CJP.DNAg.TD.999.R3.doc - Finite Gravitational Time Dilation...

Finite Gravitational Time Dilation in Black Holes
Using Dynamic Newtonian Advanced Gravity
(DNAg)
Andrew Worsley 1
Joseph Worsley 2
1
UCL, Gower Street, London WC1E 6BT, UK. [email protected]
2
Bristol University UK. [email protected]
Key words/PACS: gravitation, 95.30.Sf; time dilation, 06.30Ft; black holes, 04.70Bw; Cygnus X-1.
1

Abstract
In this paper we use a dynamic form of modified Newtonian
gravity to reformulate the equations for gravitational time
dilation. Here we introduce the generic equations for gravitational
time dilation. It is shown that these equations agree exactly with
gravitational time dilation in satellite navigation systems. The
equations are also in agreement with a reanalysis of observations
of gravitational red shifts in black hole accretion discs. Using
these equations, we translate the time dilation into a finite value
at the black hole event horizon. Thus this reformulation resolves
the difficulties of the existence of black hole singularities.
Importantly these dynamic gravitational equations provide
testable predictions in the vicinity of black holes.
2

1. Introduction:
The current description of gravity in modern classical physics has
been successful at predicting gravitational observations in low and
medium gravitational fields. However, there remain some outstanding
issues particularly with regards galactic rotation curves, and the presence
of dark matter. We have previously shown that using a dynamic form of
Newtonian advanced gravity (DNAg), it is possible to explain both the
galactic rotation curves [1, 2], and the presence of dark matter [2, 3].
It is however, essential to have an understanding of time dilation
and black hole gravitational physics and in particular the effects of
gravitational time dilation in the vicinity of black holes. The generic
equations for time dilation for this model have not previously been
published. Here we present original equations for gravitational time
dilation.
The current interpretation of the gravitational equations predicts
that the effects of time dilation, leads to infinite time dilation at the event
horizon and the formation of singularities [4, 5] .
Classical model of gravitational time dilation :
2
2
1 1 1
Rc
GM
z    (1)
where M is the gravitational mass, c is the speed of light and G the gravitational constant,
R is the radius for space.
However, it is possible that an adaptation to the gravitational equations
can be made, which can account for time dilation in the presence of black
holes. To do this it is first possible to parameterize the gravitational
equations for space and time separately.
3

For space, the perihelion advance, in Straumann [6] is given by:
2
2 6
tan
L
m 
      (2)
where m= GM/c 2 and L 2 /m = a(1- e 2 ), a is the semi-major axis and e is the eccentricity .