In the following section we use mathematics to arrive at the conclusion that if we shorten the length of the light clock while keeping the frequency constant, such a clock will indeed keep proper time with the faster running source clock higher in the field. The reason why this occurs is that as light 'blue shifts' in gravity field (increases in frequency) the wave length also becomes shorter in length. We can compensate for this difference by making the clock 'shorter' while continuing to use the source frequency and the clock will then keep proper time with the faster running source clock located higher in the gravitational field.

For this reason it seems reasonable to assume that it is not required that we think of the universe as coming with 'clock included', for it is possible to explain the passage of 'time' as an 'emergent property' of field geometry. 'Quantum oprocesses' exchange information in small packets of energy which travel at 'the speed of light' and when the geometry of the field changes, we would expect this to effect the exchange of information between these quantum processes. Given that all reality is a manifestation of these underlying quantum processes, we would then expect this to result in a relative change in the passage of 'time' between two reference frames.

The following mathematical discussion employs the equations for Gravitational Time Dilation dervied from Einstein's theory of General Relativity. For a brief introduction to these equations see the following page -> An Introduction to the Equations of Gravitational Time Dilation

Scilab Program

Visit my Scilab page for information on downloading Scilab as well as simple instructions on loading and running Scilab programs.

We begin by investigating the field geometry. We define the required constant values.


stacksize('max');
Gc = 6.67259E-11; //grav constant
Er = 6371000; //radius earth meters 6378137 equator
Em = 5.9742E24; //mass earth kg
c = 299792458; //speed of light meters per second


We define two matrix arrays, for graphing purposes.


x = (1:10000); // array to store original unshifted frequency of light for x axis of graph
df = (1:10000); // array to store frequency increase ie if fsource = 100 and freceive = 90 df = 10


We define two more variables, one for the altitude above the earth’s surface (where the source frequency is emitted) and one for the frequency in hertz.


h = 999000; // height from surface of earth in meters
ef = 100000000; // first frequency 1 billion hz


We then create a loop wherein we will investigate the blue shifting of 10000 different frequencies when sent to the surface of the earth from a height of 999 kilometers (999000 meters).


for i = 1:10000
x(i) = ef; // save frequency in array for graph purposes
// calc blueshifted frequency using equation derived from General Relativity
// where ‘of’ is the blueshifted frequency received at the surface of the earth
// and ‘ef’ is the original emitted frequency sent from 999 km above

of = sqrt( (1 - ( (2 * Gc * Em) / ((Er + h ) * c^2))) / (1 - ( (2 * Gc * Em) / (Er * c^2)))) * ef;

// save the frequency difference in array ‘df’ for graph purposes
df(i) = of - ef; // store difference in frequency count between blue shifted and original freq

// increase the frequency of emitted light and try again in next run through loop
ef = ef + 100000000; // add another one billion hz for next calculation in loop
end; // end of the repeating code loop


We can use the following code to create a graph displaying the results.


my_handle = scf(100001); // create graphics handle
clf(my_handle,"reset"); // reset the graphics
my_plot_desc = "Gravitational Blue Shift Frequency Difference 999km, x = emitted, y = freq difference";
my_handle.figure_name = my_plot_desc;
plot2d(x,df); // x axis is original frequency y axis is difference blue shifted frequency
xtitle(my_plot_desc," "," "," "); // graph title


We can use the following code to create a text file displaying the results of the calculation.


fd = mopen('C:\df.txt', 'w'); // open a file for write
for i = 1:10000 // loop through the arrays
mfprintf(fd, "%25.0f %25.40f \n", x(i), df(i)); // print freq and diff in formatted columns
end; // end loop
mclose(fd); //close the text file


We can also save the results of the calculation for reuse in further calculations using the following code.


save('C:\fdiff.dat',x,df);
// can be reloaded by command -> load('C:\fdiff.dat',’x’,’df’);


The result of this calculation is the creation of the graph of a linear function (the graph is a straight line). The x axis of the graph is the original frequency and the y axis of the graph is the small difference in frequency between the received blue shifted frequency and the original frequency (for example if 90 was emitted, 100 received then diff (y axis) would be 10).





To ensure that this is in fact a linear function we can increase the size of the loop variable ‘i’ in the code above, while also making a matching increase in the size of the arrays ‘x’ and ‘df’ to make millions of calculations instead of just 10000, and the result would be a graph such as the one below, which remains linear, even at higher energy levels.





The graph is linear and for this reason we can use the standard ‘point slope formula’ to generate a linear equation.

y = (m * x) + b;

Y is the small frequency difference in the received frequency, x is the original frequency, m is the slope of the line, and b is a constant value.

The equation for calculating the slope (m) of a line is,

// given any two points on the line (x1, y1) and (x2, y2)
m = (y2 – y1) / (x2 – x1);

When we have found the value ‘m’ and by using any one point on the line (x, y) we can then calculate the value of the constant ‘b’ for

b = y – (m * x);


However when the frequency is ‘zero’ the ‘frequency difference’ is also zero. Since ‘b’ is the value of the ‘y intercept’, we know that ‘b = 0’, and that our equation is therefore of the simple form, ‘y = m * x’.

When we have discovered the values of ‘m’ we now have a simple linear equation we can then use to generate a second graph. We generate a loop in the code as follows.


ef = 100000000; // first frequency 1 billion hz
for i = 1:10000
x(i) = ef;
df(i) = (m * ef) ;
ef = ef + 100000000; // add another one billion hz for next calculation in loop
end;


The result of this linear calculation produces a graph identical to the first graph generated using the equation from General Relativity.





