CALCULATION OF THE MASS DENSITY OF THE CORIOLIS FIELD
by
Edmond S. Miksch
ed_miksch@yahoo.com
Copyright: October 17, 2007
Abstract
A review of the Coriolis field is provided, and a discussion of Coriolis fields due to relativistic effects that are caused by moving masses. Parallels are drawn between the magnetic field and the Coriolis field. A calculation is made of the Coriolis field caused by moving mass streams, and it appears that the Coriolis field has a negative mass density. A factor of 2 discrepancy in the calculation remains to challenge the reader.
Index of Coriolis Page
- Abstract
- Index
- Internal Links
- Prerequisites
- Terminology and Conventions
- Introduction to the Coriolis Field
- Relativistic Coriolis Fields
- News Flash: Coriolis Fields are generally non-uniform
Other negative mass web sites.
- Go to Negative-Mass home page.
- Go to time rate gradient page.
- Why Gravitational Radiation is Believed to have Negative Mass
- Annotated References and Links
Prerequisites
An understanding of basic physics including electromagnetism, and an
understanding of the basic principles of special and general relativity, will be
helpfull for understanding the following.
Terminology and Conventions
We use the term "mass" to refer to 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. To clairify some equations, asterisks are used to denote multiplication. SI units are employed.
Vector quantities, which have both direction and magnitude, are denoted by bold face characters. When the syimbol 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. When an integral sign is needed, we employ the phrase (integral of) to avoid printing problems.
We do not, however, disparage purists who prefer to consider geodesics in Minkowsky space, rather than considering gravitational forces as contributing to the acceleration of a mass. Our attempt is to stay as close as possible to everyday experience. SI units are employed. These include the meter, second, kilogram, Joule, Newton, Volt, Ampere, Coulomb, etc.
For the magnetic field, we employ the magnetic flux density B, rather than H, because B relates more easily to observable quantities. In particular, B relates to the force on a charge moving across a magnetic field. For the Coriolis field, we employ the vector Ω. Thus, the equation for the force on a mass m moving at velocity V across a Coriolis field Ω is F = 2mV X Ω. The symbol X denotes the vector cross product.
Introduction to the Coriolis Field
The Coriolis field is a curious matter that combines the mystery of gravity with
the mystery of magnetism. It derives its name from a French mathematician,
Gaspard de Coriolis (1792-1843), who noted that in a rotating coordinate frame,
for example, a frame attached to the rotating Earth, a mass moved in a direction
oriented at an angle relative to the axis of rotation of the coordinate frame
tends to accelerate in a direction perpendicular to its motion, and
perpendicular to the axis of rotation.
For example, a wind in the Northern Hemisphere, away from the equator, blowing
toward the north, tends to veer toward its right, hence toward the east.
Likewise, a wind in the Southern Hemisphere, blowing toward the south, tends to veer toward its left, hence also toward the east. This causes the prevailing westerly winds (which blow toward the east) in the temperate zones both north and south of the equator. Both the Gulf Stream
and the Japanese Current are in the Northern Hemisphere, flow toward the north,
and veer toward the east due to the Coriolis Field. Likewise, north of the
equator, air flowing toward a low pressure region tends to veer towards its right
and this creates a counterclockwise flow around the low pressure region. Thus,
tornadoes and hurricanes in the Northern Hemisphere rotate counterclockwise
around their low pressure regions. The opposite is true in the southern
hemisphere.
In latitudes away from the equator, at high altitudes where terrain does not interfere with air flow, air does not flow from high pressuure to low pressure. Instead, air flows in a direction perpendicular to the pressure gradient, hence, parallel to the isobars. Force to react the pressure gradient comes from the wind interacting with the Coriolis field of the Earth.
In a science museum, one may see a pendulum that is free to move back and fourth
in any vertical plane passing through its center of support. One observes that
the plane in which it moves slowly rotates, or precesses, as the day proceeds.
This, also, is an effect of the Coriolis field. If a spinning flywheel is exposed to a Coriolis field, its bearings experience a torque that is perpendicular to the Coriolis field and perpendicular to the axis of the flywheel. A turn signal in an airplane operates by this principle to measure the magnitude of the vertical component of the Coriolis field seen by the airplane, thus showing the rate at which the airplane is turning.
