** **

**On
the extension of Newton's second law**

** to theories of gravitation in curved
space-time**

*Archives of Mechanics *(Warszawa)** 48**, No. 3, pp. 551-576 (1996)

** **

** **

M. ARMINJON (GRENOBLE)

**Abstract-** We investigate the possibility of extending Newton's
second law to the general framework of theories in which special relativity is
locally valid, and in which gravitation changes the flat Galilean space-time
metric into a curved metric. This framework is first recalled, underlining the
possibility to uniquely define a space metric and a local time in any given
reference frame, hence to define velocity and momentum in terms of the local
space and time standards. It is shown that a unique consistent definition can
be given for the derivative of a vector (the momentum) along a trajectory. Then
the possible form of the gravitation force is investigated. It is shown that,
if the motion of free particles has to follow space-time geodesics, then the
expression for the gravity acceleration is determined uniquely. It depends on the
variation of the metric with space and time, and it involves the velocity of
the particle.

**1. Introduction**

This work comes from an attempt to
explore the possibility of extending the "logic of absolute motion",
which prevails in the Lorentz-Poincaré interpretation of special relativity
[8-9, 15, 20-24], so as to obtain a consistent theory of gravitation. Thus, a
theory with a preferred frame has been tentatively proposed [1-4]. Just like
general relativity (GR), this theory endows the space-time with a curved
metric. Just like in GR, special relativity (SR) holds true locally in this
tentative theory. However, an extension of Newton's second law, or rather of
its modified expression valid in SR, has been defined for a test particle (mass
point or photon) in the most general situation within this investigated theory
[4]. As it will be reported here, the way used in this theory to define
Newton's second law in a "curved space-time" turns out to be both
natural and general in its principle. Hence, it has been tried to find in the
literature such a natural and general extension, but this quest has not been
really successful. Apart from approximate equations occurring in
"post-Newtonian" treatments, two exact extensions of Newton's second
law to relativistic theories of gravitation can be found among well-known
textbooks: Landau & Lifchitz [11, §88] define this law for a *constant* gravitation field, and Møller
[18, §110] "tries to write [the equations of space-time geodesics] in the
form of three-dimensional vector equations" in a general case but, as his
sentence suggests, and as will be discussed below (note ^{1} and Sect.
4), his attempt is not fully satisfactory. Jantzen *et al*. [10] review and unify the various attempts, including the
important work of Cattaneo [6-7], to "split space-time into space plus
time" and to rewrite the relativistic equations of motion with
"spatial gravitational forces". It appears from their review that
three different definitions have been introduced, by various authors, for the
time-derivative of the momentum. These definitions will be examined in Sect. 4.
It will appear that one does not obey Leibniz' rule, while none of the other
two does involve *only* the *separate* ingredients "space
metric" and "time metric" in a given reference frame- as should
be true for a natural extension of Newton's second law. However, it seems that
one has good reasons to search for such extension and hence to find this
"missing link" [17] between classical and relativistic mechanics.

Indeed,
the Lorentz-Poincaré construction of special relativity [15, 20-21], fully
developed by Jánossy [8-9] and Prokhovnik [22-24], obtains the
"relativistic" effects as being all consequences of the
"true" Lorentz contraction assumed to affect all bodies in motion
with respect to the "ether". As it has been recently reestablished
[27] against contrary statements, it is impossible to consistently measure the
anisotropy in the *one-way* velocity of
light. This makes the Lorentz-Poincaré version empirically undistinguishable
from the Einstein version of SR [22]. The Lorentz-Poincaré interpretation
allows to concile special relativity with our intuitive notion of distinct
space and time, and thus with the most crucial concepts of classical mechanics.
However, special relativity does not describe gravitation: for gravitation,
general relativity is the current tool. But in GR, the laws of motion become a
consequence of the space-time curvature, e.g. the "free" particles
are assumed to follow the geodesic lines of the space-time metric. Thus, at
least as long as the geodesic formulation of motion has not been derived from a
generalization of Newton's second law, one is enforced to give a physical
status to space-time in GR. On the other hand, despite the experimental success
of GR, it remains unsolved problems as regards gravitation. We may mention the
problem of the singularity occurring with the gravitational collapse of very
massive objects, and the need to postulate "dark matter" in order to
explain stellar motion in galaxies. We should also mention the questions on the
influence of the coordinate condition in GR, which were raised a long time ago
(e.g. Papapetrou [19]), but that have been newly discussed by Logunov *et al.* [13-14]. Logunov *et al.* present detailed arguments
against the usual agreement that, in GR, the choice of the coordinate condition
has no physical consequence. It thus may be worth to investigate alternative,
speculative theories and to ask questions on the formulation of motion.

In
this paper, an extension of Newton's second law will be given for theories of
gravitation in curved space-time in which SR is locally valid, *including GR*. In doing so, care will be
taken to maintain *space* covariance in
a given reference frame, in order that the force be properly defined. However,
no attempt will be made to investigate the transformation of the force from one
reference frame to another. Section 2 will be focused on the definition of the
right-hand side of Newton's law, i.e. the time-derivative of the momentum: it
will be shown that this may be defined from rather compelling principles, up to
the same parameter *l* as in the tentative theory [4], and
which also must be *l* = ½ if Leibniz' rule is to apply.
In Section 3, it will be investigated which form of the gravitation force is compatible
with Einstein motion (for "free" particles), i.e. motion along
space-time geodesics. In a first step, Leibniz' rule will not be imposed but it
will be assumed, in analogy with Newtonian theory, that the gravitation force
depends linearly on the spatial derivatives of the metric and does not depend
on its time-derivative. In a second step, Leibniz' rule will be assumed, but no
restriction to the gravitation force will be imposed. In Section 4, the three
anterior definitions of the time-derivative of a spatial vector, reviewed by
Jantzen *et al*., will be examined from
the point of view of "consistency" (validity of Leibniz' rule), and
"naturalness" (space plus time separation).

