Can anybody help me understand the calculation of the geodesic effect as
presented in Section 4.7 of the Foster & Nightingale text "A Short Course in
General Relativity"?

I can reproduce all of the calculations in the section and obtain a facile
understanding, but have trouble with interpretation and details that are
rather finessed by the authors.

In the opening paragraph they state: "The orthogonality condition simply
means that lamba^mu has no temporal component in an instantaneous rest frame
of an observer traveling along the geodesic."

But later we see that lambda^0 is a constant times lambda^3, which is not
zero. So is the above statement only true in flat spacetime, or is the frame
that the lambda vector is expressed in not the instantaneous rest frame of
the observer traveling along the geodesic?

Also, when I try to calculate the length of the lamda vector using the
Schwarzschild metric I do not obtain a constant value. Sould I
calculate -ds^2 with a nonzero lambda^0 component or should I simply
calculate the spatial part using lamda^1 to lamda^3? In either case the
length is not constant.

David Park

http://home.earthlink.net/~djmp/

David Park wrote:
Can anybody help me understand the calculation of the geodesic effect as
presented in Section 4.7 of the Foster & Nightingale text "A Short Course in
General Relativity"?

I don't have that book, and you're unlikely to get a useful response
around here without providing enough detail so someone doesn't need the
book.


I can reproduce all of the calculations in the section and obtain a facile
understanding, but have trouble with interpretation and details that are
rather finessed by the authors.

In the opening paragraph they state: "The orthogonality condition simply
means that lamba^mu has no temporal component in an instantaneous rest frame
of an observer traveling along the geodesic."

I am unable to guess what "lamba^mu" is. As it has no temporal component
in that frame I'm tempted to guess it is the components of the
4-acceleration projected onto that observer's coordinates; but as the
path is a geodesic the 4-acceleration is identically 0 and this would
have been phrased differently, and it's inconsistent with your next
paragraph.

How do Foster & Nightingale define lamba^mu?


But later we see that lambda^0 is a constant times lambda^3, which is not
zero. So is the above statement only true in flat spacetime, or is the frame
that the lambda vector is expressed in not the instantaneous rest frame of
the observer traveling along the geodesic?

Also, when I try to calculate the length of the lamda vector using the
Schwarzschild metric I do not obtain a constant value. Sould I
calculate -ds^2 with a nonzero lambda^0 component or should I simply
calculate the spatial part using lamda^1 to lamda^3? In either case the
length is not constant.

That's incomprehensible to me without knowing what lambda^mu represents.

Note: in your earlier paragraph, lambda^mu appears to be path-related,
but in this last paragraph it does not. Which is it?


Tom Roberts

Tom,

I was somewhat hoping I might get a response from someone who had worked
with the Foster & Nightingale book, a pretty good introductory text. I don't
think I will try to reproduce the detailed equations in a posting.

Lamda is a vector (such as the axis of a gryoscope?) that is parallelly
transported around a circular orbit in the Schwarzschild metric. Basically,
in the calculation they calculate the phase change in one orbit, but also
take into account that the local frame is rotating. One has a period in
proper time and the other in coordinate time and it is by taking account of
this difference that the geodesic effect arises. The vector is slightly
rotated after one orbit.

But I was trying to visualize the lambda vector and see that it was constant
length, and also what they meant by saying it had no temporal component
(since the solution they gave seems to have all four components.) When I
tried to calculate the length, using the metric, it did not seem to be
constant.

David Park



"Tom Roberts" wrote in message
y.com...
David Park wrote:
Can anybody help me understand the calculation of the geodesic effect as
presented in Section 4.7 of the Foster & Nightingale text "A Short
Course in
General Relativity"?

I don't have that book, and you're unlikely to get a useful response
around here without providing enough detail so someone doesn't need the
book.


I can reproduce all of the calculations in the section and obtain a
facile
understanding, but have trouble with interpretation and details that are
rather finessed by the authors.