We are curious about the Coriolis field because an observer who sees a Coriolis field always sees it to be accompanied by a centrifugal acceleration field, which points outward. In contrast, the Earth has an acceleration field, which we call the gravitational field, which always points inward. According to the general theory of relativity, there is no physical distinction between an acceleration field seen by an observer and a gravitational field seen by that observer. A centrifugal acceleration field may, therefore, be regarded as a gravitational field. In the language of vector algebra, the Earth is associated with a negative divergence, or sink of the acceleration (or gravitational) field vector. A Coriolis field, conversely, is associated with a positive divergence, or source of the acceleration (or gravitational) field vector.
It is instructive to rewrite Newton's law of gravity in differential form as follows, where MV is the mass density of space, div is the divergence operator, g: is the gravitational field vector, and G is the gravitational constant, which equals 6.672 * 10-11.
If we wish to apply Equation C1 to the acceleration field, denoted a, associated with the Coriolis field, we first calculate the divergence of the centrifugal acceleration field a. It is well known that if an observer rotates at an angular rate of omega radians per second, that observer will see a centrifugal acceleration field as follows, where r denotes a cylindrical radius vector from the axis of rotation of the observer to any point in space, and a is the acceleration vector at that point:
Applying the divergence operator in cylindrical coordinates, we obtain:
Simplifying the equation, we obtain for the divergence of the acceleration field vector, a due to a Coriolis field Ω , the following:
Combining Equation C1 with Equation C2, we obtain for the mass density of the Coriolis field:
We now note, with astonished consternation, that not only is the mass density of the Coriolis field negative, it is also enormous! If we consider a unit Coriolis field, which coresponds to a rotation rate of one radian per second, and insert values into Equation C3, we find that the mass density of that modest Coriolis field is about -2.385 billion Kilograms per cubic meter. Hence, if I just turn in my swivel chair, I fill the entire universe (as seen from my reference frame) with a negative mass density of billions of kilograms per cubic meter!
So what good is this? Shouldn't I learn my lesson and never view the universe from a frame of reference that sees a Coriolis field? Well, here's the problem. According to the theory of relativity, moving masses generate Coriolis fields in their vicinity. This is sometimes referred to as "frame dragging" or the "Lense-Thirring effect". A satellite is now in orbit, the Gravity Probe B, which contains four spinning quartz spheres, held in position by superconductors, to reveal precession caused by the Coriolis field caused by rotation of the massy Earth. Note that this is not the Coriolis field that is of concern to meteorologists. That Coriolis field is due to rotation of the observer relative to distant galaxies, and does not depend on the mass of the Earth.
The Coriolis fields generated by moving masses are non-uniform. Hence, I cannot select a coordinate frame for which the Coriolis field is everywhere zero. I must accept the Coriolis field as a reality that I cannot avoid, and I may need to consider the mass density of the Coriolis field.
Various investigators refer to the Coriolis field by a variety of terms. These include the
gravimagnetic field, the kinemassic field, the gravinetic field, the gravnetic
field, gravomagnetic field, gravitomagnetic field, gravitomagnetism, and the magnetic-like
gravitational field. In the following section, we provide an explanation of the origin of the Coriolis fields due to moving masses, and we show that such Coriolis fields have negative mass.
Coriolis fields caused by moving masses have some similarity to magnetic fields generated by moving electric charges. In this section, we will be providing a relativistic derivation of the magnetic field as a basis for the relativistic derivation of the Coriolis field. We do well, however, to ignore some common aspects of magnetism, which are not fundamental enough to be relevant. Those aspects having to do with magnets comprised of ferromagnetic materials, and the attraction of feromagnetic materials to the magnets are not helpfull. These are more complicated problems in which the movement of electric charges is due to the spin of electrons.
For a more simple view of magnetic effects, we consider the works of Ampere, Biot and Savar, who found that electric currents exert forces on each other, even without ferromagnetic materials being present. In particular, parallel currents attract, antiparallel currents repel, and a circular turn carrying an electric current tends to expand. The magnetic field acts as an intermediary between two or more currents, or between one part of a current and another part of the current. The magnetic field is created by the current(s) and exerts forces on the currents. Any one observer will see the magnetic field to have a particular magnitude and direction at each point in space. An observer moving at a different velocity may see a different magnetic field.