**2. Definition of Newton's second law for a (pseudo-) Riemannian space-time
metric**

2.1 *Some clarification on the kind of theories considered*

We suppose that,
according to some gravitation theory, the physical standards of space and time
are influenced by a gravitation field, but that SR holds true locally (GR is
the prototype of such gravitation theories, of course). It will be useful to
recall in some detail what is meant by this, not the least because it will make
clear that this framework does not preclude to consider a preferred-frame
theory, nor does this framework imply that a fundamental physical meaning must
be given to the mathematical concept of space-time. It will also give the way
to separate the force into a gravitational force or rather a mass force, and a
non-gravitational force.

(**i**) According to a theory of this kind,
our space and time measurements may be arranged so as to be described by a
metric **g** with (1,3) signature on a 4-dimensional,
"space-time" manifold. This may be done as follows. Any possible *reference frame* F,
physically defined by a *spatial network of
"observers"* (each one equipped with a ruler and a clock, all made
in the same factory, say), allows one to define (in many ways, actually) an
associated coordinate system (*x** ^{a}*) (

The assumption that SR applies
locally is the one which allows to define a (1,3) space-time metric. This
assumption means, in the first place, this: in any reference frame, the
velocity of light, as measured on a to-and-fro path between infinitesimally
distant positions, is always the same constant *c*. Under this condition, the link between physical space and time
measurements and the metric **g** may be described as in Landau &
Lifchitz [11], it is based on using the Poincaré-Einstein synchronization
convention for infinitesimally distant clocks. Thus the proper time along the
trajectory of a mass point ("time-like" line in space-time), i.e. the
time *t* measured by a clock bound to the moving point, is
directly given by metric **g** :

(2.1)* ds*^{2}* = c*^{2}* d**t*^{2} = *g _{ab } *

Also, the distance *dl* between neighbouring observers (of a
given frame F, specified by a coordinate system), as they find by
using their rulers, or by measuring the interval *d**t* of their proper time that it takes
for a light signal to go forth and back, is expressed by a space metric tensor **h** = **h**_{F }(it depends on
the frame F ):

(2.2)
*dl*^{2} = (*c d**t* /2)^{2 }=
*h _{ij} dx^{i} dx^{j}*,

Moreover, a synchronized local time *t*** _{x}**(

(2.3) _{}.

As emphasized by
Cattaneo [6], the interval *dt*** _{x}** is invariant under any
coordinate transforma-tion that leaves the reference frame unchanged
("internal transformation") and has thus an objective physical
meaning. If the

(2.4) *dt*_{x}* /dt* = Ö*g*_{00 }º *b*

The expression (2.4) of the local
time has the immediate physical meaning of showing how clocks are affected by
the gravitation field (usually they are slowed down, i.e. *g*_{00}
decreases towards the gravitational attraction). The property "*g*_{0i} = 0" holds
true after any coordinate transformation of the form *x' *^{0} = *f *(*x*^{0}), *x' ^{ i }*=

(**ii**) The other assumption involved, in saying that SR applies
locally, is that the laws of non-gravitational physics are "formally
unaffected" by gravitation, in the following sense: in the absence of
gravitation, any such law must (or should) be formulated in the frame of SR.
Then, in the absence of gravitation, it may be expressed in a generally
covariant form, in replacing the partial derivatives, valid in Galilean
coordinates, by the covariant derivatives with respect to the *flat* space-time metric **g**^{0}
[Galilean coordinates are ones in which the flat metric **g**^{0} has
the canonical diagonal form, *g*^{ 0}* _{mn}* =

2.2 *Extended Newton law for a constant gravitation field*

Let us first consider the *static* case, i.e. the case where a frameF exists, defined by a coordinate
system (*x** ^{a}*), in
which all components

*x' *^{0} = *a* *x*^{0}
+ *f
*(*x*^{1}, *x*^{2}, *x*^{3}), *x' ^{ i }*=

thus a different range for the time
transformation than for the second property, discussed above. Then, the *right-hand side* of Newton's second law,
valid for SR, i.e. *d***P**/*dt*
with **P** the momentum including the
velocity-dependent mass, is easy to extend to any such theory of gravitation.
The velocity **v** of a test particle
(relative to the frame F ) is measured with the local time *t*** _{x}**
of the momentarily coincident observer in the frame F, and
its modulus

(2.5)* v ^{i}* º

The momentum is hence for a time-like
test particle (mass point) :

(2.6) **P**
º *m*(*v*) **v**, *m*(*v*) º *m*(*v*=0).*g** _{v}* º

(using the mass-velocity relation of
SR)[1]. For
a light-like test particle (photon), one substitutes the mass content of the
energy for the inertial mass *m*(*v*). Then we must define the derivative
of the momentum with respect to the local time. Thus in general we have to
define the derivative of a vector **w**
= **w**(*c*) attached to a
point

**x**(*c*) = (*x ^{i}*(

which moves, as a function of the
real parameter *c*, in some Riemannian space : here
this space is the 3-D domain N = N_{F} constituted by the spatial network which
defines the considered frame F. Hence the points in N are specified by their
constant space coordinates *x ^{i}*,

(2.7) _{},

where the G^{ i}* _{jk}* 's are the
Christoffel symbols of metric

(2.8) **F**_{0} + **F**_{g} = *D***P**/*Dt*** _{x }**.

