1,5 The Metric Equation

      This document explains how relativistic effects are calculated from the metric equation, which is based on the metric tensor. The metric equation is derived as follows. Consider two points, (1), (2) separated by distances Dx, Dy, Dz in the x, y, z directions. The Pythagorean theorem shows that the distance Dd between the two points is

      (Dd)2  =  (Dx)2 + (Dy)2 + (Dz)2                                         (1)

Assume that a light pulse is emitted at point (1) and travels to point (2) in time Dt. The distance Dd between the points is equal to (cDt), which is  the speed of light c multiplied by the time interval Dt. Since  Dd and (cDt) are equal, the difference [(cDt)2 - (Dd)2] must be zero.  Hence the following relation holds:

      0  =  (cDt)2 - (Dd)2  =  (cDt)2 - (Dx)2 - (Dy)2 - (Dz)2           (2)

To handle more general experiments, the difference between (cDt)2 and (Dd)2 is defined as (Ds)2. This yields the metric equation, which is  

      (Ds)2  =  (cDt)2 - (Dd)2  =  (cDt)2 - (Dx)2 - (Dy)2 - (Dz)2      (3)  

A relativistic event is characterized by the time of the event and the three-dimensional spatial coordinates where the event occurs. The generation of the light pulse at point (1) is event (1), and the reception of the light pulse at point (2) is event (2). In this case the quantity (Ds)2 is zero. 

      Assume that a missile is fired from point (1) to point (2). Event (1) is the firing of the missile at point (1), and event (2) is the missile impact at point (2). Since the missile speed is less than the speed of light, the time difference Dt between the events is greater than in the first experiment, and so (Ds)2 is greater than zero. A case where (Ds)2 is greater than zero is called time-like. In a time-like case, one event can influence the other.

      Now assume that point (2) has an early-warning system, which detects the flash of the missile firing at point (1). In response to this flash, point (2) fires a missile in return. Event (1) is the firing of the missile from point (1) and event (2) is the firing of the second missile from point (2). The difference between these two events is still time-like. The quantity (Ds)2 is greater than zero, and so one event can (and did) influence the other. 

      Now assume that two warring parties have a truce that ends at noon. Not trusting the other party, each one decides to launch a preemptive strike exactly at noon. Event (1) is when point (1) fires its missile and event (2) is when point (2) fires its missile. In this case (Ds)2 is negative, and the condition is called space-like. In a space-like case where (Ds)2 is negative, there is not enough time between the two events for a light pulse to travel between them, and so one event cannot influence the other.

      Let us apply the metric equation for (Ds)2 to a space travel experiment, which is discussed in Story [2], Chapter 7. Two observers traveling at different speeds are measuring the speed of light. The two observers measure different values for the time and spatial variables involved in the experiment. However if (Ds)2 is zero for one observer, it is zero for the other, because the speed of light is the same for both observers. Since the two observers always measure the same value for (Ds)2, the quantity  (Ds)2 is called invariant. By recognizing that (Ds)2 is the same for all observers one can derive the equations of Special Relativity.

Effect of Gravity on the Metric Equation  

      In a gravitational field the speed of light is not constant, and so the simple metric equation does not apply and must be generalized. The metric equation is first expressed in terms of the general coordinates x0, x1, x2, x3. Coordinate x0 is normalized time (which is equal to ct and denoted t) and x1, x2, x3, are the spatial coordinates, x, y, z. The metric equation becomes

      (Ds)2  =  (Dt)2 - (Dx)2 - (Dy)2 - (Dz)2 

                =  (Dx0)2 - (Dx1)2 - (Dx2)2 - (Dx3)2               (4)

In a gravitational field, the terms of the metric equation are modified by the metric tensor elements g00, g11, g22, g33 to give:

      (Ds)2  =  g00(Dx0)2 + g11(Dx1)2 + g22(Dx2)2 + g33(Dx3)2                  

    =  g00(Dt)2 + g11(Dx)2 + g22(Dy)2 + g33(Dz)2                  (5)

This assumes that the metric tensor is diagonal. For a non-diagonal metric tensor with 16 elements, the metric equation has 16 terms, which includes cross-product terms, such as  g13Dx1Dx3. However, our discussion is limited to diagonal metric tensors. (After all, Einstein could only consider diagonal metric tensors. A gravitational field equation yielding a non-diagonal metric tensor cannot be solved without a computer.)

