On Jan 29, 2013, at 12:51 AM,
Yes Paul, it is possible to treat inertia, and inertial forces in particular, in GRT just as one does in Newtonian mechanics -- that is, inertia (and its measure, mass-energy) is a primary quality of matter not requiring further explanation.
Jack: "Inertia", "Inertial" is used in self-contradictory ways, often by the same author in the same paper or text book.
Wheeler & Co seem to use it consistently. In the book "Gravity & Inertia" by the "origin of inertia" they do not mean the rest mass "m" of the observed test particle in Newton's second law F = ma in inertial frames. They mean the pattern of force free geodesics. Note, that in non-inertial frames
F + Ffictitious = ma'
where F is the external non-gravity force spacelike 3-vector, a is the proper acceleration 3-vector in the inertial frame. In a rotating non-inertial frame for example in the v/c << 1 limit with rotation 3D pseudo-vector w with apparent velocity v' and radial vector r' in the non-inertial frame.
Ffictitious ~ -mw x w x r' - 2m w x v' - GMmr'/r^3 - mdw/dt x r'
= 0 in an inertial frame
The apparent acceleration (aka "kinematical") of the test particle in the non-inertial frame is a'
In the formalism of Einstein, Ffictitious is contained in the Levi-Civita connection term of the covariant derivative.
Case 1 test particle on a geodesic
F = 0, a = 0
Here the fictitious force is really fictitious as far as the test particle m is concerned, though it may be caused be real forces on the detector/observer.
Case 2 test particle is off-geodesic - case of uniform circular motion, with M = 0
Specializing to the now non-inertial co-rotating rest frame of the test particle
F =/= 0, a =/= 0, dw/dt = 0, a' = 0, v' = 0
All we have left is
F = -mw x w x r'
Where F is the real radially inward non-gravity centripetal force, with an equal and opposite centrifugal force on the source of F.
Case 3 gravitational geodesic orbit - no real force on the test particle.
-mw x w x r' - 2m w x v' - GMmr'/r^3 - mdw/dt x r' = ma'
m cancels out of the equation.
w and v' are the rotation pseudo-vector of the source mass M and relative velocity of the detector with the test particle of mass m.
The proper acceleration of the test particle a = 0.
Jim: That is why I included Abraham Pais's quote of Einstein on Mach's principle that is the header for Chapter 2 in the book. But I think you will agree that the discussion of the past year and a half about inertia has been about substantially more than just semantic disagreements. Unless there is some backsliding, we now seem all to be talking about the same thing, though there is still no agreement on the CAUSE of inertia and inertial "effects" [that is, inertial reaction (third law) forces].
Jack: Jim your above remark is mostly metaphysics not physics. Your remark on the "CAUSE of inertia" is not Popper falsifiable. Again, do you mean "m" or "geodesic pattern"? Probably the former. The cause of Newton's third law is not a mystery. It's mainstream physics Noether's "first" theorem applied to the translation group T3 in an isolated non-dissipative system.
Noether's (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918.[1] The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system's behavior can be determined by the principle of least action.
Noether's theorem has become a fundamental tool of modern theoretical physics and the calculus of variations. A generalization of the seminal formulations on constants of motion inLagrangian and Hamiltonian mechanics (developed in 1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian alone (e.g. systems with a Rayleigh dissipation function). In particular, dissipative systems with continuous symmetries need not have a corresponding conservation law. ...
Examples
Time invariance
For illustration, consider a Lagrangian that does not depend on time, i.e., that is invariant (symmetric) under changes t → t + δt, without any change in the coordinates q. In this case,N = 1, T = 1 and Q = 0; the corresponding conserved quantity is the total energy H[5]
Translational invariance
Consider a Lagrangian which does not depend on an ("ignorable", as above) coordinate qk; so it is invariant (symmetric) under changes qk → qk + δqk. In that case, N = 1, T = 0, andQk = 1; the conserved quantity is the corresponding momentum pk[6]
In special and general relativity, these apparently separate conservation laws are aspects of a single conservation law, that of the stress–energy tensor,[7] that is derived in the next section.
Rotational invariance
The conservation of the angular momentum L = r × p is analogous to its linear momentum counterpart.[8] It is assumed that the symmetry of the Lagrangian is rotational, i.e., that the Lagrangian does not depend on the absolute orientation of the physical system in space. For concreteness, assume that the Lagrangian does not change under small rotations of an angle δθ about an axis n; such a rotation transforms the Cartesian coordinates by the equation
Since time is not being transformed, T=0. Taking δθ as the ε parameter and the Cartesian coordinates r as the generalized coordinates q, the corresponding Q variables are given by
Then Noether's theorem states that the following quantity is conserved,
In other words, the component of the angular momentum L along the n axis is conserved.
