Static Equilibrium in a Region having a Time Rate Gradient
by
Edmond S. Miksch
ed_miksch@yahoo.com
(412) 373-4919
First Copyright: January 18, 2007
Latest Copyright: April 6, 2009
Introduction
To explain this mystery, we consider the conditions for static equilibrium of an experimental subject in a relativistic time rate gradient due to a gravitational field. Figure TRG1 schematicly illustrates the logic of the analysis of the systems. An experimental subject in a gravitational field is acted upon by a force at its top and a force at its bottom. The force is applied during a time interval that is Ttop at the top and Tbot at the bottom.
The measurements of time and force at the top are made by a local observer at the top. Likewise, the measurements of time and force at the bottom are made by a local observer at the bottom. Such time intervals are sometimes referred to as "proper time" or "clock time". Due to the time rate gradient effect, otherwise known as gravitational time dilation, Ttop is greater than Tbot.
The numerical values for force and time obtained by the local observers are then combined and we look for equilibrium conditions. It is found that force balance and momentum balance cannot both prevail for a body in static equilibrium in a region of space having such a time rate gradient. Explanations are then provided as to how the weight of various electric and magnetic fields is supported. It is found that these fields can be self supporting in a gravitational field if momentum balance, not force balance, is employed. An equation is then derived for the self divergence of the gravitational field vector, which is in agreement with an equation derived earlier from energy considerations.
Contents
- Introduction
- Contents
- Prerequisites
- Terminology and Conventions
- Mystery of the Leaning Timber
- How is the Weight of a Uniform, Vertical Electric Field Supported?
- How is the Weight of a Uniform, Vertical Magnetic Field Supported?
- How is the Weight of a Unidirectional, Horizontal Magnetic Field Supported?
- Should we Reword Newton's First Law of Motion?
- Acknowledgement
- References and Links
Prerequisites
Basic physics, including elementary mechanics and vector operators, notably the divergence and curl, is a prerequisite, as well as familiarity with the theory of relativity.
Terminology and Conventions
We use the term "mass" to refer to the conserved quantity, and never use it for the rest mass of a particle, since rest mass is not conserved. Mass has gravitational properties and inertial properties. We see no reason for questioning the equivalence of gravitational and inertial mass. Energy is viewed as comprising mass in any volatile form. When we speak of "force balance" in reference to a body, we mean that the forces acting on the body are either zero or are equal and opposite. When we speak of "momentum balance" in reference to a body, we mean that the impulses acting on the body are either zero or are equal and opposite. The term "impulse" is taken from mechanics, and equals force times time. When a net impulse is applied to a body, it causes a change in momentum of the body. The change in momentum is equal to the impulse. Force, impulse and momentum are all vector quantities.
Vector quantities, which have both direction and magnitude, are denoted in the text by bold face characters. In the figures, they are denoted by overlying arrows. When the symbol for a vector quantity is not in boldface, it denotes the magnitude of the vector quantity. Asterisks are sometimes used to denote multiplication. Weight is considered to be a real force, as is centrifugal force and the Coriolis force. In this, we rely on the relativistic principle that we may view the universe from any coordinate frame we choose.
Measurements are always made by local observers. That is to say, a length anywhere in space is measured by a local measuring rod, such as a nearby atom that emits an electromagnetic wave of known wavelength. The atom may be bound to other atoms to reduce its recoil, as was done in the Harvard Tower Experiment. Time intervals between two nearby events are measured by a local clock, such as the frequency of a known electromagnetic wave from a nearby atom. Acceleration of an object is measured by local measuring rods and local clocks, and mass is defined by the negative of the source term of the gravitational field vector, in accordance with the vector form of Newton's law of gravity. Mass may also be determined by energy / c2 . An exception to the use of local instruments is that an observer at one elevation may measure the rate of a clock at another elevation by examining the frequency shift of electromagnetic radiation sent from the other clock. (We note that we will not be considering relatively moving systems that would cause Doppler effects and the time dilation effects of special relativity due to the Lorentz transformation.)
The Mystery of the Leaning Timber
The gravitational field g is directed downwardly in the figure, and the strut is disposed obliquely relative to g. F1 and F2 denote horizontal forces on the ends of the strut, and F3 and F4 denote vertical forces on the ends of the strut. The latter do not interest us. We are interested only in the balance of horizontal forces, F1 and F2, or the corresponding momentum balance.
