## Wednesday, December 15, 2004

Black Holes

In Newton's gravity theory, gravity is a force like any other, e.g. the electric force. Every crackpot scheme I have seen about how Einstein is wrong (about gravity let's leave special relativity aside for now) is based on the amateur's inability to get past this cognitive barrier. Z is no exception to this rule. Neither is the so-called PV theory pushed by Eric Davis in the recent USAF teleportation report.

For example in Newton's theory using Galileo's relativity of absolute simultaneity, which is OK when v/c << 1 we can approximate the Earth's surface as a Newtonian Global Inertial Frame (NGIF) with a test particle of mass m described by the Lagrangian

L = (1/2)mv^2 - mgz

z is altitude from surface of the Lab inertial frame.

A typical solution is the parabola (for initial conditions only along z) in the x-t plane

z(t) = z(0) + v(0)t - (1/2)gt^2

Note that m cancels out of the problem because of Galileo's equivalence principle.

If we put in an initial horizonal speed u(0) along x then

x(t) = x(0) + u(0)t

Solve for t and eliminate t in z(t) replacing it by x and one gets another parabola for z(x).

In Newton's theory this parabola is a NON-INERTIAL motion in the INERTIAL FRAME caused by a real gravity force Fg where Newton's second law Fg = ma is obeyed.

The picture does a bait and switch Gestalt Shift in going from Newton to Einstein that most people simply do not get - including some physicists.

In Einstein's theory of gravity, called geometrodynamics, Fg is eliminated. It is a "force without Force". The parabola is an INERTIAL MOTION called the "geodesic", i.e. the straightest possible path not in 3D space but in 4D CURVED spacetime. The Earth's surface is not an INERTIAL FRAME. Small pieces of the Earth's surface are now REST LOCAL NON-INERTIAL FRAMES (REST LNIF). At rest relative to center of mass of Earth because of quantum pressures from Pauli's exclusion principle and the electric force so that detectors at rest on Earth's surface move on non-geodesic world lines. What we think of as the "gravity force of weight" is really only the electrical force pushing us off the natural force free weightless "free float" geodesic. This is why Astronauts are continually weightless in their free float elliptical orbits round Earth with rockets off. Those closed orbits are the straightest possible paths in the curved space-time caused by the Earth's mass M.

If the distance from center of Earth to surface is r, the curvature scale at the surface is ~ GM/c^2r^3 = g/c^2r with dimensions 1/(Area) where here (Area)^1/2 ~ 1AU ~ 10^13 cm ~ local radius of curvature of space-time from the Earth at its surface.

The "g-force" is only felt by local observers pushed off geodesics by non-gravity forces. Local observers on geodesics are weightless with zero g-force.

The g-force is not a tensor because if a tensor is zero in one local frame it is zero in all other local frames connected to it. You cannot transform a non-zero tensor to a zero tensor or vice versa within the rules of Einstein's geometrodynamics from 1916. You cannot even say that the g-force has a tensor piece that is not identically zero. Such a tensor piece of g-force is called "non-metricity" and that is zero in Einstein's 1916 theory.

Example
Schwarzschild approximation to the curved spacetime above the surface of Earth. The local frame invariant differential separation between two neighboring events is

ds^2 = (1 - 2GM/c^2r)(cdt)^2 - (1 - 2GM/c^2r)^-1dr^2 - dl^2(flat spherical polar)

The REST LNIF time clicked off by a clock at rest at distance r is

dT(LNIF) = (1 - 2GM/c^2r)^1/2(cdt) Alice's REST LNIF clock at fixed r

Bob free falling past Alice in contrast measures

dT(LIF) = dT(LNIF)/(1 - 2GM/c^2r)^1/2

Note that

2GM/c^2r < 1

This is the LOCAL GRAVITY RED SHIFT

dT(LIF) > dT(REST LNIF) at SAME EVENT in FREE SPACE!

This same red shift is also seen by a distant REST LNIF observer at r -> infinity

That is

dT(LIF at r) = dT(REST LNIF at r -> infinity)

Hal Puthoff's PV theory only has half of Einstein's equivalence principle in his "Tables I & II" with an incorrect ontology and measurement theory. That is, he only has the RHS of the above equation. Hal Puthoff does not believe we need to use tensors in his approach to the "metric engineering" for propellantless propulsion, i.e. "warp drive".

What is the permanent gravity field?

We need the idea of the Killing vector field.

Curved spacetimes can, but need not, have special symmetries. If the metric guv(LNIF) is invariant under {LC} parallel transport along a family of worldlines, the field of tangent vectors to those world lines is called a Killing vector field.

The Schwarzschild metric has the Killing vector field T^u = (1,0,0,0).

Tu is timelike inside local light cones for r > 2GM/c^2

Tu is lightlike i.e. T^uTu = 0 on the Event Horizon of the vacuum black hole at r = 2GM/c^2

Tu is spacelike outside the local light cone for r < 2GM/c^2r where the coordinate patch for r > 2GM/c^2 breaks down.

The permanent g-force, more precisely, the g-acceleration for REST LNIF observers is the

SURFACE GRAVITY = a(1 - 2GM/c^2r)^1/2

a = THE NORM OF THE PROPER 4-acceleration needed to keep the LNIF observer at a fixed r. It cannot exist physically without some non-gravity force!

The SURFACE GRAVITY at the EVENT HORIZON of the BLACK HOLE is ~ c4/GM