Re: Misinterpretation of the radial parameter in the Schwarzschild solution?

LEJ Brouwer wrote:
The field equation definitely is not invariant under exchange of r and
t, as the Schwarzschild metric solving it is clearly not.

You obviously do not know what the field equation is. As I keep saying: you _REALLY_ need to learn the basics. What you think is the "field equation" is not -- it is the field equation _projected_ onto specific coordinates. More on this below.

Here is the Einstein field equation (omit the cosmological constant):

G = T where G is the Einstein curvature tensor, and T is
the energy-momentum tensor, and I have chosen
convenient units (c=1 and (8pi)k=1, where k is Newton's
gravitational constant)

Note that neither r nor t appear there; indeed no coordinates appear at all, and this equation is completely independent of coordinates. So exchanging r and t has no effect whatsoever, because the field equation is completely independent of them.

This equation is valid _throughout_ the manifold (we'll come back to that). For the Schwarzschild spacetime T=0 (i.e. vacuum), but I'll retain it for generality.

It is common to project this equation onto an arbitrary (but specific) coordinate basis, obtaining a set of equations:

G^ij = T^ij [i and j are component indexes ranging 0,1,2,3]

Of course this can be done only within the region of the manifold for which the specific coordinates used are valid. One can further express the {G^ij} in terms of the metric components {g_ij} and their derivatives with respect to the coordinates {x^i}; one then has a set of nonlinear second-order partial differential equations of the {g_ij}. One can re-label the coordinates: {x^i} => {t,r,theta,phi} -- and write down the equations as you are familiar with in terms of such coordinate labels.

The key point is that when written in terms of tensor components and coordinates, the equation is valid only in the region of the manifold in which the coordinates are valid. The _actual_ field equation (above), expressed in terms of the tensor themselves, has no such restriction.

[It should come as no surprise that when projecting a
tensor equation onto a set of coordinates, the resulting
set of equations depends on those coordinates,
_including_ their region of validity.]

In solving for the metric components of the Schwarzschild manifold, one normally selects spherical coordinates {t,r,theta,phi) at spatial infinity and solves the set of equations, giving the usual Schw. line element. Sloppy mathematicians (aka most physicists) don't bother to point out the limitations of this procedure: it is valid _ONLY_ within the region for which those coordinates are valid. Indeed, a careful analysis shows that those coordinates are only valid in the region:
-infinity<t<infinity, r>2M, 0<theta<pi, 0<phi<2pi
This is known as the exterior region.

One can, however, look at the line element and notice that it is also valid in the region:
-infinity<t<infinity, 0<r<2M, 0<theta<pi, 0<phi<2pi
But in this region r is timelike and t is spacelike, with the future in the -r direction (hence the singularity at r=0 is in the _future_ of every point in this region). This is known as the interior region.


All your blathering is simply ignoring the basic mathematical structure presented above. You _really_ need to learn the basics. There is no shortcut, and the papers you quote are equally ignorant of the basic mathematics used in GR and described above. <shrug>

In short, you think the portion of the manifold r<2M "does not exist", while in reality it is the _COORDINATES_ you use that do not exist (i.e. are invalid), but the interior region of the manifold is perfectly valid, albeit rather strange to naive Euclidean sensibilities.


Tom Roberts
.

Relevant Pages

  • Re: Number of gravitational field components?
    ... In general, a second rank tensor field on a four-dimensional Lorentzian manifold has 16 components at each event, but the electromagnetic tensor field F_is antisymmetric, so it only has 6 algebraically -independent- components at each event. ... in general relativity, E gives the "electrogravitic" part of the curvature, which describes the tidal forces on a small object. ... In a vacuum spacetime, a kind of "trace" of the Riemann tensor, the Ricci tensor, vanishes by virtue of the Einstein field equation. ... In both Maxwell's theory of electromagnetism and Einstein's theory of gravitation, we can classify fields by their algebraic symmetries, and the various types can be characterized by saying that for an "adapted" congruence X, the decompositions will have distinctive special forms. ...
    (sci.physics.research)
  • Re: Doubts about relativity dogmas
    ... term "propagation of gravity" is unfortunate but that is people mean. ... In GR, gravitation is a manifestation of the geometry of the manifold, and that simply _IS_ -- it does not "propagate" or change or anything, it simply is a property of the spacetime manifold. ... To an inhabitant of the manifold these fields will change, but to an external analyst such changes are completely determined by the solution to the field equation that the manifold represents. ...
    (sci.physics.relativity)
  • Re: GR1916, available online.
    ... valid manifold modeling any physical world, ... The field equation of GR does not describe trajectories, it describes the geometry of the manifold. ... and is an important component in determining trajectories in general. ... in physics. ...
    (sci.physics.relativity)
  • Re: GR1916, available online.
    ... components of the electric and magnetic fields to the 10 components ... A Lorentz transformation or any other coordinate ... "static" Einstein Field Equation from that, ... The Riemann Christoffel curvature tensor ...
    (sci.physics.relativity)
  • Re: Misinterpretation of the radial parameter in the Schwarzschild solution?
    ... then you would have to 'merely' shuffle them in the ... Einstein field equations too - which means that what you are now ... I must admit I haven't check whether the field equation is invariant. ... the manifold in the region r<2M is every bit as much a manifold as that ...
    (sci.physics.relativity)