The gravitational field we contemplate is to be precisely unidirectional. The gravitational field vector anywhere in the region of interest is to be precisely parallel to the gravitational field vector anywhere else in the region. We would not use Earth's field because, even above a featureless prairie, the gravitational field vectors are not precisely parallel, they point towards the center of the Earth.
The field we require would be found, however in Einstein's elevator. The acceleration (or gravitational) field seen by an observer who accelerates in field-free space sees a field with precisely parallel field vectors. This is because if acceleration is in the Z direction, the X and Y coordinates are not involved, and that is because the field can be thought of as being due to an ongoing Lorentz transformation involving Z and T, but not involving X and Y. The unidirectional field we require could also be approximated in a small region above the Earth provided rings of dense material were placed below and around the experimental region.
It is well established, from general relativity, that, due to the gravitational field, g, clocks at low elevations run more slowly than clocks at higher elevations. This effect is seen in the spectra of stars such as the sun, in which known spectral lines are shifted to longer wavelengths, which indicates light of lower frequency. The effect has been observed in terrestrial laboratories, for example, in the Harvard Tower Experiment by Pound and Rebka. (See Link 1 below.) Gamma rays sent from the top to the bottom of the tower showed an increase in frequency, and those sent from the bottom to the top showed a decrease in frequency. It is understood that for any pair of identical clocks that properly measure time at their own locations, a clock at a low elevation will run more slowly that its duplicate at a high elevation. The fractional change in rate, based on general relativity and confirmed by experiment, is as follows.
In the preceding expression, ΔZ is the difference in elevation between the clocks, g is the magnitude of the acceleration of gravity and c is the velocity of light. We consider a thought experiment lasting a finite time, Tbot seconds as indicated by the lower clock. The time interval indicated by the upper clock is then:
To be valid, Equation TRG1 requires that the magnitude of (ΔZ * g) be much less than c2. To be precise, we should say that Equation TRG1 is valid in the limiting case wherein the magnitude of (ΔZ * g) is vanishingly small compared to c2. This condition was met in the Harvard Tower Experiment, and we assume it for our thought experiment. We note that the appearance of the clocks in Figure TRG1 is inconsistent with this limiting case, since the clocks indicate a very large difference in elapsed time. The figure is drawn as it is to emphasize that the clock at the top indicates more elapsed time than the clock at the bottom. Ttop is greater than Tbot.
The term "impulse", as used in mechanics, is used to indicate the product of force and time. The impulse applied by force F1 acting for the time Ttop is ( F1 * Ttop ). The impulse applied by force F2 acting for the time Tbot is - ( F2 * Tbot ) . It is noted that F1 and Ttop are measured by instruments placed at the upper end of the strut. Likewise, F2 and Tbot are measured by instruments placed at the lower end of the strut.
We now ask for the conditions for equilibrium. We look for the case in which the strut does not accelerate either toward the right, nor toward the left. From our freshman physics class, we would expect that equilibrium would prevail if F2 = F1.
But, now we consider momentum balance. We consider a thought experiment having a duration of Tbot seconds as seen by an observer at the lower end of the strut, and Ttop seconds as seen by an observer at the upper end. The time interval, Ttop at the top end of the strut begins at the same instant as Tbot at the bottom end of the strut. Likewise, the time interval Ttop at the top end of the strut ends at the same instant as Tbot at the bottom end of the strut. This is possible, even though Ttop does not equal Tbot because space-time is curved due to the gravitational field. If F1 = F2, then the net impulse applied to the strut during the duration of the thought experiment is given by:
By applying Equation TRG1 to Equation TRG2, we obtain:
With considerable consternation, we note that, having assumed force balance, that is F1 = F2, we are left with a finite net impulse. Thus, if the forces F1 and F2 are balanced, then, during the duration of our thought experiment, the strut is given a finite momentum, which causes it to begin moving toward the right. If we had assumed a momentum balance at the outset, then we would have found that the forces F1 and F2 were not equal. Something has to give. Either we abandon force balance, or we abandon momentum balance. We suspect that, for the strut to be in static equilibrium, then there must be momentum balance. In the following section we will apply that guess to explain how the weight of a uniform, vertical electric field is supported.
How is the Weight of a Uniform, Vertical Electric Field Supported?