Using the same equations (2.3) and
(2.5) to (2.7), the same definition may and must be used in the *stationary* case, in which the *g _{ab}*

2.3 *Extended Newton law for a general gravitation field*

In the general case where the
gravitation field is not constant in the frame F, the
new feature is that now the space-time metric **g** depends also on *x*^{0}. Hence also the space
metric **h** [Eq. (2.2)] varies, not only as as a function of the
space coordinates *x ^{i}* (as
is natural for a general Riemannian metric on a space depending on these
coordinates), but also as a function of the time coordinate

*h*_{c }* _{ij }*[(

Moreover, we have a preferred
parameter *c* = *t*** _{x}** on the trajectory. It is
easy to convince oneself that nothing needs to be changed in Eqs. (2.3), (2.5)
and (2.6), because they involve only the local components of the metric (which
now become its local and "current" components), not its variation. In
order to properly define an extension of (2.7), let us list the properties that
should be satisfied by this searched derivative of a vector on a trajectory in
a manifold equipped with a variable metric:

(a) It must be a (space) vector,
i.e. it must be contravariant for any coordinate transformation of the form *x'*^{ i} = *x'*^{ i} (*x ^{ j}*).

(b) It must be linear in **w**. More precisely, it must obviously
have the form

*D***w**/*D**c** = *(*d***w**/*d**c*)_{c}_{ = }_{c}_{0}_{ }+ **t'****.w**(*c*_{0}),

with *c*_{0} the
point of the trajectory where the derivative is to be calculated, and where **t'** is a
mixed second-order (space) tensor, transforming a (space) vector into another
one.

(c) It must reduce to (2.7) if the
metric field **h*** _{c}* does

(d) It should account for the
variation of metric **h*** _{c}* as a function of

(e) It must be multiplied by *d**c**/d**z*
if *c* is changed to *z* =*f* (*c*).

(f) It must satisfy Leibniz'
derivation rule for the derivative of a scalar product, i.e.

(2.9) _{},

in which it is understood that, on
the left, the variation of metric **h** with *x*^{0}
is accounted for, as becomes obvious if one writes down explicitly the scalar
product:

(2.10)
**h*** _{c}* (

(Hence, it is likely that (f)
implies (d).)

First,
we note that definition (2.7) still makes sense, and satisfies requirements
(a), (b), (c) and (e). Of course, it is now specified that the Christoffel
symbols of metric **h** are those at the relevant position *and "time"*, thus in (2.7)

(2.11) G^{ i}* _{jk}* = G

The
"candidate" thus defined by Eq. (2.7) will be now denoted by *D*_{0}**w**/*D**c*. It
does not satisfy (d) [nor (f), in fact], for it amounts in substituting the
metric **h**_{c}_{0
}of the "time" *a *= *x*^{0}(*c*_{0}) for
the variable metric** h*** _{c}*. From (b) and (c), it follows that
we have to search an expression in the form

(2.12) *D***w**/*D**c** = D*_{0}**w**/*D**c* + **t****.w**(*c*_{0}),

in which **t** is a
mixed second-order space tensor [indeed, the ordinary derivative *d***w**/*d**c* = (*dw ^{i}/d*

_{}.

Thus tensor **t** must
contain either *h*_{ij}_{,0} terms
or *h*^{ij}_{,0} ones,
with (*h** ^{ij}*) the
inverse matrix of (

(2.13) *t*^{ i}* _{k }*=

or any linear
combination of these two tensors. But since

*h*^{ij}*h*_{jk}_{ }= *d** _{ik}*,

we have **t** +** t'** = 0,
so that, without imposing Leibniz' rule, we are left with a one-parameter
family of candidates:

(2.14) *D*_{l }**w**/*D**c** *º* D*_{0}**w**/*D**c* + *l* **t****.w**.

Finally, nearly
the same short calculation as in ref. [4] shows that *Leibniz' rule (2.9) imposes* *l**=1/2*, hence only one definition of the derivative remains:

(2.15) *D***w**/*D**c** *º* D*_{0}**w**/*D**c* + (1/2) **t****.w**, **t** º _{},

or in
coordinates:

(2.16) _{}.

Thus, a theory of
the kind considered should provide an expression for the mass force **F**_{g}, and this expression
would depend on what the theory considers as "the gravitation field"
(this may include the space-time metric **g**, in any case it
must determine **g**). Then one and only one
"Newton law" can be consistently stated in such a theory: it is Eq.
(2.8), where the momentum **P** is given
by Eq. (2.6) and its derivative *D***P**/*Dt*** _{x }**is calculated using

2.4 *Comments and link with the investigated
preferred-frame theory*

It is seen that
the derivative of the momentum is defined in any possible reference frame (and
it depends on the frame). Hence, if a theory gives a covariant expression for **F**_{g} and **g**, the
extended second Newton law does not restrict the covariance of the theory. On
the other hand, a preferred-frame theory may give **F**_{g} and **g** in one reference frame only; if one
were able to calculate the transformation law of the derivative *D***P**/*Dt*** _{x}**,
then this same law would apply to the force, so the law of motion would be
reexpressed in a covariant form.

The investigated ether theory [1-4],
which is indeed non-covariant, starts from a heuristic interpretation of
gravity as Archimedes' thrust in a perfectly fluid "micro-ether" (the
rigid ether frame E considered by
Lorentz and Poincaré would be defined by the average motion of this
"micro-ether" at a very large scale). The transition to account for
"relativistic" effects is based on a formulation of Einstein's
equivalence principle, natural in this preferred-frame theory: the equivalence
is stated to exist between the absolute metric effects of uniform motion and
gravitation. This leads to postulate a gravitational contraction (resp. a
dilation) of the space (resp. time) standards, depending on the field of the
"ether pressure" *p _{e}*,
thus getting a curved (Riemannian) space metric

(2.17) *f *= *b*^{ 2} = (*p _{e}*
/

where* p _{e}*

(2.18) **F**_{g} = *m*(*v*) **g**,

with **g** the gravity acceleration, given by (2.19)

_{}

where **g** = **h**_{E } is the physical space metric in the frame E, and
where grad** _{g}** (resp. grad

