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**Here: “Gravitation as a
pressure force…”, part 3 = Section 7**

** **

**7.
Gravitationally-modified Maxwell equations and their link to photon
trajectories**

As emphasized by Will [40],
any viable theory of gravitation should be "complete", in the sense
that it should match with the whole of physics. However, even GR does not match
with quantum physics, because a consistent theory of "general relativistic
quantum gravity" does not exist yet, in fact it is not sure that such
theory is still expected seriously. Hence, accounting for the present state of
physics, a reasonable demand of completeness is that, in addition to a
mechanics, an alternative theory of gravitation should involve also a
description of the electromagnetism in the presence of gravitation [40]. This description
should lead to a set of "gravitationally-modified" Maxwell equations,
and a further requirement is that these modified equations should be consistent
with the geometrical optics of the theory as defined by the trajectories of
light-like particles. In other words, the theory should tell us how the
electromagnetic *rays* may be defined
from some particular electromagnetic *waves*,
and it should prove that these rays do move as light-like particles do it under
the gravitation. In GR and other metric theories, the modification of the
Maxwell equations is rather easy, because it is an application of the
well-known rule "comma goes to semi-colon": due to the commonly
accepted formulation of Einstein's equivalence principle, one just has to
substitute the curved physical metric **g** for the flat one into the covariant expression of
any law of non-gravitational physics. (However, this rule is generally
ambiguous when the expression contains higher-order derivatives. So in the case
of Maxwell's equations, one is carefully warned not to apply the rule to the
equations for the potential, and instead to apply the rule to the equations for
the field tensor **F**.) Further, in GR and other
metric theories, the transition from wave optics to geometrical optics is most
rigorously made with the help of the discontinuities equations for an
electromagnetic shock wave, with the result that the electromagnetic rays are
the bicharacteristics of Maxwell's equations, corresponding to "null"
fields, and following light-like geodesics (*cf.*
e.g. Synge [37]).

In
the present theory, the electromagnetic
field tensor **F** is stated to derive from a 4-potential in the usual
way: in particular it is antisymmetric, *F** _{mn}* = -

*F*_{lm}_{ }_{,n} +* **F*_{mn }_{, l} +* **F*_{nl }_{,m} = *F*_{lm }_{;n} +* **F*_{mn }_{; l} +* **F*_{nl }_{;m} = 0. (46)

However, we cannot use the
"comma goes to semi-colon" rule, because this theory is not a metric
theory (although it does endow the space-time with a curved physical metric **g**). Specifically, the form of
the second group of Maxwell's equations in metric theories,

_{} (47)

(where *J ^{ µ}* is the 4-current), implies that,

*T*_{ field}^{
}* ^{mn}* º (-

obeys Eq. (24), instead of
Eq. (37) - and Eqs. (24)
and (37) are equivalent only for a constant gravitational field. In the
proposed theory, the Maxwell equations are obtained as an application of
Newton's second law for a charged dust subjected to the gravitation force and
to the Lorentz force. Hence, we begin with a short outline of continuum
dynamics in the presence of gravitational and non-gravitational forces [8].
Just the same method of induction from a dust to a general continuum is used as
in Sect. 6, the difference being that now the dust particles are subjected, in
addition, to a non-gravitational *external*
force. The density **f** of the latter
is assumed given, and is expressed in terms of the physical volume measure* d**V* = Ö*g* *dx*^{ 1}* dx*^{ 2}* dx*^{
3} (with *g* º det (*g** _{ij}*)). Thus, the
non-gravitational external force over a volume element of the dust is

*d ***F**_{0} = **f*** d**V*.* *(49)

(In accordance with the
spirit of field theory, the particles are still assumed to not interact
directly: instead, their presence produces a field that exerts a force on them.
This is more intuitive with the concept of a space-filling fluid ether, of course.)
Adapting the energy equation (28) and then the expression (36) of the
4-acceleration for an individual particle to the situation with a
non-gravitational force, and using the same method as in Sect. 6, one gets

_{} , _{}_{}, (50)

where *b ^{
µ}* (

The expression of the Lorentz force on a point particle
with charge *q* may be derived uniquely from the
requirements that (i) it must be an invariant space vector by the group (7),
and (ii) this vector must reduce to the classical expression in the absence of
gravitation (in Galilean coordinates for the flat metric **g** = **g**^{ 0}),

_{}_{}. (51)

By this method, the
expression of the Lorentz force in the presence of gravitation is found to be
[8]

_{}. (52)

Considering then a continuous medium (a
"charged dust"), we define *r*_{el} º *d**q/**d**V* and *J ^{ µ}* º

_{}. (53)