If n is arbitrary, i.e., if the system is insensitive to any rotation, then every component of L is conserved; in short, angular momentum is conserved.
http://en.wikipedia.org/wiki/Noether's_theorem
Jack: NO NEED FOR ASTROLOGY!
Jim: As far as I am concerned, this is real progress. I gave up hope of even getting this far a year ago. That is why my participation in this discussion over the past few months has been minimal. The other part of the issue of inertia, on which I still hold out no hope of getting understanding and agreement, is that we now -- 90 years after Einstein said what was reported by Pais in the mentioned quote -- know a lot more about cosmology. Indeed, we know that critical cosmic matter density obtains, and accordingly that space is flat at cosmic scale AS MATTERS OF FACT.
Jack: This is irrelevant it seems to me.
Jim: That means that, as Jack is now calling it, Sciama's "screening factor" -- the coefficient of time derivative of the gravitomagnetic vector potential in the gravielectric field equation (in the approximation where only the g_oo and g_oi potentials need be considered) is one. And that means that inertial "effects" are accounted for as the gravitational action of chiefly cosmic matter (where "matter" is everything that gravitates).
Jack: Jim, I think your sentences here are very ambiguous. First of all g00 and g0i are local observer frame-dependent. They are not invariant geometric objects. Also the universe on the large scale is not rotating so g0i = 0 in the usual representations. For example, in the usual comoving observer representation g00 = 1 and g0i = 0, so what happened to your theory here?
General metric for LIF geodesic co-moving observers
The FLRW metric starts with the assumption of homogeneity and isotropy of space. It also assumes that the spatial component of the metric can be time-dependent. The generic metric which meets these conditions is
where ranges over a 3-dimensional space of uniform curvature, that is, elliptical space, Euclidean space, or hyperbolic space. It is normally written as a function of three spatial coordinates, but there are several conventions for doing so, detailed below. does not depend on t — all of the time dependence is in the function a(t), known as the "scale factor".
[edit]Reduced-circumference polar coordinates
In reduced-circumference polar coordinates the spatial metric has the form
k is a constant representing the curvature of the space. There are two common unit conventions:
k may be taken to have units of length−2, in which case r has units of length and a(t) is unitless. k is then the Gaussian curvature of the space at the time when a(t) = 1. r is sometimes called the reduced circumference because it is equal to the measured circumference of a circle (at that value of r), centered at the origin, divided by 2π (like the r of Schwarzschild coordinates). Where appropriate, a(t) is often chosen to equal 1 in the present cosmological era, so that measures comoving distance.
Alternatively, k may be taken to belong to the set {−1,0,+1} (for negative, zero, and positive curvature respectively). Then r is unitless and a(t) has units of length. When k = ±1, a(t) is the radius of curvature of the space, and may also be written R(t).
A disadvantage of reduced circumference coordinates is that they cover only half of the 3-sphere in the case of positive curvature—circumferences beyond that point begin to decrease, leading to degeneracy. (This is not a problem if space is elliptical, i.e. a 3-sphere with opposite points identified.)
http://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric
Jack: Not only that, but our future universe is asympotically de Sitter which in the static LNIF representation is
Static coordinates
We can introduce static coordinates for de Sitter as follows:
where gives the standard embedding the (n−2)-sphere in Rn−1. In these coordinates the de Sitter metric takes the form:
Note that there is a cosmological horizon at .
http://en.wikipedia.org/wiki/De_Sitter_space
Jack to Jim: Indeed you did not seem to know that g00 = 0 is a condition for a horizon. Note that here g0i = 0.
Jack: Machian Screening Factor?
Analogy to Debye Screening
Debye length
From Wikipedia, the free encyclopedia
In plasmas and electrolytes the Debye length (also called Debye radius), named after the Dutch physicist and physical chemist Peter Debye, is the scale over which mobile charge carriers (e.g. electrons) screen out electric fields. In other words, the Debye length is the distance over which significant charge separation can occur. A Debye sphere is a volume whose radius is the Debye length, in which there is a sphere of influence, and outside of which charges are screened. The notion of Debye length plays an important role in plasma physics, electrolytes and colloids (DLVO theory).
http://en.wikipedia.org/wiki/Debye_length
For off-geodesic motion F = ma Newton's 2nd law
Jim based on an obscure model of Sciama & Berry? (who denies it in email to me) says
F = ma should be replaced by
F = (cosmological screening factor) ma
Exactly what it is and how to compute it using real Einstein GR as opposed to a completely unjustified EM analogy is not clear to me. I hope it is clear in Jim's book?
Suppose it is really there? Even so, it says nothing about m itself. You can think of m as the cosmologically bare mass of the test particle. m is still determined locally by Higgs field, + QCD + standard nuclear, atomic, molecular, solid state physics of binding energies.
Next step is to look at the Hoyle-Narlikar Wheeler-Feynman theory of gravity.