(2.20) *ds*^{ 2} = *b*^{ 2} (*dx*^{0})^{ 2} - *dl* ^{2}, *x*^{0} =
*cT*,

where *dl*^{2} is the line element of
metric **g**. This has the following simple expression in
"isopotential" coordinates (*y** ^{a}*),
i.e. coordinates such that, at a given time

(2.21) (*g** _{ij}*) =
diag (

For a
time-dependent field *p _{e}*,
such coordinates are

**3. Extended Newton law and geodesic motion**

* *

*3.1 A possible form for the gravitation force in a
globally synchronized reference frame*

We now investigate the possible form of the gravitation force. In order
to make some meaningful induction from Newtonian theory, it is very useful to
work in a reference frame F, in which the *g*_{0i} components of metric **g** are zero (Subsect. 2.1). The
concept of global simultaneity is indeed so deeply involved in any Newtonian
analysis, that any induction from Newtonian theory to the general situation
with curved space-time, where a simultaneity is defined only along a
trajectory, would seem dangerous. Whereas, if one works in a frame such that *g*_{0i} = 0, the only change in the time
concept is that now the clocks go differently at different positions and times
[Eq. (2.4)]. We note that the existence of a frame F,
in which the *g*_{0i} 's are zero, is not a physically
restrictive assumption, since it breaks down only for rather pathological
space-times: in "normal" space-times it is even possible to select a
"synchroneous" frame which not only enjoy this global
synchronization, but in which the *g*_{00}
component is uniform, i.e. the local time flows uniformly (Landau &
Lifchitz [11], Mavrides [16]). Thus it "normally" exists *many* different frames such that *g*_{0i} = 0. Which form of the gravitation
force could one consistently state in such a reference frame?

For the class of
theories considered in Section 2, what is considered by any such theory as
"the gravitation field" has been assumed to determine the space-time
metric **g** (for
non-covariant theories, we should add that this has only to be true in some
preferred reference frame which is like E, i.e. such that *g*_{0i
}= 0). Here,
we will assume, in a more restrictive way, that *the metric field* **g** *contains the gravitation
field* (at least in the preferred frame). This is
true in any reference frame for GR and for the "relativistic theory of
gravitation" (RTG) proposed by Logunov *et
al.* [13-14], and this is true in the ether frameE
in the tentatively proposed theory. On the other hand, in order that SR
holds true locally and that the inertial and (passive) gravitational mass
coincide, the gravitation force must have the form

(3.1) **F**_{g} = *m*(*v*) **g**,

with **g** a space vector in the
considered frame. If we want that the metric field plays the rôle of a
potential, we must ask that **g**
depends linearly on the first derivatives of **g**, and bearing in mind Newtonian theory we
should add that only the *spatial*
derivatives *g _{mn}*

*g** _{ij}* = -

with **h** the space
metric in this frame, i.e. the metric **g** reduces to the joint data **g** = (*f*, **h**) with *f *= *g*_{00}. Thus, we are looking for a space vector **g** depending linearly on the spatial
derivatives of *f* and **h**. To be contravariant by general
space transformation, **g** must depend
linearly on the *covariant derivatives*
of *f* and **h** (with respect to the space metric **h**!). But, as is known, the covariant derivatives of metric **h** with respect to **h **itself are all zero (in other words,
one may cancel all spatial derivatives *h*_{ij}_{,k} at any given point by a purely
spatial coordinate transformation). Hence, **g**
should have the form

(3.2) ** g**
= *a*(*f*, **h**) grad_{h}*f*,

where *a* must be a given
function of the *values* of the metric
fields at the considered point (*x** ^{a}*)
in space-time,

Now
we add the condition that *geodesic motion
(Einstein's assumption) must apply to free particles (***F**_{0 }= 0*) for a
static gravitation field*. This is exactly equivalent to assuming the
following expression for the gravitation force *in the static case*:

(3.3) _{}, where *b* º Ö*g*_{00},

Indeed, it was already proved (and it will be proved again below, in a
different way) that Eq. (3.3), which
occurs naturally in the ether theory, implies geodesic motion for mass
particles in the static case [2]; this is also true for photons [3],
substituting in that case the mass content of the energy *e* = *h**n* for the inertial mass *m*(*v*).
Conversely it is proved in Landau & Lifchitz [11] that geodesic motion
implies the expression (3.3) for
the force in
the static case, defined as the
derivative (2.7) of the momentum (2.6) [4]. Thus the reason for assuming geodesic
motion in the static case is that it is indeed so for the tentative ether
theory as well as, of course (and in any situation) for usual theories of
gravitation with curved space-time, in particular GR and the RTG. So we must
have, by Eqs. (3.1), (3.2) and (3.3):

(3.4) _{}, i.e. *a*(*f*,
**h**)_{},

when *f*_{,0}
= 0 and **h**_{,0} = 0. But since *a*(*f*, **h**) depends only on the local values
of *f* and **h**, not on their variation, Eq. (3.2) implies then that **g** keeps the form (3.4) and thus Eq.
(3.3) holds true in the most general situation.

* *

*3.2 Expression of the 4-acceleration for a "free" particle
using the extended Newton law*

In theories with a (pseudo-) Riemannian
space-time metric, two well-known space-time vectors may be defined for a
time-like test particle (i.e. a mass point). These are the 4-velocity **U**, which is the velocity on the world
line of the particle in space-time, when the world line is parametrized with
the proper time *t* of the particle,

(3.5) *U** ^{a}* º

and the 4-acceleration **A**, which is the absolute derivative *D***U***/**Dt* of the former relative to the *space-time* metric **g**. Thus

(3.6) *A** ^{a}* º

the G ' ^{a}* _{mn}*
's being the Christoffel symbols of metric

(**i**)
Spatial components of the 4-acceleration in a globally synchronized reference
frame.