We refer to the classic physics book: Principles of Physics II Electricity and Magnetism by Francis Weston Sears (1947). In Section 8-8, page 212, the author states that the energy of a charged capacitor is considered to be associated with the electric field of the capacitor rather than the charges on the plates. We consider a capacitor with vacuum between the plates, and obtain for the energy density of the electric field the following:
The quantity ε0 is the permittivity of free space and equals 8.85 * 10-12 . E is the electric field intensity in Volts per meter, and the energy density is expressed in Joules per cubic meter. By applying the famous relation: E = m * c2, we obtain for the mass density of the electric field:
Now, if we consider an electric field in a region of space where there is a gravitational field, g , the electric field has a weight density equal to:
The foregoing equation, TRG6, applies to any electric field in a vacuum and in a gravitational field. Neither the electric field nor the gravitational field need be homogeneous or isotropic. Equation TRG6 expresses the weight density of the electric field at any point in a vacuum where there is both a gravitational field g and an electric field E having a magnitude E. For a specific numerical example, for an electric field of 108 Volts/meter, in a gravitational field of 9.8 meters/second2 , the weight density of the field is about 9.65 * 10-12. Newtons/cubic meter.
We may not be concerned that such a weight is too much to bear. A bridge designer would not be frightened by such a deadweight load. However, suppose we set up a thought experiment in which the electric field is precisely uniform in both magnitude and direction. For such a case, we ask how the weight of the field can be supported. Figure TRG3 illustrates an experimental setup wherein the electric field is precisely uniform in both magnitude and direction. The electric field is established by a pair of parallel plates, an upper plate with a negative charge and a lower plate with a positive charge. An electric field E is in the space between the plates, and a gravitational field, g is also present. The electric field is assumed to be precisely uniform, but the gravitational field may have some variation in magnitude and/or direction. In this ideal situation, we ignore edge effects, with the thought that it is useful to consider this ideal case. We consider a test volume indicated as the rectangle in the center of the figure. It has a top surface and a bottom surface separated by ΔZ. The shape of the test volume, viewed from above, is arbitrary. It can be a circle, square, etc., because we are only dealing with forces or momenta applied to the top and bottom surfaces, per unit area.
Our reason for selecting a uniform, vertical electric field is that in other configurations, the weight of the electric field can be borne by a slight curvature of the electric field lines. For a horizontal electric field, or the field of a spherical charge, there is no mystery. The electric field just sags a bit so its stress tensor provides an upward force. The mystery, for a uniform, vertical electric field, lies in the fact that a precisely uniform, vertical electric field can exert no net force on our test volume. We have the same mystery for an electric field that is directed either upwardly, as shown, or downwardly.
The total force (if any) exerted on the test volume is given by the electric terms in the Maxwell stress tensor. These terms have been described by saying that the electric field conveys stress as would taught rubber bands parallel to the field lines, the rubber bands repelling each other laterally. In the figure, the electric field, E fills the entire space between the plates. It has the same value at both the top and bottom surfaces of the test volume. On the top surface of the test volume, the electric field exerts an upward force per unit area of ε0 * E2/2. On the bottom surface, the electric field exerts a downward force per unit area of ε0 * E2/2. We have here a force balance; the two forces precisely canceling. The total force exerted by the electric field on the test volume is zero. How, then, can the weight of the electric field be supported?
We now consider momentum balance as we did above for the mystery of the leaning timber. Figure TRG3 shows two clocks separated by the elevation difference of ΔZ, which is the elevation difference between the top surface and the bottom surfaces of the test volume. As discussed above, these run at different rates. As before, we consider a thought experiment lasting a finite time, Tbot seconds as indicated by the clock at the bottom of the test volume, and Ttop as indicated by the clock at the top. We repeat Equation TRG1 to have handy the relation between Ttop and Tbot. As before, Equation TRG1 is valid in the limit wherein (ΔZ * g) is vanishingly small compared to c2. In our limiting case, Ttop is very nearly equal to Tbot. The difference is only important when we compare, as by subtraction, a quantity dependent on Ttop with a quantity dependent on Tbot. (The positions of the hands of the clocks is exaggerated to emphasize that the clock at the top indicates a larger elapsed time, Ttop than the clock at the bottom, which indicates Tbot.)
The net impulse per unit area, during the time of the experiment is:
We denote by IE the net impulse per unit area during the time of the experiment, and obtain:
Applying Equation TRG1 to Equation TRG8, we obtain:
To simplify our discussion, we now assume that the area of the top surface of the test volume and that of the bottom surface of the test volume is unity. The volume of the test volume is therefore ΔZ.