As a background for understanding Coriolis fields caused by moving masses, Table C1, which follows, presents comparisons of the electric field and the gravitational field. Table C2 presents comparisons of the magnetic field and the Coriolis field.
The first row in Table C1 cites the basic laws behind electrostatics and gravity, Coulomb's law and the law of gravity. Coulomb's law states that: "The force that one particle exerts on another particle is directly proportional to the product of their charges and inversely proportional to the square of their separation." The law of gravity is similar, except that the force is due to masses, rather than charges. Formulas for these laws are presented in the second row of Table C1. Row 3 of this table provides a label for Row 4, which presents vector field laws derived from Coulomb's law and the law of gravity. It is noted that a positive electric charge is a source of the electric field vector. A positive mass is a sink for the gravitational field vector.
Row 5 is a label for Row 6, which presents the force that an electric field exerts on an electric charge and the force that a gravitational field exerts on a mass. In row 7, it is noted that charges of like sign repel each other, and that masses of like sign attract each other.
Table C2, below, presents a brief comparison of the magnetic field and the Coriolis field. Row 1 reminds us that magnetic fields are generated by electric charges in motion (relative to the observer who sees the magnetic field). For example, a wire carrying an electric current causes a magnetic field that is due to moving electrons in the wire. Likewise, masses moving relative to the observer generate a Coriolis field as seen by that observer. Row 2 reminds us that the magnetic field exerts forces on charges in motion relative to the observer. A magnetic field exerts no force on a charge at rest relative to the observer. Likewise, the Coriolis field exerts forces on masses in motion relative to the observer. A Coriolis field exerts no force on a mass at rest relative to the observer.
Rows 3 and 4 remind us that the magnetic force (on a moving charge) is perpendicular to the magnetic field, and perpendicular to the velocity of the charge. Also, the Coriolis force (on a moving mass) is perpendicular to the Coriolis field and perpendicular to the velocity of the mass.
Row 5 points out that the magnetic force between parallel charge flows (electric currents) is attractive, and that the Coriolis force between parallel mass flows is repulsive.
In Row 6, the origins of the magnetic force and Coriolis force are briefly explained. For the magnetic force on a moving test charge, we consider an observer moving with the charge. That observer sees no magnetic force on the charge, but may see an electric force on the charge. The electric force arises because of an electric field (seen by that observer) which is due to unbalanced electric charges in the current conductors that generate the magnetic field. The charge density in those conductors, as seen by the test charge, can be calculated by applying the Lorentz contraction, as seen by the observer moving with the test charge, to the charges in the conductors.
Row 6 also briefly explains the Coriolis force on a moving test mass due to a Coriolis field due to streams of masses moving relative to the test mass. Again, the Lorentz contraction of the streams of masses is relevant and also the relativistic increase of mass with velocity of the streams of masses, as seen by an observer moving with the test mass. More later.
Relativistic Coriolis Fields
Comparison of Electric and Gravitational Fields
Electric Field Gravitational Field
1
2
3
4
5
6
7
8
9
Comparison of Magnetic and Coriolis Fields
Magnetic Field Coriolis Field
1
Magnetic field is generated by charges in motion.
Coriolis field is generated by masses in motion.
2
Magnetic field exerts forces on charges in motion.
Coriolis field exerts forces on masses in motion.
3
Magnetic force is perpendicular to magnetic field.
Coriolis force is perpendicular to Coriolis field.
4
Magnetic force on a moving electric charge is
perpendicular to velocity of the moving charge. Coriolis force on a moving mass is pepencidular
to velocity of the moving mass.
5
Magnetic force between parallel charge flows is
attractive.
Coriolis force between parallel mass flows is repulsive.
6
To calculate the magnetic force on a moving charge, Q,
consider an observer moving with Q. Apply the Lorentz
contraction to streams of charges moving relatively to Q.
Obtain force on Q from the electric field seen by the
observer moving with Q.
