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Einstein relativity theory shows its high capability of promoting itself to solve the long stand physical problems. The so-called generalized special relativity (GSR) was derived later, using the beautiful Einstein relation between field and space-time curvature. In this work we re-derive (GSR) expression of time by incorporating the field effect in it, and by using mirror clock and Lorentz transformations. This expression reduces to that of (GSR) the previous conventional one, besides reducing to special relativistic expression. It also shows that the speed of light is constant inside the field and is equal to C. This means that the observed decrease of light in matter and field is attributed to the strong interaction of photons with particles and mediates which causes successive absorption and reemission processes that lead to time delay. This absorption process makes some particles appear to move faster than light within the field or medium. This new expression, unlike that of GSR, can describe time and coordinate relativistic expressions for strong as well as weak fields at constant acceleration.

Einstein’s special relativity and general relativity represent one of the biggest achievements that change radically the space-time concept [

SR shows that time, length, mass, and energy are velocity dependent [

These remarkable successes of SR and GR motivate some authors to promote SR within the frame of GR to produce the so-called Generalized Special Relativity (GSR) [

Mubarak model, the matter energy-momentum tenser relation on the generalized Lorentz factor

In this work two mirrors of certain length, acting as time clock, are under the action of gravity. The motion of the two mirrors under gravity is utilized to find a useful expression of time in the presence of the gravitational field. The speed of light is assumed to be constant in the gravity field. This assumption is confirmed by obtaining the light speed in accursed space time [

This paper, which is concerned with the derivation of time dilation in the gravitational field, consists of 3 sections; apart from introduction, Section 2 is devoted for presenting EGSR theory. The derivation of time dilation in the gravitational field for any field is done in Section 3. The speed of light in the gravitational field is found in Section 4. Sections 5 and 6 are devoted for discussion and conclusion (Generalized Special Relativity [

According to GSR the expressions of time

where

The expression of time in the presence of gravitational field can be found by considering two as a clock, with time intervals to representing the travel between the two mirrors as shown in

It moves with velocity up ward under gravity of acceleration g, the velocity of the lower mirror is given by

where

The vertical displacement

The speed

Thus

where:

where

Inserting (9) in (6) yields as shown in

Since the point

Speed

In view of Equations (7)-(10), one gets

Thus:

Using relation (10)

If one consider

Then relation (10) becomes

And relation (13) becomes

For weak field

Thus

Which coincides completed with the expression of time in the presence of the gravitational field obtain within the framework of GSR. Relation (14) it is important to note that of the factor

According to Equations (9) and (10) this relation holds for any field other than the gravitational field.

The length contraction can be obtained by considering a clock falling by sliding on a rod of a height

where the rod is moving and accelerated w.r.t him, thus his length is

Thus, with the aid of Equations (18) (19) and (13).

This is in agreement with the corresponding expression in GSR.

Using Lorentz transformation, the event at point

(21)

Since the space is homogeneous it follows that

Consider ampoule of light emitted from a source when the origins 0 and 01 are in coincidence at

In this care

Substituting Equation (23) in Equations (22) and (21) yields

Inserting Equation (25) in Equation (24) yields

With the aid of Equation (8) and Equation (10)

When one consider the expression for the maximum speed in Equation (15), one gets

It is very striking to observe that when no field exist, i.e.

This is the usual SR expression.

For weak field

Therefore

Thus Equation (25) becomes

This is the usual expression of GSR for a weak field.

In Section 3 time dilation expression in the presence of any field has been derived by using mirror, see Equations (11) and (14). The only assumption which causes a limitation to this expression is the constancy of acceleration. The mirror motion is described by using Newton’s Law of linear motion with constant acceleration.

The speed of light is assumed to be constant. The resulting expression for

It also reduced to GSR form for a weak field as shown in Equation (16).

The expression of length contraction has been obtained as well. Again this expression reduced to SR and GSR also [

Lorentz transformation is also utilized to find a relativistic expression for

The expressions for time and length obtained by using Lorentz transformation and using mirror clock, for fields at constant acceleration, and by assuming the speed of light to be constant indicate that GSR rests on a solid ground. It also indicates that space and time are affected by any field, not gravity only. Unlike the curved space- time derivation, where the field is assumed to be week, this derivation holds for strong fields as well. By reducing to SR and GSR for a weak field, it indicates its self-consistency.

M. H. M. Hilo,R. Abd Elgani,R. Abd Elhai,M. D. Abd Allah,Amel A. A. Elfaki, (2014) Deriving of the Generalized Special Relativity (GSR) by Using Mirror Clock and Lorentz Transformations. Natural Science,06,1275-1281. doi: 10.4236/ns.2014.617117