We now consider the impulse due to gravity applied to the electric field in the test volume during the time of the experiment. By simply multiplying density times volume times g, we obtain:
(We could equally well have employed Ttop because we are considering the limiting case wherein ΔZ * g << c2 and Tbot approaches Ttop ). By applying Equation TRG6 for the weight density of the electric field, we obtain:
The minus sign in Equation TRG11 is because, of course, the weight of the electric field is directed downwardly.
We now compare IE in Equation TRG9 with Ig in Equation TRG11. We find that they are exactly equal and opposite. That is:
We find that during the time of the experiment, the downward impulse delivered to our test volume by the gravitational field acting on the weight density of the electric field is exactly equal to the net upward impulse delivered to the test volume by the electric field. We note that if the electric field had been directed downwardly rather than upwardly, we would have obtained exactly the same result. The weight density of the electric field would have been unchanged, the force per unit area exerted on the top and bottom surfaces of the test volume by the electric field would have been unchanged, and the time rate gradient due to the gravitational field would have been unchanged.
How is the Weight of a Uniform, Vertical Magnetic Field Supported?
We refer, again, to the physics book: Principles of Physics II Electricity and Magnetism by Francis Weston Sears (1947). From Section 15-13, page 358, we obtain the energy density of a magnetic field in vacuum the following. ( We have added the subscript 0 to μ because we want the energy density of a magnetic field in a vacuum. The constant μ0 is the permeability of a vacuum. )
Reasoning as before for the electric field, we note that since the magnetic field has an energy density, it must have a mass density. If a gravitational field is also present, then the magnetic field must have a weight density. The weight density is given by the following:
Figure TRG4 is an illustration of a solenoid for producing a uniform, vertical magnetic field. As before, a test volume having a height Δ Z is shown in the uniform, vertical magnetic field. The magnetic field is denoted B and the gravitational field is denoted g. For our thought experiment, B can be either antiparallel, as shown, or parallel to g. On the right side of the figure, current in the coil flows in the direction from the reader toward the page. On the left side of the figure, current flows in the direction from the page toward the reader. As before, a clock at the top of the test volume runs rapidly and indicates many seconds. A clock at the bottom of the test volume runs slowly and indicates few seconds.
Applying current to this coil becomes an interesting matter when we consider time rate gradient effects. If we were to employ a coil with a uniform pitch, with the same current flowing through all turns of the coil, then an observer at low elevations, employing a local magnetometer, would measure a stronger field than an observer at a high elevation. This is because a clock at the lower observer's elevation runs more slowly than a clock at the upper observer's elevation. Hence, the number of electrons per second flowing through a turn at the lower observer's elevation would be greater than the number of electrons per second flowing through a turn at the upper elevation. More electrons per second means more electrical current, and, by the laws of magnetism, a greater current generates a stronger magnetic field. We suspect that a radial component of the magnetic field would exist and this would make it very difficult to calculate the forces available to support the weight of the magnetic field.
To obtain the simple problem we wish to consider, we must apply the same number of Ampere-turns/meter at all elevations. We can do this by employing a coil having a variable pitch, all turns of which carry the same current. The pitch would increase downwardly, in accordance with the prevailing time rate gradient. In this manner we could have the same number of Ampere-turns/meter at all elevations, and hence energize the coil with a single electric current. Alternatively, we can build the coil with individual loops, each driven by a separate constant-current generator. We assume that the turns are sufficiently closely spaced and far enough from the test volume that the magnetic field in the test volume is everywhere vertical and of uniform magnitude.
Having taken these pains to set up our thought experiment, the result is very simple and is quite similar to the electrical example. Equation TRG14, for the weight density of the magnetic field, is similar in form to Equation TRG6 for the electric field. Also, the stress tensor is similar. The stress tensor of the magnetic field, like the stress tensor of the electric field, can be described as taught rubber bands that repel each other laterally. The magnetic field exerts an upward force per unit area on the top surface of the test volume of B2/( 2 * μ0 ). Likewise, at the lower surface, it exerts a downward force of the same amount, B2/( 2 * μ0 ). These cancel. The upward force on the top surface is balanced by the downward force on the bottom surface. Force balance, therefore, cannot explain how the weight of the magnetic field in the test volume is supported.