It is recalled that we use coordinates (*x** ^{a}*)

(3.7) *h** _{ij}* = -

hence

(3.8) _{}.

In this equation, we note that, in view of Eq.
(3.7)_{1} (and since *h*^{ij}^{ }= - *g ** ^{i j}* is always true):

(3.9) _{}.

By (2.4) and (3.5) we get:

*U*^{ 0} = (*dx*^{0}*/dt*** _{x}**) (

but, using Eqs. (2.1)-(2.3) and (2.5), it may
be proved (cf. Landau & Lifchitz [11]) that, independently of the fact that
*g*_{0i} = 0, one has always:

(3.10) _{},

as was already noted [2] for the tentative
theory. Hence we obtain

(3.11) _{},

so we reexpress another term in Eq. (3.8),
calculating _{}as for _{}in Eq. (3.9)
and using again Eq (2.4):

_{}.

We recognize here the component *g ^{i}* of the assumed gravity
acceleration [Eq. (3.4)], thus

(3.12) _{}.

It is now possible to calculate (*D***U***/**Dt*)* ^{i}* with the Newton law, for a
"free" particle [Eq. (2.8) with

(*D*_{l }**P**/*Dt*** _{x}**)

and we have by Eqs. (2.5), (2.6) and (3.10):

(3.13) *P ^{i}* =

so the "unspecified" Newton law
writes

(3.14) (*D*_{l }**u'**/*D**t*)^{i}
= *g*_{v }^{2}* g ^{i}*,

where **u' **º (*U ^{ i}*) means the spatial part of the 4-velocity

(3.15) _{}.

Hence, the unspecified Newton law imposes the
following values to the spatial components (in coordinates bound to a globally
synchronized frame F ) of the 4-acceleration of a free
test particle [Eq. (3.8) with (3.9) and (3.12)], depending on the parameter *l*:

(3.16) _{}_{}.

In particular,* the spatial part of the equation for space-time geodesics is satisfied
for a variable gravitation field (**h*_{jk,}_{0
}¹0)* if and only if the parameter **l** has the value **l** = 1*.

(**ii**) Time component of the
4-acceleration in a globally synchronized frame

For the time component, we have simply

(3.17) *A*^{0} = _{}.

Using Eq. (3.7)_{1} and the fact that *g*_{00 }= *b*^{ 2} [Eq.
(2.4)], the G ' ^{0}* _{mn}*
's are easily calculated:

_{}.

By Eq. (3.11), which implies also that *U ^{ k }*= (

*A*^{0}_{} .

At this point, we may insert the energy balance deduced from the
"unspecified" Newton law for the free test particle (Eq. (4.21) in ref. [4]):

(3.19) _{} + *b g*_{v }_{}

with *v ^{i}* = (

_{}

_{},

so that some cancellation occurs in (3.18). We obtain finally:

(3.20)
*A*^{0 }=_{}.

(3.21)

In particular, the time part of the equation
for space-time geodesics, as well as the spatial part, is satisfied for a
variable gravitation field* *(*h*_{ij,}_{0 }¹0) if and only if the parameter* **l** *has the value* **l** =* 1. However, it is recalled that the value* **l** *= 1 specifies the Newton law in an incorrect
manner, since it means that Newton's second law is based on a vector time
derivative which does not obey Leibniz' derivation rule.

Let us summarize the results of
Subsects. 3.1 and 3.2, which concern Newton's second law and geodesic motion:

(NGM1) *Consider a theory with
curved space-time metric* **g** *and locally valid SR, and
assume that in some "globally synchronized" reference frame* F *(**g*_{0i }= 0*),* *the gravitation force
(3.1) involves a space vector* **g** *depending only on the metric field* **g***. More precisely, assume that* **g** *does not depend on the time variation of*
**g** *and is linear with respect to the space variation of* **g**. *In order that free particles
follow space-time geodesics in the static case* *(**g _{mn}*

* *

(3.21) _{},

*with ***h*** the space metric in **F**. This expression implies Eqs.
(3.16) and (3.20) for the 4-acceleration, thus it implies that, for a
time-dependent field, geodesic motion corresponds exactly to the incorrect
Newton law (**l** = 1).*

3.3 *Characteristic form of the
gravitation force associated with geodesic motion*

* *

The assumption that the metric field **g **plays the rôle of a potential for the gravity
acceleration **g** seems quite natural,
if one thinks of a "soft" generalization of Newtonian gravity. The
foregoing result implies, among other things, that Einstein's assumption of a
motion following space-time geodesics is not such soft extension. But, after
all, in Maxwell's theory the electric field involves also time derivatives of
the electromagnetic potential, besides the usual space derivatives. Moreover,
the Lorentz force depends on the velocity of the charged particle. A more
general expression than we assumed for the gravity acceleration might hence be
correct also, the more so as we now have empirical reasons to think that the
gravity interaction indeed propagates, as does the electromagnetic field, and with
the same velocity (Taylor & Weisberg [25]). That gravitation propagates
with the velocity of light was first envisaged by Poincaré in his
"electromagnetic", Lorentz-invariant theory of gravitation [20-21]
and, as is well known, it is predicted by Einstein's theory.

Thus we now investigate
the possible form of the vector **g**,
subjected to the unique constraint that *geodesic
motion should occur with the correct form of Newton's second law, i.e. **l** = *½. We
continue to work in a globally synchronized reference frame and, in order to
simplify the expressions, we take **g**
in the form

(3.22) _{}.

Starting from Eq. (3.6) as before, nothing changes until Eq. (3.12),
which now becomes

* *

(3.23) _{}.

And again nothing changes until Eq. (3.16), which is modified into

(3.24) *A ^{i}* =

Hence, the spatial components of the 4-acceleration cancel with *l* = ½, if and only if

(3.25) _{}, i.e.
_{}.