To calculate the Coriolis force on a moving mass, M,
consider an observer moving with M. Apply the Lorentz
contraction and also the relativistic mass increase to
streams of masses moving relatively to M. Obtain force
on M from the gravitational field seen by the observer
moving with M.
7
Magnetic fields do not cause curvature of space.
Coriolis fields cause space to become curved.
Calculate Magnetic Field Due to Moving Charges
as Guide for Calculating Coriolis Field Due to Moving Masses
In accordance with Row 6 of the preceeding table, we plan to calculate the relativistic Coriolis field between a pair of parallel massy plates moving in opposite directions. As background for that calculation, we first calculate the magnetic field between a pair of parallel plates having an equal positive surface charge density; the plates moving in opposite directions at the same speed. This particular thought experiment is selected because it does not involve negative charges, since for the Coriolis case, we do not want to require negative masses.
Figure C1 (below) illustrates our thought experiment for generating a magnetic field. Two positively charged plates are considered that are planar and parallel to each other with an experimental space therebetween. The plates have equal and uniform surface charges, and move in opposite directions at equal speeds, U, as seen by an observer, whom we shall refer to as Observer 1. The plates are assumed to extend large distances to the left and right, and in the direction perpendicular to the plane of the figure, so that edge effects need not be considered.
The top plate moves toward the right, and the bottom plate moves toward the left, relative to Observer 1, at speed U. A positively charged test particle, P moves toward the right at velocity V, as seen by Observer 1. Indicia deliniate equal lengths, L along the top and bottom plates, and the charge between indicia is denoted Q.
To simplify the math, we consider a limiting case in which U and V are both much less than the speed of light, and the discussion is simplified by assuming that V is less than 2U. (Although we are looking for relativistic effects, velocities approaching the velocity of light are not required. We will be invoking the Lorentz contraction which occurs for any nonzero velocity.) Because of symmetry, and other considerations, Observer 1 sees no electric field in the experimental space between the plates due to the charges on the plates, since fields due to the charged plates cancel in this space.
Figure C2 illustrates the system from the viewpoint of a second observer, denoted Observer 2. Observer 2 moves toward the right, relative to Observer 1, at velocity V, which is also the velocity of the test particle, P. When Observer 2 views the bottom plate, he sees that plate to be moving toward the left at an increased velocity, U + V . Because of the increased speed of the bottom plate, as seen by Observer 2, that observer sees the indicia on the bottom plate to be more closely spaced due to the Lorentz contraction, than was seen by Observer 1. The distance between indicia on the bottom plate, as seen by Observer 2, is denoted LB.
In like manner, Observer 2 sees the indicia on the top plate to be spaced farther apart due to a reversed Lorentz contraction caused by the reduced speed of the top plate, as seen by Observer 2. Figure C2 reminds us of these changes by the spacing of the indicia on the top and bottom plates. The distance between indicia on the top plate, as seen by Observer 2, is is denoted LT, and the charge between indicia on each plate is denoted Q. (The charge between indicia does not depend on the speed of the observer.)
Observer 1 sees no electric force on particle P because that observer sees zero electric field in the space between the top and bottom plates. Observer 2 sees zero magnetic force on particle P because the velocity of particle P, as seen by Observer 2, is zero. That observer, however, sees an electric field, E2 in the space between the plates because the bottom plate has more charge per unit area than the top plate, as seen by that observer.
The increased charge per unit area of the bottom plate is due to the reduced distance, LB between indicia on the bottom plate, as compared to the increased distance, LT between indicia on the top plate. The electric field E2 exerts an upward force on particle P, which we have assumed to have a positive charge. That force is interpreted by Observer 1 to be a magnetic force because it is proportional to and perpendicular to the velocity V of particle P. We obtain the magnetic field in the experimental space between the plates by calculating E2 and dividing by V.
To obtain E2, we require the difference in surface charge density of the two plates, as seen by Observer 2. To simplify the calculation, we utilize the requirement that U and V are both much less than the velocity of light, c. This is the same assumption which is used when we approximate the kinetic energy of a massy body as being equal to its mass divided by 2 and multiplied by the square of its velocity. We ignore terms in (U/c)4,(V/c)4, U2V2/c4, as well as higher terms. We also ignore corrections of the order of U2/c2 in comparison with unity, which would be applied equally to both the top and bottom plates.