Again, we return to momentum balance. We consider a thought experiment lasting a finite time, Ttop at the top surface and Tbot at the bottom surface. Equation TRG1 provides the relationship between Ttop and Tbot . We repeat Equation TRG1 to have handy the relation between Ttop and Tbot . As before, Equation TRG1 is valid in the limit wherein (ΔZ * g) is vanishingly small compared to c2. In our limiting case, Ttop is very nearly equal to Tbot. The difference is only important when we compare, as by subtraction, a quantity dependent on Ttop with a quantity dependent on Tbot.
The net impulse per unit area, due to the stress tensor of the magnetic field acting during the time of the experiment on the top and bottom surfaces of the test volume is:
We denote by IB the net impulse per unit area due to the stress tensor of the magnetic field acting during the time of the experiment, and obtain:
Applying Equation TRG1 to Equation TRG16, we obtain:
To simplify our discussion, we now assume that the area of the top surface of the test volume and that of the bottom surface of the test volume is unity. The volume of the test volume is therefore ΔZ.
We now consider the impulse due to gravity applied to the magnetic field in the test volume during the time of the experiment. By simply multiplying volume, time, and weight density, we obtain:
(We could equally well have employed Ttop because we are considering the limiting case wherein ΔZ * g << c2 and Tbot approaches Ttop ). By applying Equation TRG14 for the weight density of the magnetic field, we obtain:
The minus sign in Equation TRG19 is because, of course, the weight of the magnetic field is directed downwardly. We now compare IB in Equation TRG17 with Ig in Equation TRG19. We find that they are exactly equal and opposite. That is:
We find that during the time of the experiment, the downward impulse delivered to our test volume by the gravitational field acting on the weight density of the magnetic field is exactly equal to the net upward impulse delivered to the test volume by the magnetic field. We note that if the magnetic field had been directed downwardly rather than upwardly, we would have obtained exactly the same result. The weight density of the magnetic field would have been unchanged, the force per unit area exerted on the top and bottom surfaces of the test volume by the magnetic field would have been unchanged, and the time rate gradient due to the gravitational field would have been unchanged.
It is noted that the terms in Equation TRG17 for the impulse applied by the magnetic field come from quite different sources from the corresponding terms in TRG19 for the impulse applied by the gravitational field. For example, the factor 1/c2 in Equation TRG17 arises from the time rate gradient. The corresponding factor in TRG19 arises from the ratio of energy to relativistic mass.
How is the weight of a horizontal, unidirectional magnetic field supported?
We now consider a case that has quite different mathematics from the previous case, but which comes to a similar conclusion. We ask how the weight of a horizontal, unidirectional magnetic field is supported. We consider the field that would be generated inside an elongate solenoid having a horizontal axis. The energizing current flows serially through all the turns of the coil, and the coil has a constant pitch. Figure TGR5 illustrates a cross section of an elongate solenoid having a rectangular cross section. A right-handed coordinate frame is shown. X is indicated by the approaching arrow symbol, Y is to the right and Z is upward. The axis of the solenoid is in the X-direction. The turns of the coil are equally spaced from each other in the X-direction.
The coil is energized by a clockwise current, as indicated by the arrows in the windings. From basic magnetism, we know that a magnetic field is generated inside the coil. The direction of the field is in the direction from the reader toward the figure, as indicated by the receeding arrow symbol next to the symbol B. This is the -X direction. The coil is immersed in a gravitational field indicated by g. The gravitational field g is directed in the -Z direction. The coil is assumed to be very long compared to its transverse dimensions, so the return field outside the coil is vanishingly small. Hence, outside the coil, we assume that B = 0.
Because the energizing current flows serially through all the turns of the coil, the coil current at a low elevation on the coil, where clocks run slowly, has a greater value than the coil current at a relatively high elevation, where clocks run more rapidly. Hence, the magnetic field inside the coil at the low elevation has a larger value than the magnetic field at the relatively high elevation. (We remember that the field outside the coil has zero strength, because we have assumed the limit of a very long coil.)
It appears to be reasonable that the magnetic field inside the coil has zero variation in the Y direction, because if there were variation in the Y-direction, the stress tensor of the magnetic field would exert a lateral volume force, and neither gravity, time rate gradient effects or any other know effect would balance it. (We remember that the stress tensor of the magnetic field indicates a repulsive force between adjacent field lines.) The magnetic field is expected to vary in the Z direction because gravity and time rate gradient effects can balance forces due to the variation in the Z direction. The magnetic field just inside the coil has a value given by the surface current (Amperes/meter) of the adjacent coil. The magnetic field is antiparallel to the X-axis and has the value B(Z) = - μ0 * J / Δ X , where J is the Amperes in the adjacent coil windings and Δ X is the spacing between adjacent turns of the coil.