But does this expression also cancel the time part of the
4-acceleration? To check this, one must reexamine the energy balance derived in
ref. [4]. Proceeding in the same way, we find easily that the energy balance
resulting from the expression (3.22, 3.25) of **g** is (with *l* = ½)

(3.26) _{} _{},

instead of Eq. (3.19). Thus, with the correct Newton law (*l* = ½), the same expression is now
obtained as it was obtained before with the incorrect Newton law (*l* = 1). Therefore, the time part of
the geodesic equation, *A*^{0 }=
0, is satisfied for *l* = ½, as it was previously for *l* = 1. We have proved the following:

(NGM2) *Consider a theory with curved
space-time metric* **g** *and locally valid SR, and
assume the correct time derivative (2.15) in the extension (2.8) of Newton's
second law. In order that free particles [***F**_{0} = 0* in Eq. (2.8)] follow
space-time geodesics, it is necessary and sufficient that, in any globally
synchronized reference frame* F
*(**g*_{0i }= 0*),* *the gravitation force
(3.1) involve the following expression for the gravity acceleration (space
vector * **g ***) ***:**

(3.27) _{}_{},_{},

* *

*with ***h*** the space metric in **F ** and ***v*** the velocity vector
[Eq. (2.5)].*

This result provides the general link between
Newton's second law and Einstein's geodesic assumption.

** **

**4. Comparison with the literature **

** **

4.1 *Møller's
work and the relation between covariant and contravariant form of Newton's law*

Among attempts to define Newton's second law in
the case of a variable gravitation field, a well-known one is that of Møller
[18]. However, Møller uses the absolute derivative with respect to the
"frost" space metric, thus *l* = 0 in Eq. (2.14), so that Leibniz' rule is not satisfied with the
actual, time-dependent metric. In connection with this, he notes that this
derivative does not commute with raising or lowering the indices with respect
to the space metric **h**. As a consequence, when he rewrites the equations for space-time
geodesics in the form of Newton's second law with gravitational forces, the
latter look very different in covariant and in contravariant form. We show that
this difficulty is absent with our definition.

Indeed,
it is easy to adapt our line of reasoning so as to define the time-derivative
of a spatial *covector* **w***. One finds in exactly the same way
that, apart from Leibniz' rule, a one-parameter family of time-derivatives may
defined as:

(4.1)* D*_{l }**w***/*D**c** *º* D*_{0}**w***/*D**c* - *l* **t****.w***, with

(4.2) (**t****.w***)* _{i}*
º

and where *D*_{0}**w***/*D**c*
is the absolute derivative using the "frost" metric. And one
finds that Leibniz' rule imposes *l* = ½. It is also easy to verify that, for this correct value *l* = ½ and, for a time-dependent
metric **h**, *only* for this value, the time-derivative
*D** _{l }*/

(4.3) *D*_{1/2 }(**h.****w**)/*D**c* = **h.**(*D*_{1/2 }**w**/*D**c*).

Therefore, if one takes the covariant
components of the momentum instead of the contravariant ones, thus substituting

**P*** = **h.****P**

for **P**,
then the corresponding "covariant Newton law" will involve just the
covariant components of the force,

**F*** = **h.****F** = **h.**(**F**_{0 }+ **F**_{g})
in Eq. (2.8).

4.2 *Newton's
second law with the
"Fermi-Walker" time-derivative*

From now on, we will discuss the work on
"Newton's second law in relativistic gravity" as reviewed and unified
by Jantzen *et al*. [10]. They define
the equivalent of what we call a frame (spatial network) by a 4-velocity vector
field **u**, and they name it
"observer congruence". What they call "observer-adapted
frames" is a very different notion from that of adapted coordinates as
defined by Møller [18] and Cattaneo [6-7]. Here we continue to work in adapted
coordinates, i.e. such that the observers of the network (or congruence) have
constant space coordinates. In such coordinates, the contravariant and
covariant components of **u** are given
by

(4.4) _{}

(we keep our notations, except for the fact
that we set *u*^{a}^{ }º *dx*^{a}*/ds* and adopt the (3,1) signature as in
refs. [6-7] and [10], until** **the end of this Section). It
follows that the spatial projection tensor

_{},

has a simple expression:

(4.5) *P*^{ i }_{j}_{
}= *d*^{ i }* _{j}*,

It corresponds to the projection of the local tangent space to space-time
onto the hyperplane which is **g**-perpendicular to the local 4-velocity **u** of the observer congruence. In connection with this, what is
called a "spatial tensor" by Cattaneo [7] and by Jantzen et al. [10]
is also a very different notion from that used by Møller [18] and in the rest
of this paper. For us (and for Møller), a spatial tensor is just an element of
a tensor space at the relevant point of the spatial network (3-D Riemannian
manifold) N, thus its components depend on the three spatial (Latine) indices
only, *i* =1, 2, 3, in adapted
coordinates. In refs. [7], [10] and in the remainder of this Section, a spatial
tensor is a *space-time tensor* which
is *equal to its projection*, the
latter being generally defined by Eq. (2.2) of ref. [10]. E.g. for a 4-vector
(space-time vector) **X**, the
projection writes:

(4.6)
(**P****.X**)* ^{a}* =

Hence in adapted coordinates, by (4.5):

(4.7) (**P****.X**)* ^{ i }*=

so that the "time" component *X*^{ 0} is *not* equal to zero for a "spatial vector" (except for a
"normal congruence", i.e. the case where* g*_{0 j }= 0 in adapted coordinates). We note
also that the "rescaled time" *t*_{(U,u)} considered in ref. [10] (for a time-like test
particle with 4-velocity **U**), as well
as the "standard time" *T*
considered in refs. [6-7], is the same variable as our "local time" *t*** _{x}**,
synchronized along the trajectory of the test particle, with their

(4.8) _{}.

