AH, but as we have pointed out, this
situation is supposed to be analogous to the one in Diagram 5-3 (where the
ship is sitting stationary on the Earth). If so, then we could argue that
the observer and the ball in the first part of Diagram 5-3 (which are in
free-fall in the Earth's gravitational field) are what we would now call our
inertial observers in the presence of gravity.
So, let's look at one last illustration in which the whole ship is in
free-fall as well as the observer and the ball. Diagram 5-5, shows such a
situation. Notice that the observer, the ball, and the ship all accelerate
at the same rate towards the earth. They each move the same distance during
each step shown. Now, look at just the ship and everything in it at each
step shown. The observer, the ball, and the sides of the ship are not moving
with respect to one another because they are all falling at the same rate.
At each step, the ball and the observer are at the same position inside the
ship. Therefore, until the ship in Diagram 5-5 reaches the surface of the
Earth, the observer wouldn't notice any difference between this situation
and the one in Diagram 5-2 (where the ship is floating in space).
Diagram 5-5
--------------------- time -------------------->
1
+------+ 2
| | +------+
| O | G | | |
| | r | | O | 3
| O | a | | | +------+
|/|\ | v | | O | | |
| | | i | |/|\ | | O |
|/ \ | t | | | | | |
| | y | |/ \ | | O | 4
| | \|/ | | |/|\ | +------+
| | | | | | | | |
| | | | |/ \ | | O |
+------+ | | | | | |
+------+ | | | O |
| | |/|\ |
| | | | |
+------+ |/ \ |
| |
| |
| |
| |
+------+
========== ========== ========== ==========
Earth's Earth's Earth's Earth's
Surface Surface Surface Surface
Ship Falling in Earth's Gravitation
It certainly seems, then, that a frame which is freely falling in the
presence of gravity is actually an inertial frame of reference. As one final
test, let's go back to the experiment mentioned earlier in which light rises
in the presence of Earth's gravity. As it turns out (though I won't go into
the proof) if the light is detected while it is still relatively close to
the Earth, and we consider the experiment in a frame of reference which is
freely falling near the Earth's surface, then in that frame, the light does
not loose energy. Thus, in the freely falling frame of reference, Diagram
5-1 (which depicts an inertial frame of reference) can correctly depict the
geometry of the situation.
And so, things are looking deceptively hopeful. In every case we have
studied, it seems as if we can continue to use special relativity as-is,
even in the presence of gravity, if we simply define "inertial frame" to
mean a frame which is in free fall. Then the space-time diagrams we have
drawn throughout our discussions would work just fine in the presence of
gravity, as long as we understand that they are drawn in free falling
frames. However, as I warned earlier, there is a problem here which we
haven't solved.
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