The figure illustrates a test volume disposed within the coil. It has a top surface and a bottom surface separated by an elevation difference of ΔZ. This elevation difference, of course, causes time rate gradient effects. Time runs more rapidly at the top of the test volume than it does at the bottom due to the time rate gradient effect. A clock located at the top of the test volume runs rapidly and indicates a large elapsed time, T top as indicated by the sketched clock at the elevation of the top of the test volume. A clock at the bottom of the test volume runs relatively slowly and indicates a relatively small elapsed time. Figure TRG5 shows a surface current Jtop at the elevation of the top of the test volume, and a surface current Jbot at the elevation of the bottom of the test volume. The surface current Jtop generates a magnetic field Btop at the level of the top of the test volume, and the surface current Jbot generates a magnetic field Bbot at the level of the bottom of the test volume. We repeat Equation TRG1, which provides the relation between Ttop and Tbot.
Since the charges (electrons) flowing through the turns of the coil are conserved, the currents very reciprocally as the time. Hence, since we are assuming that ( ΔZ * g / c2 ) << 1 , we obtain:
Again, we consider a thought experiment lasting a finite time, Ttop at the top surface of the test volume and Tbot at the bottom surface. The area of the test volume, viewed from above, is unity, and the volume of the test volume is ΔZ. The net impulse applied by the magnetic field to our test volume is then:
We now substitute values for Ttop and Btop from Equations TRG1 and TRG 22 into Equation TGR23 and get TRG24, which follows:
Then, since ( ΔZ * g / c2 ) << 1, we get:
Thus:
We now consider the impulse due to gravity applied to the magnetic field in the test volume during the time of the experiment. By simply multiplying volume, time, and weight density, we obtain:
(We could equally well have employed Ttop because we are considering the limiting case wherein ΔZ * g << c2 and Tbot approaches Ttop ). We repeat Equation TRG14 for the weight density of the magnetic field.
We are using the same equation for weight density as we did above for the case of a vertical magnetic field, because the energy density and hence the weight density of a magnetic field is independent of its direction. By applying Equation TRG14, we obtain:
Again, as for the case of a vertical magnetic field, we obtain momentum balance by employing the time rate gradient. Note that we still do not have a force balance. If we compare the force per unit area on the top and bottom of the test volume, we obtain an upward force that exceeds the weight of the magnetic field in the test volume. When we looked at force balance for the homogeneous vertical magnetic field, we found zero upward magnetic force and the weight was not balanced. Now, we have too much magnetic force. We thus confirm, again, that equilibrium is based on momentum balance, not force balance.
Another interesting aspect of this thought experiment is that if we apply the vector operator known as the curl to the magnetic field shown, we obtain a nonzero value. We obtain a vector pointing in the + y direction. We have this value for curl B even though no volume currents or time-changing electric fields exist. Including this curl may provide a link between gravitational equations and electromagnetic equations.
Should we Reword Newton's First Law of Motion?
Newton's first law of motion states that an object at rest will remain at rest unless acted upon by an external and unbalanced force. Likewise, an object in motion will remain in motion unless acted upon by an external and unbalanced force.
But, we have found in the preceding examples, that force balance fails to conserve momentum when time rate gradient effects are taken into account. Force balance, also, cannot explain how the weight of a vertical electric field, or a vertical magnetic field, is supported. We have shown that various force fields can be supported provided impulse balance, rather than force balance, is employed. Accordingly, it appears that we should reword Newton's first law as follows:
We understand, of course, that impulse is the integral of force multiplied by time.
Acknowledgement
The author of this web site cannot recall the reasoning process that led to the inquiry concerning the mystery of the leaning timber, or how the weight of a vertical electric field is supported. It was probably a revelation from the Lord Jesus Christ. Thank you, Lord.
References and Links
Reference 1: Negative Mass: A Thesis Presented to Reed College by Edmond Miksch: May, 1954 Page 4.
Reference 2 Negative Mass: A Thesis Presented to Reed College by Edmond Miksch: May, 1954 Page 1.
Link 1: The Pound-Rebka Experiment.
Link 2: Negative-Mass Home Page.
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