The charged dust obeys the
equation for continuum dynamics in the presence of gravitation and the Lorentz
force, Eq. (50) with **f** given by Eq.
(53). On the other hand, the *total*
energy-momentum is the sum **T** = **T**_{dust} + **T**_{field} , and this total tensor must obey the equation
for continuum dynamics in the presence of gravitation and without any non-gravitational
external force, Eq. (37). Due to the linearity of the expression (38) of *b ^{ µ}*

_{} , _{}_{}. (54)

In words: *the electromagnetic field may be considered
as a "material" continuum subjected to the gravitation and to the
opposite of the Lorentz force*. *This
gives the gravitational modification of Maxwell's second group in the present
theory*. Indeed, Eq. (54) may be rewritten as

_{} (55)

and this, in the case of an
invertible 4-4 matrix (*F*^{ m }* _{n
}*), may in turn be written in the form

_{}. (56)

Equation (56) is what
corresponds, in the present theory, to Eq. (47) of GR, and *it reduces to Eq. (47) for the case of a constant gravitational field*
(in the preferred frame). The condition det **F **¹ 0 is equivalent to **E.B** º - *e _{ }^{mnry}*

_{} . (57)

However, it is difficult to
give estimates without having recourse to a numerical work (in contrast to the
situation for matter production, Sect. 6), hence this interesting and dangerous
prediction is yet to be precised.

Equation (55) is sufficient to make the transition from
wave optics to geometrical optics in the theory. The geometrical optics *in vacuo* is defined, in the present
theory, as the study of the trajectories of light-like particles, which are
ruled by the proposed extension (19) of Newton's second law, without any
non-gravitational force, thus **F**_{0}
= 0 in Eq. (19). As we have seen, the modified second group of the Maxwell
equations, Eq. (55) or Eq. (56), is just the expression of Newton's second law
for that continuous medium which is defined by the energy-momentum tensor of
the electromagnetic field. *In vacuo*,
a volume element of the field continuum will be subjected to the gravitation *and*, in general, to *internal forces* exerted by the neighbouring elements. Therefore,
the condition **F**_{0} = 0,
which must apply to an isolated photon, has to be replaced by the double
condition that (i) the * *field continuum is subjected to zero
non-gravitational external force, i.e., **f**
= 0 in Eq. (50), and (ii) it is subjected to zero internal force. Condition
(ii) means that the continuum behaves like a dust (a dust of photons), i.e.,
that its energy-momentum tensor has the form

*T ^{ }*

One may indeed show that,
independently of the exact physical nature of the material or field, Eq. (58)
is the necessary and sufficient condition ensuring that any volume element of
the continuum is subjected merely to the gravitation and to the
non-gravitational *external* force
density **f**, thus with zero* internal* force [8]: in that case, the
spatial part of the dynamical equation of the theory, Eq. (50)_{2}, is
equivalent to

**f***
d**V* + _{} **g** = _{}, (59)

where the purely material
energy of the element is defined, consistently with Eqs. (29) and (30), as

*d**E* = *d**e*_{ m}/*b* = *T*^{
0}_{ 0}*
d**V*^{ 0}_{ }/*b* = *T*^{
0}_{ 0}*
d**V*, (60)

and where the absolute
velocity of the element, **u** = *d***x***/dT *, is defined by

*u ^{ i}* º

which is the velocity of the
flux of the total matter energy, as Eq. (31) shows.

Now, in the case of the energy-momentum tensor of the
electromagnetic field (Eq. (48)), condition (ii) turns out to be equivalent to
saying that the e.m. field is a null field (with both invariants equal to
zero), whereas, due to Eqs. (53) and (54), condition (i) is equivalent to *J ^{ µ}* = 0, i.e. to the
requirement that an "empty" domain is considered (i.e. a domain in
which the e.m. field is the only form of "matter"). Thus,

_{} , _{}_{}. (62)

As we have recalled above,
Eq. (62)_{2} may be rewritten, for a null field, as

_{} **g** = _{}, (63)

where **u**_{ field} is defined by Eq. (61), thus

*u*_{ field}* ^{
i}* º

[and with

*d**E*_{ field} º *T*^{ }_{field} ^{0}_{ 0}* d**V* ,

Eq. (60)]. This definition
of **u**_{ field}, together with
Eq. (58) and the fact that **T**_{
field} has zero trace, imply that

|**v**_{field}| = *c*

[8]. This and Eq. (63) allow
us to conclude: *in the case of a null
field* *in vacuo,* *each trajectory of the e.m. energy flux is a
photon trajectory of the present theory*, as defined by Eq. (19) for a free
light-like particle (**F**_{0} = 0). This is the
link between wave optics and ray optics in the proposed theory.

To
part 4 of “Gravitation as a pressure force…”