      Compare the general metric formula of Eq. 5 with Eq. 4, which applies in the absence of a gravitational field. This shows that with no gravitational field, g00 is equal to +1 and g11, g22, g33 are equal to -1. Within our solar system the gravitational field is very weak, and so the values of the metric tensor elements are very close to these ideal values.

Calculating Relativistic Effects from the Metric Equation

      With the Yilmaz theory, the metric tensor elements g11, g22, g33 in rectangular coordinates are equal. Consequently the speed of light and the contraction of distance is the same for all directions. This allows us to simplify the metric equation by considering a single spatial dimension, which we call x. The metric equation reduces to

      (Ds)2  =  g00(Dt)2 + g11(Dx)2                              (6)                     

      For the Schwartzschild solution, the speed of light and the contraction of distance are not the same in all directions. This confusing issue is discussed in Appendix G of Believe [1] and in this Website [3], Addendum (Page 5), Chapter 2. In this document, we ignore it. For the Schwartzschild solution, we only consider motion in the radial direction.

      If Ds in the metric equation is set to zero, a light pulse can just travel between the two events. Hence this condition yields the speed of light.

      If Dx is set to zero, (Ds)2 is positive because g00 is positive. Hence the interval between the events is time-like. In this case the value for Ds is called the proper time between the two events, which we denote Dtp. Proper time is the time difference read by a clock that is moved at constant velocity between the two events.

      If Dt is set to zero, (Ds)2 is negative because g11 is negative. Hence the interval between the events is space-like. In this case the value for Ö[-(Ds)2] is called the proper distance between the two events, which we denote Dxp. Proper distance is the distance read on a ruler stretched between the two events, where the ruler is at rest in the coordinates for which the two events are simultaneous.

      Applying these principles to the metric equation (Eq. 6) gives the results in Table 1. In the first case, Ds is set to zero, and the ratio Dx/Dt is calculated, which (as we will see) gives the apparent relative speed of light. In the second case, Dx is set to zero, and the value for Ds is the proper time, which is denoted Dtp. In the third case, Dt is set to zero, and the value for Ö[-(Ds)2] is the proper distance, which is denoted Dxp. The formulas that are calculated from the metric equation for the variables of the second column are given in the third column.

Table 1: Derivation of relativistic parameters from  metric equation

condition        variable            formula             meaning

  Ds = 0        Dx/Dt               Ö[-g00/g11]        speed of light
Dx = 0        Dtp = Ds           Ö[g00] Dt          proper time
Dt = 0        Dxp = Ö[-(Ds)2]  Ö[-g11] Dx        proper distance

Table 2: Relativistic effects obtained from metric equation.

meaning                   physical            equation                formula
                                 ratio                 ratio

  speed of light             cap/c              Dx/Dt              Ö[-g00/g11]
spatial contraction    Dxap/Dxp        Dx/Ö[-(Ds)2]      1/Ö[-g11]
 clock period              Dtap/Dtp         Dt/Ds              1/Ö[g00]
wavelength ratio        l(inf)/l             Dt/Ds              1/Ö[g00]

      The variables Dt and Dx in Table 1 are usually called coordinate time and coordinate distance. These are time and distance intervals that are measured at infinity relative to a star, and so are the values that appear to exist to an observer on earth. We use the term "apparent" to represent the term that is usually called "coordinate" in the relativity literature. The variables Dt and Dx in Table 1 are called apparent time and apparent distance intervals, and are denoted Dtap and Dxap.

      Since Dt is equal to cDt, the ratio Dx/Dt is equal to (1/c)(Dx/Dt). The velocity Dx/Dt for this case is the apparent speed of light, which we denote cap. Hence the ratio Dx/Dt is equal to cap/c, which is the relative value of the apparent speed of light.

