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4 Kinetic Energy of Gravitational Motion.It was shown in [2] that in D0, Pseudo-Euclidean Space-Time, the increase in the rest mass of an accelerated body to its energy mass at some spatial velocity, was due to the storage of kinetic energy generated by the externally applied force.It was also shown In [1] that in the gravitational Relativistic Domain D1, the increase in mass that occurs when a body is in motion under the sole influence of the gravitational source, was due exclusively to it's motion through the varying temporal rate generated by the source. As there is no artificially applied accelerative force under this latter condition, the question arises as to the nature of kinetic energy of the gravitationally accelerated mass.
In [1] it was shown that the total energy of the gravitating body remained
constant throughout the entire time that the motion continued. This was
stated in [1] as Eq.(3.16) and is repeated below for convenience
The total energy of the body therefore remains exactly the same as it was at the instant before motion started, see [1], Eq.(3.21). There can only be one consequence of this - in purely gravitationally accelerated motion, kinetic energy does not exist. The sole reason for this is that the gravitationally applied acceleration generates a force within the body precisely proportional at all times to it's energy mass, (see (3.10) and the ensuing discussion). This force is not therefore the cause of the motion but the consequence of it, and does not result in a transference of energy in the form of increased mass. However, this is only true for purely gravitationally induced motion. When an external force is also applied, kinetic energy is generated in D1 as it is in D0. This is examined in the following Section.
4.1 The Kinetic Energy Generated by an Externally Applied Force in D1.The kinetic energy generated in an accelerated body by an externally applied force may be developed directly from (3.17). Re-arranging (3.17) gives
Insertion of (3.14) and multiplying out gives
Now put
and substitution of (4.4) into (4.3) then gives after minor reduction
Kinetic energy is given by the integral of the applied force over the
distance it acts so that
In (4.6) the term on the right hand side is an exact differential so that it
can be integrated by inspection to be
which from the first term in (4.4) becomes
and so
which from (3.14) becomes
and the kinetic energy is clearly the difference between the total energy of the mass at the point of observation and, at the point at which motion started. This is exactly the same as in D0, e.g. putting u = u0 = 1 reduces (4.11) to the kinetic energy of D0, Pseudo-Euclidean Space-Time.
Also note that (4.11) can be re-arranged to show that
which therefore shows that the energy mass under this condition is now made up of the original mass at the location that motion started, translated to the point of observation via the square of the ratio of the respective temporal rates, i.e. the gravitational variation of mass, plus an element due to the storage of kinetic energy imparted to the mass by the action of the artificial accelerative force F. Again this latter effect is the same as in D0.
4.2 Dissipation of Energy when Bringing a Gravitating Mass to Rest.In view of the result that gravitationally induced motion does not involve a gravitating mass accumulating kinetic energy, it is necessary to explain the apparent dissipation of energy when a gravitating mass is brought to rest.
The gravitationally accelerated motion that exists within the Relativistic
Space-Time Domain D1, is the natural state of existence within that
Domain and, for a gravitating body does not involve an exchange of energy.
To bring a gravitationally accelerated mass to rest requires the application
of an artificially generated opposing force. The energy dissipation that
takes place during this process occurs due to two causes. First, and most
obvious is that the generation of the artificial force can only be effected
by some mechanical, electrical, chemical or nuclear process. All of these
require the dissipation of energy to achieve the objective. However, there
is a second more important cause. Because the gravitationally accelerated
state of the body is its natural state of existence in D1, bringing it
to rest via the application of an external force is causing it to decelerate
against this natural state of existence. This has the opposite effect to
that in the previous example, it extracts energy from the gravitating body
by reducing its mass. This process can be demonstrated as follows.
Under these conditions, the solution to (3.13) is, with the non zero initial velocity
and with this the solution to (4.2) becomes
Inserting the initial conditions in (4.14) determines k to be
and thus
which from (4.13) is
If the prior motion due to gravitation started at a position s0
where u=u0 and m=m0 then from (3.2) and (3.4)
Inserting (4.19) into (4.18) then gives
and because Ek < 0 the mass of the body at the point where it has been
brought to rest is less than it would have been had it been allowed to
continue gravitating. The loss of energy through this process is therefore
effected by a reduction in the mass of the decelerated body and this energy
loss is absorbed in both the arresting and gravitating bodies as a
mechanical deformation. The mass loss can be determined by equating (4.17)
and (4.20). This gives
and, if the mass had been allowed to continue gravitating to the point of
observation its mass would have been given by (3.4). The mass loss is
therefore the difference of (4.22) and (3.4) thus
which clearly must be negative. Note that (4.21) is identical in form to (4.12) and that, it is also clear that (3.18) must apply in this case in that the apparent mass under deceleration must be the inertial mass of the decelerated body.
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P.G.Bass, March 2004
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