Science > Physics > Baron Munchausen meets Rube Goldberg in Einstein's Elevator
| Topic: |
Science > Physics |
| User: |
"Jack Sarfatti" |
| Date: |
03 Sep 2006 10:49:30 PM |
| Object: |
Baron Munchausen meets Rube Goldberg in Einstein's Elevator |
PS No matter how hard the Baron fires his rocket motor, his non-geodesic
world line must be inside the forward light cone at each point on that
world line. The field of light cones is not parallel in curved
space-time so that the bundle of non-geodesics available to the Baron is
not identical to what it would be in globally flat space-time.
Therefore, one must take the connection field of the source in some
global coordinates like the asymptotic flat "Book Keeper coordinates"
(Wheeler's term) in
ds^2 = (1 - 2M/r)dt^2 - (1 - 2M/r)^-1dr^2 + r^2(dtheta^2 + sin^2thetadphi^2)
to make the problem simpler let theta = pi/2 equatorial orbit
x^1 is radial r, x^3 is azimuthal phi, the only non-zero LC-connection
components then are
(^010) = (M/r^2)(1 - 2M/r)^-1
(^133) = -r(1 - 2M/r)
(^100) = (M/r^2)(1 - 2M/r)
(^313) = 1/r
For the fiducial hovering LNIF shell observers at fixed r
dt(shell) = (1 - 2M/r)^1/2dt
dr(shell) = (1 - 2M/r)^-1/2dr
dphi(shell) = dphi
So for an arbitrary motion of the Baron subject to the above causal
light cone restriction, one needs to find the GCT from the shell
observer to the Baron to do find the Baron's coincident connection field
relative to the shell observer.
i.e. symbolically we have the CONTINGENT inhomogeneous non-tensor
transformation from non-geodesic LNIF Shell Observer to the non-geodesic
LNIF Baron
(Baron) = (GCT)(GCT)(GCT)(Shell Observer) + (GCT)Grad(GCT)
This is not intrinsic. This is not of any fundamental theoretical
interest. It is contingent even though one can formally relate it to
curvature, it's Fool's Gold. It's a Chimera. It's The Siren beckoning.
Also it is a very complex calculation in general not worth the effort
since the Baron would do better to directly measure his local g with a
scale.
On Sep 3, 2006, at 8:23 PM, Jack Sarfatti wrote:
The word "fictitious" is a bad one like "hidden variables" in Bohm's
reinterpretation of quantum theory. You certainly "feel" a "fictitious"
g and it will cause a pointer on a suitable detector to move. Therefore,
fictitious forces, or inertial forces, are physical in that they are
detectable. However, they are not tensors relative to the relevant
symmetry groups, therefore, one cannot construct objective
frame-invariants from them under those symmetry groups. In this subtle
sense I meant "g-forces are not physical" or "g-forces are fictitious"
although in a pragmatic experiential sense they are real!
<pastedGraphic.jpg>
The covariant equation for the NON-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) = F^u(non-gravity)/m(test)
If Alice is LIF her connection vanishes and she sees the special
relativity F = ma
D^2x^u(test)/ds^2 = d^2x^u(test)/ds^2 = F^u(non-gravity)/m(test)
What does the Baron see in his own LNIF rest frame when he fires a small
rocket motor on his cannon ball?
D^2x^u(Baron)/ds^2 = (Connection
Baron)^u00(dx^0(Baron)/ds)(dx^0(Baron)/ds) = F^u(Baron)/m(Baron)
(Connection Baron)^u00(dx^0(test)/ds)(dx^0(test)/ds) -
F^u(non-gravity)/m(Baron) = 0
(Connection Baron)^000(dx^0(test)/ds)(dx^0(test)/ds) -
F^0(non-gravity)/m(Baron) = 0
(Connection Baron)^i00(dx^0(test)/ds)(dx^0(test)/ds) -
F^i(non-gravity)/m(Baron) = 0
i = 1,2,3 spacelike
The experienced "fictitious" inertial nongeodesic g-force in the Baron's
frame is simply
g-force = {(Connection Baron)^i00}
= {F^i(non-gravity)/m(Baron)}
Notice that curvature is completely irrelevant i.e. gradients of the
connection play no role whatsoever.
Let the Baron in his LNIF rest frame look at a Alice who is on a
geodesic. The Baron's version of Alice's geodesic equation is
D^2x^u(Alice)/ds^2
= d^2x^u(Alice)/ds^2 + (Connection
Baron)^uvw(dx^v(Alice)/ds)(dx^w(Alice)/ds) = 0
This simplifies to
= d^2x^u(Alice)/ds^2 + (Connection
Baron)^u00(dx^0(Alice)/ds)(dx^0(Alice)/ds) = 0
= d^2x^u(Alice relative to Baron)/ds^2
+ F^i(non-gravity on Baron)/m(Baron)(dx^0(Alice relative to Baron)/ds)^2 = 0
Now above assumes Minkowski space-time.
Suppose all of the above happens in the vacuum curvature field of an SSS
source
g00 = (1 - 2M(source)/r) = - 1/grr etc.
in the usual asymptotic coordinates.
Let Alice and the Baron be momentarily close to a given r,theta, phi, t.
For example, in the equatorial plane theta = pi/2 "1" = "r"
(SSS connection)^100 = M/r^2(1 - 2M/r) etc.
One would then have to find the GCT connecting the nearly coincident
static shell observer to the Baron. If, for example, the Baron adjusted
his rocket motor to be a shell observer, then the GCT is the trivial
identity transformation and one can then use
(SSS connection)^100 = M/r^2(1 - 2M/r)
Actually doing a detailed calculation is not trivial.
From: Jack Sarfatti <adastra1@mac.com>
Date: September 3, 2006 4:58:01 PM PDT
To:
Subject: Re: Zielinski fails to grasp the subtle beauties of Einstein's
Vision
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
<pastedGraphic.jpg>
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.
<pastedGraphic.jpg>
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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| User: "" |
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| Title: Re: Baron Munchausen meets Rube Goldberg in Einstein's Elevator |
04 Sep 2006 07:19:54 AM |
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Baron Munchausen meets Rube Goldberg in Einstein's Elevator.
**********************
Did they exchange pleasantries?
.
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