      The above principles are applied to Table 1 to give the ratios shown in Table 2. The second column shows the variables in terms of physically meaningful terms, and the third column shows the values specified in the metric equation. The fourth column gives the formulas for these ratios that are calculated from the metric equation. A gravitational field causes the speed of light to decrease, it causes a dimension to contract, and it causes a time interval to expand. Because of the expansion of time interval, a clock runs slower.

      Since a gravitational field causes a clock to run slower, an excited atom on the surface of a star oscillates at a lower frequency, and so its spectrum has a longer wavelength. The wavelength ratio l(inf)/l is equal to Dt/Ds, where l is the normal wavelength and l(inf) is the wavelength that is observed on earth (at an infinite distance from the star). This wavelength ratio is shown in the last row of Table 2.       

      Table 3 shows the g00 and g11 elements for the Einstein and Yilmaz theories given in Story [2], Appendix E.. The values for the Einstein Schwartzschild solution are obtained from Table E-3 and those for the Yilmaz solution are obtained from Eqs. E-19, E-20.   

Table 3: Metric tensor elements for Einstein Schwartzschild solution and Yilmaz solution of the gravitational effects of a star

                       Einstein               Yilmaz

           g00            1 - 2(m/r)               exp[-2m/r]
          g11          -1/[1 - 2(m/r)]         - exp[2m/r]  

      These g00 and g11 values are applied to the formulas of Table 2 to obtain the specific formulas in Table 4 for the Einstein Schwartzschild solution and for the Yilmaz solution of a single star. The clock rate in Table 4 is equal to the reciprocal of Dtap/Dtp in Table 2, which is the clock period. These specific formulas are plotted in Story [2], Chapter 10, Figs. 10-1 to 10-3.

Table 4: Relativistic effects caused by a gravitational field for Einstein and Yilmaz theories

Characteristic                General            Einstein              Yilmaz

Speed of light                 Ö[-g00 /g11]      [1 - 2(m/r)]          exp[-2m/r]
Clock rate                       Ö[g00]        Ö[1 - 2(m/r)]          exp[-m/r]
Spatial contraction             1/Ö[-g11]      Ö[1 - 2(m/r)]          exp[-m/r] 
Wavelength ratio (l(inf)/l)  1/Ö[g00]      1/Ö[1 - 2(m/r)]        exp[m/r]  

   To generalize the Yilmaz solution, the m/r ratio is replaced by the relativistic gravitational potential f. The resultant values of g00 and g11 for the general Yilmaz solution are shown in the second column of Table 5. For the Yilmaz cosmology model, 2f is equal to (r/r0)2.. The third column of Table 5 gives the metric tensor elements for the Yilmaz cosmology model.

Table 5: Metric tensor elements g00 and g11 for the general Yilmaz solution and for the Yilmaz cosmology model  

element     General Yilmaz      Yilmaz Cosmology Model       

g00                    e-2f                         exp[-(r/r0)2]      
g11                    - e2f                        - exp[(r/r0)2]       

      The metric tensor values of Table 5 are applied to Table 2 to obtain the values of the relativistic effects for the Yilmaz cosmology model shown in Table 6.  

Table 6: Effects of gravitational field on speed of light, spatial contraction, and  clock rate reduction, for the Yilmaz cosmology model and the general Yilmaz solution

meaning           variables          formula             Yilmaz            cosmology model         

speed of light     cap/c         Ö[-g00/g11]             e-2f              exp[-(r/r0)2]
contraction      Drap/Dr        1/Ö[-g11]                e-f              exp[-(r/r0)2/2]    
clock rate         Dt/Dtap        Ö[g00]                   e-f              exp[-(r/r0)2/2]    

 

References for Document 1,5

[1]  Adrian Bjornson, A Universe that We Can Believe, Addison Press, Woburn, MA, 2000, ISBN 09703231-0-7.

[2]  Adrian Bjornson, The Scientific Story of Creation, Addison Press, Woburn, MA, 2000, ISBN 09703231-1-5.

[3]  Website. This internet website, www.olduniverse. com 

References 1 to 3 are referred to as Believe [1], Story [2], and Website [3].

 

      

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