The Topsy Turvy World of Curved Space-Time Horizon Hologram Computers

Where we are at r = 0 - the frequency shift formula here is

dt = invariant

therefore,

ds(0)/goo(0)^1/2 = ds(r)/g00(r)^1/2

frequency = 1/ds

1/f(0)goo(0)^1/2 = 1/f(r)g00(r)^1/2

f(r)/f(0) = [goo(0)/goo(r)]^1/2 ---> (1 - / ^2/3)^-1/2 ---> infinity at our future horizon

This is an infinite blue shift for the static LNIF at our future horizon where 1 - / ^2/3 = 0 detecting light signal from us at r = 0 along its past light cone that connects with our future light cone. Similarly the advanced wave back from our future will be infinitely redshifted. However, there is complete cancellation in the Cramer transaction, the return wave in the Novikov loop is always the same frequency as the offer wave no matter what set of observer-participators do the measurements. Each set will have a consistent description - though not the same description.

This is in contrast to us outside a black hole where we as static LNIF are at r ---> infinity

f(r)/f(infinity) = (1 - 2GM/c^2r)^-1/2 ---> infinity at the horizon, i.e. we see zero frequency at r = infinity, i.e.  infinite red shift for a retarded wave coming from the black hole horizon along our past light cone.

There are other sets of observers - not all of them are physically interesting. The math allows more choices than are physically convenient. Note that Wickedpedia does not use the conformal observers that Hoyle and Narlikar use here in 4)

4) Note the issue of the red and blue shifts is very tricky depending on the state of acceleration of the absorber detectors.

The cosmological red shift z is, for the de Sitter (dS) metric relative to us at proper time zero

1 + z = (wavelength at co-moving absorber)/(wavelength at comoving emitter) = e^/(proper time at absorber) ---> infinity at our future horizon.

To see the connection with the conformal time diagram Fig 1.1

Conformal time tau = /^-1/2[1 - e^-/^1/2proper time)]

infinite proper time at our future horizon is finite conformal time 

tau = /^-1/2

The conformally flat dS metric is

ds^2 = (1 - /^1/2tau)^-1[Minkowski metric]

---> infinity at the future event horizon consistent with zero frequency.

This is for co-moving observers in the accelerating Hubble expansion flow.

Static LNIF observers see something entirely different at fixed r where

g00 = 1 - / ^2 = -1/grr

static LNIFs see an infinite blue shift of light coming at r = 0 as they adiabatically approach r --- /^-1/2

indeed, their real tensor covariant acceleration ~ Unruh temperature needed to stay at fixed r is

g(r) = 2c^2/ (1 - / ^2)^-1/2 ---> infinity at the future horizon.

This is an example of horizon complementarity - one has to specify precisely the total experimental arrangement to get sensible answers not only in quantum theory, but also in Einstein's theory of curved space-time gravity.