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.(3.15) and is repeated below for convenience

(4.1)

The total energy of the body therefore remains exactly the same as it was at the instant before motion started, see [1], Eq.(3.20). 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. 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, (see (3.10) and the ensuing discussion). 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.

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

(4.2)

Insertion of (3.14) and multiplying out gives

(4.3)

where has been substituted for . Now put

then

(4.4)

and substitution of (4.4) into (4.3) then gives after minor reduction

(4.5)

Kinetic energy is given by the integral of the applied force over the distance it acts so that

(4.6)

In (4.6) the term on the right hand side is an exact differential so that it can be integrated by inspection to be

(4.7)

which from the first term in (4.4) becomes

(4.8)

If initial conditions are such that at the point of application of the accelerating force, , E_k=0 and of course u=u₀, then

(4.9)

and so

(4.10)

which from (3.14) becomes

(4.11)

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

(4.12)

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.

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.

Assume that s₁ is the point of application of the decelerative force F, at which the initial conditions are u = u₁, m = m₁, and E_k = 0 by virtue of (4.1). Under these conditions, the solution to (3.13) is, with the non zero initial velocity

(4.13)

and with this the solution to (4.2) becomes

(4.14)

Inserting the initial conditions in (4.14) determines k to be

(4.15)

and thus

(4.16)

which from (4.13) is

(4.17)

This appears to be of the same form as in the previous example for accelerating a body against gravitation, (4.11). However, consider (4.16) after the mass has been brought to rest. Then and thus

(4.18)

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)

and

(4.19)

Inserting (4.19) into (4.18) then gives

(4.20)

because, in this case, < 1 then E_k < 0 , i.e. the mass loses energy during deceleration. This loss occurs as follows, from (4.17) and the first part of (4.19)

(4.21)

and because E_k < 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

(4.22)

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

(4.23)

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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