We have thus in adapted coordinates, by Eq.
(4.7):

(4.9) _{},

and the "time" part of the derivative
is not independent from the "space" part:

(4.10) _{}.

What corresponds to Newton's second law in [10]
is the evaluation of the spatial projection of the 4-acceleration **A** of the test particle. Apart from the
different notation, it amounts almost exactly to Eq. (2.8) here, with the same
definition (2.6) for the momentum, involving the same relative velocity (2.5),
though with the derivative defined by Eq. (4.8) instead of Eq. (2.15). One
difference is that the velocity **v**
and momentum **P** are now spatial
4-vectors which turn out to be the respective projections of the 4-vectors **U'** and **P'**, with **U'** the
4-velocity **U**, rescaled to the local
time, and **P'** the usual 4-momentum.
Thus the spatial components of **v** and
**P** are the same as in this work, and
the "time" components obey the general rule for a spatial vector **X**, i.e. such that **P****.X** = **X**:

(4.11) *X *^{0} = - *g*_{0 j }*X ^{
j }/*

Another difference is that the gravitational
force, which is the total force for a free particle, is hence deduced, in the
frame of GR and other "metric theories", from the geodesic equation,
i.e. **A** = 0.

Having
thus recognized that the spatial part (4.9) of the derivative (4.8) plays
exactly the same rôle in ref. [10] as the derivative (2.15) plays here, we may
comment on the difference between the two derivatives. Since the spatial
components (4.9) are just those of the space-time absolute derivative *D***X***/**Dc*, the Fermi-Walker TSCD involves space-time coupling in a generally
inextricable way, in that it cannot in general be defined in terms of only the
spatial metric **h** and the
local time *t*** _{x}**. Hence, this derivative cannot be used in an arbitrary
reference frame to define a "true" Newton law as it has been defined
here, i.e. precisely a law involving only the separate space and time metrics
in the given reference frame, thus allowing to "forget" the concept
of space-time as long as one does not change the reference frame.

4.3* The
"normal" and "corotational" Fermi-Walker derivatives obey
Leibniz' rule*

Surprisingly, the question whether the
introduced time-derivatives satisfy the Leibniz rule is not investigated in
refs. [6, 7, 10]. However, it is not difficult to show that *the two Fermi-Walker derivatives do verify
Eq. (2.9)*. The spatial metric in those works is of course the same thing as
here, except for the indices and the signature:

(4.12) _{}.

Using Eqs. (4.12) and (4.11), one verifies
that, for two *spatial* space-time
vectors **X** and **Y**:

(4.13) **h**(**X**, **Y**) = **g**(**X**, **Y**).

On the other hand, the absolute space-time
derivative obeys the Leibniz rule:

(4.14) _{}.

Since the "normal" Fermi-Walker
derivative is a spatial vector whose spatial components are just those of the
absolute space-time derivative [Eqs. (4.8) and (4.9)], we may use Eq. (4.13) to
rewrite Eq. (4.14) as:

(4.15) _{}.

This is the Leibniz rule for the two spatial
vectors **X** and **Y**.

The
"corotational" Fermi-Walker (cfw) derivative is related to
the"normal" Fermi-Walker derivative by [10]:

(4.16) _{}.

Here *w ^{ a}_{ m} *are the mixed components of the
"spin-rate" space-time tensor. This comes from the decomposition of
the covariant "spatial 4-velocity gradient",

(4.17) **k** =** k**(**u**)
= - **P****.** **Ñ**^{(}^{g}^{)}
**u*** ^{b}*,

into symmetric and antisymmetric part:

(4.18) *k** _{ab}* = -

and the mixed components* w ^{ a}_{
m}* are
obtained by raising the index

**t** = **h**^{-1}**.h**_{,0} (*dx*^{0}*/dt*** _{x}**)

tensor in our derivative (2.15) (with *c* = *t*** _{x}**) - but the
tensors

As
to Leibniz' rule, it applies as for the ordinary fw derivative. Indeed, due to
the antisymmetry of the covariant tensor **w **[Eq. (4.18)_{3}], the definition
(4.16) gives

_{}

_{}.

The Leibniz rule follows from this by (4.13)
and (4.15), the cfw derivative being also a spatial vector [10]:

(4.19) _{}.

4.4 *The
case of a globally synchronized frame and the "Lie" time-derivative*

We consider the particular case of a *globally synchronized frame* (or
"normal congruence"), in which the *g*_{0i} components of the space-time metric
are zero in some adapted coordinates. Then the spatial projection tensor **P** [Eq. (4.5)] writes simply

(4.20) _{}

in such coordinates. Hence, in such
coordinates, substituting its spatial
projection **P**(**u**)**.T** for a space-time tensor **T**
amounts exactly to taking its space components only. In particular, the
"time" component of a spatial vector **X** is now equal to zero. Moreover, the spatial Christoffel symbols
of the space-time metric are equal to the Christoffel symbols of the spatial
metric [Eq. (3.7)]. This implies that the Fermi-Walker derivative coincides,
for the case considered and for a *spatial*
vector **X** (thus *X*^{0} = 0), with the*
D*_{1/2} derivative. Indeed, using Eq. (3.9), we find:

(4.21) _{}

_{},

with **X'**
º (*X ^{ i }*).

For
the non-zero components of the **k** tensor [Eq. (4.17)], we obtain using Eqs. (4.20), (3.9) and (4.4) [and
since *h** _{jk}* =

(4.22) _{}.

Therefore, the "spin-rate" tensor **w** is nil for a normal congruence [6],
so that the corotational Fermi-Walker derivative coincides, for spatial
vectors, with the "normal" one, and thus with the proposed
derivative, *D* = *D*_{1/2}. On the other hand, we have from (4.18) and (4.22):

_{}.

