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European Journal of Applied Sciences – Vol. 12, No. 4

Publication Date: August 25, 2024

DOI:10.14738/aivp.124.17250.

Leyva, B. Q. (2024). Light Time Delay and Bending of Photon Sources Located in the Solar System Calculated with the Extended

Newtonian Inertia (ENI). European Journal of Applied Sciences, Vol - 12(4). 31-49.

Services for Science and Education – United Kingdom

Light Time Delay and Bending of Photon Sources Located in the

Solar System Calculated with the Extended Newtonian Inertia

(ENI)

Barbaro Q Leyva

Independent Work USA

ABSTRACT

An equation for the gravitational redshift was obtained which coincides with the

GR result for the case of an emitter being far from the gravitational field. The

expression was obtained from the relativistic Einstein-Doppler effect without

using general relativity. This suggested the equivalence between a change of speed

and a change of gravitational potential and triggered a generalization of the 2nd

law of Newton using high order derivative of the gravitational/inertial

acceleration. The implied gravitational time dilation (considering a variable speed

of light) was used in the ENI model previously developed by the author to calculate

the Shapiro delay for light sources coming from the solar system. The obtained

results agree very well with the results of general relativity (maximum relative

deviation of about 0.2 percent for Pluto). The use of the time dilation factor in the

Newtonian model yields also results very close to GR. Therefore, the factor of 2

commonly attributed to the difference between the Newtonian and the GR model

did not appear here. The speed of light for photons grazing the Sun (Earth and the

Milky Way) was calculated to be almost the one in vacuum but for a neutron star it

is about 30 percent higher than the one in vacuum. The ENI model yielded results

consistent with the GR models when applied to the Mariner 6 and 7 missions, and

to the Cassini probe. The ENI model for the deflection of light was extended

(without invoking the time dilation) to finite distances of the source and the

observer with results being in very good agreement with GR. The decomposition of

the starlight frequencies when approaching the Sun was hypothesized

Keywords: Newtonian gravitation, Newton’s 2nd law, Theory of relativity. Gravitational

deflection, Gravitational time delay,

INTRODUCTION

While trying to reproduce the Shapiro time delay using the ENET model [1], [2], ref [3] was

found which calculated the Shapiro time delay using numerical Newtonian simulations in

which light was considered a particle (photon) and used an approximation to the redshift

factor to calculate the photon trajectory and travel time. This and other models that use a

variable speed of light made me think that perhaps accounting for the light time dilation

(obtained from the redshift) in the ENET model, good results could be obtained when

compared to the ones from GR. Part of the work presented here was triggered by that thought.

An equation for the gravitational redshift was obtained which coincides with the GR result for

the case of an emitter being far from the gravitational field. That expression was obtained

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Services for Science and Education – United Kingdom 32

European Journal of Applied Sciences (EJAS) Vol. 12, Issue 4, August-2024

from the relativistic Einstein-Doppler effect without using the details of GR. This suggested

the equivalence between a change of speed and a change of gravitational potential.

The implied (by the redshift equation) gravitational time dilation factor was used in the ENET

model (here named ENI), previously developed by the author, to calculate the Shapiro time

delay for light grazing the Sun emitted from sources located in the solar system. The obtained

results agree very well with the results of general relativity (maximum relative deviation of

about 0.2 % for Pluto).

The use of that factor in the Newtonian model yields also results very close to GR. Therefore,

the factor of 2 commonly attributed to the difference between the Newtonian and the GR

model did not appear here.

The ENI model yielded results consistent with the GR models when applied to the Mariner 6

and 7 missions, and to the Cassini probe

In ref [2] the bending of light of photons grazing the Sun was calculated assuming that the

source and the observer were located at an infinite distance from the Sun. In this work the

model is extended to finite distance of the source/observer without invoking the time

dilation.

The paper presented here has pedagogical importance as an introduction to the classical

topics of general relativity (i.e. intrinsic perihelion precession of planets, light deflection by

massive bodies, Shapiro time delay and light redshift) at wide range of student levels which

could include (but not limited) high school and undergraduate students.

ENI MODEL FOR THE PHOTON

The Michelson-Morley experiment result along with the Lorentz-Fitzgerald transformation,

and Einstein special theory of relativity support the concept of the apparent time dilation and

length contraction given by

2 1/ 2

'(1 )

− t = t − 

: Time dilation

2 1/ 2 x = x'(1−  )

: Length contraction

Where,

 = v/c

➢ v: Is the speed of the moving reference frame and c is the speed of light in vacuum.

➢

c

: The speed of light in vacuum

The primed variables correspond to a reference frame moving with respect to a reference

frame considered at rest.

The concept of acceleration requires 2 space-time intervals (two successive-interval boosts of

contraction/dilation) to relate 3 space-time events. Considering another reference frame