In the opening paragraph they state: "The orthogonality condition simply
means that lamba^mu has no temporal component in an instantaneous rest
frame
of an observer traveling along the geodesic."

I am unable to guess what "lamba^mu" is. As it has no temporal component
in that frame I'm tempted to guess it is the components of the
4-acceleration projected onto that observer's coordinates; but as the
path is a geodesic the 4-acceleration is identically 0 and this would
have been phrased differently, and it's inconsistent with your next
paragraph.

How do Foster & Nightingale define lamba^mu?


But later we see that lambda^0 is a constant times lambda^3, which is
not
zero. So is the above statement only true in flat spacetime, or is the
frame
that the lambda vector is expressed in not the instantaneous rest frame
of
the observer traveling along the geodesic?

Also, when I try to calculate the length of the lamda vector using the
Schwarzschild metric I do not obtain a constant value. Sould I
calculate -ds^2 with a nonzero lambda^0 component or should I simply
calculate the spatial part using lamda^1 to lamda^3? In either case the
length is not constant.

That's incomprehensible to me without knowing what lambda^mu represents.

Note: in your earlier paragraph, lambda^mu appears to be path-related,
but in this last paragraph it does not. Which is it?


Tom Roberts

David Park wrote:
I was somewhat hoping I might get a response from someone who had worked
with the Foster & Nightingale book, a pretty good introductory text. I don't
think I will try to reproduce the detailed equations in a posting.

Lamda is a vector (such as the axis of a gryoscope?) that is parallelly
transported around a circular orbit in the Schwarzschild metric. Basically,
in the calculation they calculate the phase change in one orbit, but also
take into account that the local frame is rotating. One has a period in
proper time and the other in coordinate time and it is by taking account of
this difference that the geodesic effect arises. The vector is slightly
rotated after one orbit.

But I was trying to visualize the lambda vector and see that it was constant
length, and also what they meant by saying it had no temporal component
(since the solution they gave seems to have all four components.) When I
tried to calculate the length, using the metric, it did not seem to be
constant.

Certainly the spin axis of a (pointlike) gyroscope is spacelike in its
instantaneously-comoving inertial frame. So if I represent its spin
4-vector by S and the tangent vector to its path (aka 4-velocity) by U,
then U.S=0 everywhere on its path.

Note that S has a time component of 0 ONLY in its
instantaneously-comoving inertial frame. In particular, wrt Schw.
coordinates its time component will be nonzero.

Just think of a Lorentz boost in a small region near its
location at some specific time, boosting from its rest
frame to the local Schw. frame -- that boost will give a
nonzero time component to a vector that is purely spatial
in the moving frame. Remember that in GR you can indeed
do this for a SMALL REGION (the gyroscope is in freefall,
but the Schwarzschild coordinates are not, so for them
you must consider a small enough spatial region so the
curvature is negligible, and a short enough time so a
freefalling frame that starts at rest wrt the Schw. coords.
has not enough time to move significantly -- now we're
comparing two freefalling local frames, and SR applies to
them). [I discuss this qualitatively only, as doing this
numerically is algebraically difficult and prone to error.]

So I'll guess that you or they intermixed coordinates -- the "no
temporal component" applies in the instantaneously-comoving inertial
frame, but the computation seems to have been carried out in Schw.
coordinates (where the temporal component is nonzero).

S certainly must have a constant norm (just consider how it behaves in
the succession of instantaneously-comoving inertial frames -- it rotates
but does not change length, as the gyroscope never spins down). But as
the computation was performed in Schw. coordinates, don't forget to use
the Schw. metric components when computing its norm.


Tom Roberts

David Park wrote:

Tom,

I was somewhat hoping I might get a response from someone who had worked
with the Foster & Nightingale book, a pretty good introductory text. I don't
think I will try to reproduce the detailed equations in a posting.

Why not? You're quoting someone else and asking the responders to lookup the
quote. That's not very polite, to say the least.

John Anderson