This is the first area in which a departure from the historical approach to General Relativity occurs. Time dilation was well accepted early on after Einstein’s 1905 paper on Special Relativity. Consider if by some experiment in the early twentieth century it was discovered gravity caused time dilation based on radial distance from the massive object. If this was discovered before Einstein’s GR prediction, there may have been a number of theories put forth. One such hypothesis may have been the rest energy of SR is proportional to the time dilation effect of a gravitational field.
Consider a field, much like a Newtonian gravitational field, which calculates the gravitational time dilation effect based on two concepts.
- The mass generating the gravitational field reduces the vacuum energy surrounding it in proportion to the Newtonian gravitational potential.
- The sum effect the mass generating the gravitational field can have on the vacuum energy surrounding it is equal to the mass’s rest energy.
The field in postulate 1 is calculated by taking a virtual mass, equal to the mass generating the gravitational field, and evaluating the Newtonian gravitational potential at every location about the mass where the radial distance is greater than the radius of the mass.
\(\begin{equation}\tag{1}
V_g=-\frac{G M m_v}{r^2}
\end{equation}
\)
The virtual mass provides a scalar value at every location over a field. If by postulate two, the maximum effect the massive object generating the gravitational field can have on reducing the vacuum energy is it’s rest energy. Than a logical candidate equation for calculating the gravitational time dilation at a location is:
\(\begin{equation}\tag{2}
A_g = 1 – \frac{V_g}{E_0}
\end{equation}
\)
The negative sign in equation 2 is due to postulate 1, the mass reduces the vacuum energy. Thus at any location the gravitational time dilation would be predicted to be:
\(\begin{equation}\tag{3}
t’=A_g\cdot t
\end{equation}
\)
Where the primed time is time as measured locally by an observer in the field. The unprimed time is time as measured locally at a location where there is no gravitational time dilation effect due to a massive object. For clarity, this is a contravariant transformation of time. If it was a covariant transformation the Ag value would be inversed.
It is assumed time “ticks” fastest in a vacuum where there is no gravitational effect. This assumption implies there is a boundary for the maximum rate at which a clock can tick within a vacuum. The maximum rate is always located in a vacuum with not gravitational effect. At this location the value of Ag is always 1. For ease of writing, any space without the influence of a gravitational field is termed Exemplar space.
A clock for the purposes of this hypothesis is a physical clock (i.e. a Cesium based atomic clock) located in the field about the massive object. The clock is observed by a local observer at the same location as the clock. To obtain the time rate differential between two locations in the field, one must calculate each clock’s Ag value and set the equations equal. Then solve for each observers time rate relative to the each other. For example, if a clock is at location a, and another clock at location b, their scalar field values are:
Clock at position a
\(
\begin{equation}\tag{4}
A_{a}=1 + \frac{V_{ga}}{E_0}
\end{equation}
\)
Clock at position b
\(
\begin{equation}\tag{5}
A_{b}=1 + \frac{V_{gb}}{E_0}
\end{equation}
\)
Observer at position a views position b’s clock rate as
\(
\begin{equation}\tag{6}
t_b=\left( \frac{A_a}{A_b} \right) t_a
\end{equation}
\)
Equation 2 above only considers a static clock’s time rate at location in space compared to other static clocks in the field. It does not take into account any movement relative to the massive object generating the field. In order to take into account the movement, the field must have an additional inclusion to represent the kinetic energy of the virtual particle. Following the same logic as equation 2, the kinetic induced time dilation is predicted to be of the form:
\(
\begin{equation}\tag{7}
T_g=\frac{1}{2} m_v v^2
\end{equation}
\)
The scalar value within the field due to kinetic energy is
\(
\begin{equation}\tag{8}
A_g = 1 + \frac{T_g}{E_0}
\end{equation}
\)
The reason for the plus sing in Equation 8, instead of a negative sign as in equation 2, is the assumption kinetic energy adds the virtual mass’s vacuum energy. This assumption is based in part on the belief the volume of vacuum is greater per unit of time for a moving object than it is for a stationary object. Therefore a moving object has more vacuum to “draw” energy from than a stationary object.
It is important to remember the velocity is relative to the mass generating the gravitational field. Without a secondary object to calculate a relative velocity from, there is not way to measure a differential effect within a field. In summary, Equation 8 represents the kinetic energy contribution to time dilation due to movement within the scalar field relative to the massive object generating the gravitational field.
To calculate both the static (stationary field) effect and the kinetic (moving in the field) effect, both effects must be added together.
\(
\begin{equation}\tag{9}
A_g = 1 + \frac{T_g – V_g}{E_0}
\end{equation}
\)
The combined effect is
\(
\begin{equation}\tag{10}
A_g = 1 + \frac{v^2}{2 c^2}+\frac{G M}{r c^2}
\end{equation}
\)
It may be of interest to note that Equation 10 is the equivalent to Special Relativity’s kinetic energy to 1st order accuracy, and in Equation 10 the potential effect is equivalent to Schwarzschild’s time dilation effect to 1st order accuracy. Combined effects with both kinetic and potential in Equation 10 will be discussed in another section of the classical approach to a unified theory.
Next -> Newtonian Mechanics Over the Scalar Field
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