We can see that the graph of the change of frequency caused by the ‘Gravitational Red/Blue Shift’ is a simple linear function. The reason for this is that the graph for the change of total summed wavelengths is a horizontal flat line graph (with a slope of zero), for when we multiple the change in frequency (by subtracting the source frequency from the blue shifted frequency) and then multiple this by the change in wavelength, the result is exactly equal for any frequency (which then produces a flat horizontal graph).





We can calculate the Gravitational Blue Shift of a certain frequency of light, emitted from an altitude of 1,000,000 meters (1,000 kilometers) and then received at the surface of the earth, using the following Scilab code (which uses the Xnum plugin).


stacksize('max');

// list of constants in the string format used by Xnum plugin
Gc = '6.67259E-11'; //grav constant
Er = '6371000'; //radius earth meters 6378137 equator
Em = '5.9742E24'; //mass earth kg
c = '299792458'; //speed of light meters per second
csqr = xmul(c, c, 200); // c squared
n2 = '2'; n1 = '1';

x = ‘299792458000000000’; // frequency of emitting source, chosen so that wavelength is one nanometer

// altitude meters
h = '1000000';

// calc first blue shifted frequency, altitude h, using Xnum translation of equation derived from General Relativity
y = xmul(xsqrt( xdiv( xsub(n1, xdiv(xmul(n2, xmul(Gc, Em) ), xmul(xadd(Er, h), csqr) ) ), xsub(n1, xdiv(xmul(n2, xmul(Gc, Em)), xmul(Er, csqr))))) , x);
// the following line displays the same code in normal Scilab format
// y = sqrt( (1 - ( (2 * Gc * Em) / ((Er + h ) * c^2))) / (1 - ( (2 * Gc * Em) / (Er * c^2)))) * x;


The blue shifted frequency is
y = 299792458028315189.934161245165

We can calculate the wavelength of this blue shifted frequency as follows…

w = xdiv(c, y, 200);

This generates the solution,
w = 9.999999999D-10

We multiply ‘w’ (the blue shifted wavelength) by ‘x’ (the original source frequency), to generate the length ‘L’,
L = xmul(x, w, 200);
And we find that L = 299792457.97168481…

We subtract L from c, to find the remainder distance,
d = xsub(c, L, 200);
and this results in d = 0.028315189931486895

We can then use the equation Time = Distance / Velocity to generate the time kept by the shortened clock, and we can also measure the difference in the time kept by the clock by substituting the value ‘d’ for the value ‘L’.

ctime = xdiv(L, c, 200);
dtime = xdiv(d, c, 200);
ctime = 0.9999999999….
dtime = 9.44493D-11

We wish to find out if the shortened distance of the light clock results in the clock keeping perfect time with clock higher in the field which was the source of the shifted frequency. We use Einstein’s equation for gravitational time dilation to calculate the time kept by the clock higher in the field and the time kept by a clock on the earth’s surface.

M = Em; // mass of earth
R = Er; // radius of earth
// gravitational time dilation surface value
ts = xsqrt( xsub( n1, xdiv( xmul(n2, xmul(Gc, M, 200)), xmul( R, csqr, 200), 200), 200), 200);
// ts = 0.9999999993

//T = To / ts
// when T = 1
// 1 = To / ts -> To = ts
// when To = 1
// T = 1 / ts

Tinf = '1.0'; // time at infinity fastest second
trel = xdiv(Tinf, ts, 200); // when one second passes on earth surface, trel seconds passes on fastest clock in universe, located at ‘infinity’
// trel (relative to earth’s surface) = 1.00000000069618584248

//calculate time kept by clock at altitude of 1,000,000 meters
R = xadd(Er, 1000000, 200);
ts = xsqrt( xsub( n1, xdiv( xmul(n2, xmul(Gc, M, 200)), xmul( R, csqr, 200), 200), 200), 200);
// ts = 0.999999999398263465

trel = xdiv(Tinf, ts, 200); // when one second passes at altitude 1,000,000 meters, trel seconds passes on fastest clock in universe, located at ‘infinity’
// trel (relative to 1,000,000 meters above earth’s surface) = 1.0000000006

The calculated time difference between the two clocks (based upon the standard clock located at infinity) = ‘0.0000000000944493…’, which is equivalent to ‘dtime’, the value calculated for the time of the small remainder distance in our light clock above. This then leads us to conclude that the shortened clock does indeed keep time with the clock higher in the field.

We can calculate the ratio of ‘ctime’ (the second measured by the shortened light clock) relative to one full second on the surface of the earth.

r = xdiv(‘1.0’, ctime, 200);
// r = 1.0000000000944493, which is equivalent to the difference in one second between the clock higher in the field and the slower clock located on the surface of the earth

Based upon these observations we conclude that it is possible to explain the anomalous behaviour of clocks by means of changes in field geometry. This would then suggest that the universe does not actually come with ‘clock included’ for what we understand as the passage of time is just a secondary by-product, and only exists because of the fixed velocity of the transfer of energy, while the dilation and contraction of this ‘time’ is produced simultaneous with dilation and contraction of ‘space’ (we should therefore abandon the term ‘time dilation’ and instead we should begin to use the terms ‘spacetime dilation’ or ‘spacetime contraction’, for it would seem that it is impossible to ‘dilate time’ for ‘time’ and ‘space’ are bound up together in such a way as to make them inseparable).