| Topic: |
Science > Physics |
| User: |
"Jack Sarfatti" |
| Date: |
03 Sep 2006 06:58:59 PM |
| Object: |
"Geodesic" in Newton and Einstein |
On Sep 3, 2006, at 1:16 PM, xerberos2 wrote:
Jack wrote:You misread Einstein's text ...
I'm not misreading his text. Einstein's text is very clear. He is
proposing to treat a fictitious inertial field as if it were a real
gravitational field, so that he can pretend that the accelerating
frame K' is not accelerating.
By "real gravitational field" he means in the sense of Newton's theory.
K' that is non-geodesic in curved space-time is locally equivalent to a
geodesic inertial frame in flat space-time with a "real" gravity field.
This is Einstein's bridge back to Newton's theory. "geodesic" has two
different meanings in the same sentence here.
"Real gravitational g-field" is meaningful in Newton's theory in flat
Euclidean 3D space with absolute simultaneity (Galilean relativity v/c
---> 0) where the Newtonian geodesics are straight lines in flat
Euclidean 3D space with point test particles moving at constant speed
along them. That's the Newtonian geodesic. There is zero g-force on
Newton's geodesic.
"g-field" in Einstein's theory means exactly the same thing as in
Newton's theory except that the notion of geodesic has changed. An
Einstein geodesic projected down into 3D space is generally not a
straight line nor is the test particle speed constant. For example the
Earth's elliptical orbit around the Sun is geodesic relative to the
Sun's curvature field - to a good approximation. Curvature is geodesic
deviation. g-forces are non-zero only on non-geodesics created by
non-gravity (essentially electromagnetic) forces. There is no necessary
intrinsic relationship of a g-force event to the local curvature.
Now, what confuses Zielinski is the following: consider a cannon ball in
free fall as in
http://www.zonalibre.org/blog/diversovariable/archives/baron-munchausen.jpg
Newton's explanation: the Baron and the cannonball are NOT on a
geodesic, therefore, there is a real gravitational force per unit test
mass on both the Baron and the cannonball relative to the frame K'
(surface of Earth) that is "inertial" to a good approximation. It is the
same real gravitational force per unit test mass g for both
g-force(Newton) ~ GM(Earth)/r^2
Therefore, the Baron feels weightless, i.e. no pressure on his behind
from the cannonball since each are falling in exactly the same way at
every moment. That is, there is zero g-force in the common rest frame of
the Baron and the cannonball.
Einstein's explanation: the Baron and the cannon ball are on a timelike
geodesic in curved space-time.
The covariant equation for the geodesic in ALL frames is
D^2x^u(test)/ds^2 = d^2x^u(test)/ds^2 +
(Connection)^uvw(dx^v(test)/ds)(dx^w(test)/ds) = 0
This is the covariant
F = ma
with
F = 0
This form-invariant (local frame covariant) equation means.
OBJECTIVE TENSOR TEST PARTICLE ACCELERATION = 0
This is the DEFINITION of a GEODESIC!
THIS IS TRUE IN ALL FRAMES FOR ALL POSSIBLE CURVATURES INCLUDING
GLOBALLY FLAT ZERO EVERYWHERE-WHEN.
D^2x^u(test)/ds^2 is the GCT tensor acceleration of the test particle.
Its local frame-invariant scalar is
g = (D^2x^u(test)/ds^2D^2xu(test)/ds^2)^1/2
g = 0 on a geodesic - universally true!
Look more closely at the meaning of the geodesic equation. Let Baron
Munchausen be on the test particle that is the cannonball in the above
picture.
D^2x^u(Baron)/ds^2 = d^2x^u(Baron)/ds^2 +
(Connection)^uvw(dx^v(Baron)/ds)(dx^w(Baron)/ds) = 0
d^2x^u(Baron)/ds^2 = Newton's flat space + time kinematical acceleration
that is not a GCT tensor.
(Connection)^uvw(dx^v(test)/ds)(dx^w(test)/ds) = inertial "force per
test mass" that is a contingent artifact of the local frame of
reference. This term even exists in globally flat spacetime when K' is
accelerating from an electromagnetic force.
Let Alice be a nearly coincident to the Baron geodesic LIF in curved
space-time. What Alice sees is
D^2x^u(Baron)/ds^2 = d^2x^u(Baron)/ds^2
= 0
(Connection Alice LIF) = 0
The size of Alice's LIF is such that the gradients in (Connection Alice
LIF) are ignorable. You can think of the LIF as a ball at the bottom of
a potential well with a very small zero point jiggle - this is only a
rough analogy.
http://rsc.anu.edu.au/~sevick/groupwebpages/images/animations/capture_3D0282.jpg
Let Bob be a nearly coincident to the Baron non-geodesic LNIF observer,
then in Bob's POV
D^2x^u(Baron)/ds^2
= d^2x^u(Baron)/ds^2 + (Connection Bob)^uvw(dx^v(Baron)/ds)(dx^w(Baron)/ds)
= 0
(Connection Bob) =/= 0
Because an electromagnetic force is acting on Bob.
Finally in the Baron's rest frame, which in this case is also geodesic
D^2x^u(Baron)/ds^2 = d^2x^u(Baron)/ds^2
= 0
dx^i/ds = 0
i = 1,2,3 spacelike
dx^0/ds = 1
This covers all of the cases.
Homework Problem: Put an external force on the Baron. Describe all the
cases.
In answer to Z's question about Wheeler. When Wheeler says gravity is
curvature he means tensor "geodesic deviation" he does not mean
contingent non-tensor non-geodesic "g-force."
"Gravity" and "gravity field" mean different things in different
contexts. Usually this is not a problem for physicists to get the nuance
intended in each specific. It is a problem for Z.
.
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