What is called "Lie" TSCD derivative
in ref. [10], is not a Lie derivative in the usual sense but the projection of
a Lie derivative [10], and is defined in general by [10]:

(4.23) _{}

(extending again the definition [10] to an
arbitrary parameter *c*). Hence, we have here:

(4.24) _{}.

In other words, the so-called "Lie"
derivative coincides in that case with the absolute derivative with respect to
the "frost" spatial metric, and so does not obey Leibniz' rule.

**5. Concluding remarks**

1. From our bibliographical research, it would appear that it had not
yet been proposed in the literature the consistent definition of the
time-derivative of a vector moving along a trajectory in a manifold equipped
with a metric field **h*** _{c}* (the

2. We find that there is one and *only
one* *natural* extension of Newton's
second law to any theory with curved space-time metric, in the most general
situation. In particular, one may *uniquely*
identify that gravity acceleration **g**_{geod
}which is necessary to obey Einstein's assumption, i.e. to obtain geodesic
motion for free test particles. In doing so, we did not merely rewrite the three
"spatial" equations for space-time geodesics as the space-vector
relation "force = time-derivative of momentum": we also proved that
the latter relation implies the "time" equation of geodesics, and
this does not seem to have been done in earlier attempts. This
"geodesic" gravity acceleration **g**_{geod}
depends on the reference frame, as is natural in a "relativistic"
theory (since the acceleration is not Lorentz-invariant). It may seem more
surprising that **g**_{geod}
depends on the velocity of the particle [Eq. (3.27)]. However, this is also the
case for the Lorentz force which a charged particle undergoes in an
electromagnetic field. The striking difference is that the magnetic force does
not work, whereas the velocity-dependent part of **g**_{geod} does work. In the investigated case of a normal
congruence, it has the same form as the Newtonian inertial force that appears
in a reference frame undergoing pure strain with respect to an inertial frame
[1]. But here this "inertial" force comes from the straining of the
reference frame "with respect to itself" (i.e. due to the fact that
the spatial metric evolves with time) and it cannot in general be cancelled by
changing the reference frame. Thus, theories with geodesic motion inherently do
not allow global inertial frames, although such global inertial frames do
appear in their Newtonian limit. We also note that any velocity dependence of
the gravity acceleration, **g** = **g**(**x**,
**v**), implies that the definition of
the passive gravitational mass, i.e. *m*_{g}
**º** **F**_{g }*/***g** with **F**_{g} the
gravitation force, becomes indissolubly mixed with that of the gravity
acceleration itself: one may change **g**
and *m*_{g} to *a***g** and *m*_{g}*/**a* respectively, with *a* any scalar function of the velocity
(e.g. *a* = *g*_{v }* ^{n}* where

3. The identity between inertial and gravitational mass would have a
stronger meaning if **F**_{g}
would depend only on the position of a given test particle. However, for the
kind of theories considered here, this could be true only in some preferred
reference frame (this is, of course, in contrast with the Galilean situation).
To check this identity, one might then define **g** for particles at rest in the preferred reference frame, thus **g**(**x**)
º **F**_{g}(**v **= 0)*/m*_{0}, and check experimentally whether or not *m*_{g} **º** **F**_{g }*/***g** has the same velocity dependence as the inertial mass *m*(*v*).
In the scalar ether theory which has been tentatively proposed [1-4], a vector **g** depending only on the position, Eq.
(3.21), has been found to occur naturally- consistently with the notion that **g** should be determined by the local state
of some substratum. Thus this theory predicts "strong identity"
between inertial and gravitational mass and, in connection with this, geodesic
motion does not hold true in the general case in this theory. If one were to
modify this theory so as to obtain geodesic motion, one would have to postulate
Eq. (3.27) instead of Eq. (3.21). Then, the modified **g**-field would still be determined (in the preferred frame E
) by the scalar field *p _{e}* or

ACKNOWLEDGEMENT

I am grateful to Prof. P. Guélin for his useful comments on the
manuscript.

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[1] Equation (2.6) implicitly assumes
that the rest mass *m*(0) is the same
constant *m*_{0 },
independently of the gravitation field. This may be seen as an immediate
consequence of defining the inertial mass *m*
as the ratio **P**/**v** (=*P ^{i }/v^{i }*)
and assuming that the

[2] The
expression of **F**_{0} is taken
from the situation without gravitation: thus, as recalled in point (**ii**) of Subsect. 2.1, it involves the
field **g** (in the place of the flat metric **g**^{0}), and it depends on the
non-gravitational fields- in practice these are the electromagnetic field
and/or thermomechanical fields (the nuclear fields are very microscopic matter
fields and moreover their current theory does not belong to classical physics,
i.e. their influence cannot be described in terms of deterministic trajectories
of mass points). A "*free*"
particle is one which crosses a region free from matter and electromagnetic
field: for such a particle, the force **F**_{0}
will be zero *independently of the
reference frame considered*.

[3] Here,
rigid rotation and uniform motion *can*
be defined, at least if the metric manifold (M, **g**^{0}) has *zero* curvature, i.e. is Euclidean.

[4] Actually, Landau & Lifchitz
[11, § 88] derived from geodesic assumption the expression of the force in the *stationary* case, using this same
definition for the force (as is consistent with the present work, Subsect.
2.2). They found an expression involving an additional term which cancels if *g*_{0i }= 0.

[5] Eq. (3.19) is derived using the
fact that *some* derivation rule of a
scalar product can be obtained even with the "unspecified" Newton
law, although it does not obey the true Leibniz rule [Eq. (2.9)] unless *l* = 1/2. However, if *l* ¹ 1/2, this balance equation cannot be rewritten
as a true conservation equation, at least in the scalar theory [1-4].