In the following, we calculate LT and LB from the Lorentz contraction. To further simplify the calculations, we define L0 to be the distance between indicia on either plate as seen by an observer travelling at the same velocity as the plate. Employing L0 rather than L causes equal errors in both the top and bottom plates, and the errors are of the order of U2/c2 in comparison with unity. We obtain for the surface charge density on the bottom plate:
Q/LB = (Q/L0)[1 - (U2 + 2UV + V2)/c2] -1/2
Equation C4
To expand the quantity in brackets, we employ the binomial theorem. In general:
For the case at hand, a = 1 and N = -1/2, so the binomial theorem becomes:
Applying the binomial theorem to equation C4, we note that for our case, b = - (U2 + 2UV + V2)/c2 . Also, we only require the first two terms in the binomial expansion because we are neglecting (U/c)4,(V/c)4, U2V2/c4 and higher terms, in comparison to unity. Thus, we obtain for the charge density on the bottom plate:
For the charge density on the top plate, we obtain:
Q/LT = (Q/L0)[1 - (U2 - 2UV + V2)/ c2] -1/2
Equation C6
Again, we expand the quantity in square brackets by the binomial theorem, and only require the first two terms. We obtain:
The electric field in the space between the bottom and top plates is not affected by identical terms in the expressions for the charge densities on the plates. It only depends on terms which are different. From Equation C5, we see an excess positive charge on the bottom plate of
From Equation C7, we see a reduced positive charge on the top plate as follows.
From Gauss's law, we obtain the electric field between the plates due to the excess positive surface charge on the bottom plate and the reduced positive surface charge on the top plate. This is the electric field seen by Observer 2. Denoting the permittivity of free space as ε0 We get:
To obtain the magnetic field B, we use the vector equation:
In Equation C9, F is the force (a vector quantity) exerted on a charge q moving at velocity v (a vector quantity) relative to the observer and B is the magnetic field vector seen by the observer. For our case, the observer is our Observer 1. Hence, to obtain the magnitude B of the magnetic field vector B, we divide E2 in Equation C8 by V. Thus, we obtain for the magnitude of the magnetic field seen by Observer 1 the following:
Using our assumption that U is much less than C, we can substitute L for L0, thus:
We now note that (Q/L) U is an electric current because movement of the charged plates carries electric charge. Its units are Amperes per meter of distance perpendicular to the plane of figures C1 and C1. It is denoted I. We thus get:
We note that the rationalized magnetic constant μ0 = 1 / (ε0c2), so from Equation C10, we obtain:
We employ the vector relation, Equation, C9, to obtain the direction of the magnetic field B. B is normal to the plane of figures C1 and C2. It is directed from the reader into the figures. The magnitude and direction of B are uniform throughout the experimental space between the plates. Equation C12 is in accordance with the basic equations of electromagnetism.
Calculate the Energy in Magnetic Field B
We now seek the energy associated with the magnetic field B caused by movement of the charged plates. To simplify the arithmetic, we assume that the plates are separated by a unit distance (1 meter) and we assume that the charged plates are first at rest, and then accelerate uniformly for one second to reach a final speed UF , which corresponds to a current IF . We consider the energy in a cubic meter of space - one meter left to right, one meter between the plates, and one meter perpendicular to the figure. We employ the Faraday law of induction to calculate a back emf which opposes acceleration of the plates. At each instant during the second, Equation C12 applies. At the end of the one second acceleration, the magnetic field between the plates has a final value, BF , which is determined by substituting IF for I in Equation C12.
The Faraday law of induction states that the emf around a closed circuit equals the negative of the time rate of change of the magnetic flux passing through an area bounded by the circuit. In its differential form, the Faraday law of induction is
Equation C13 is also known as one of Maxwell's equations.
Applying the Faraday law of induction, we see that the back emf opposing the current I is constant during the one second acceleration of the plates, and it is numerically equal to BF. Hence, the power at each instant needed to drive the currents is IBF. Note that we are not calculating the kinetic energy of the plates. We are calculating the energy needed to overcome the back emf that opposes the current I. To calculate the energy required during the second, we integrate on time. We are employing the phrase (integral of) to replace the integral sign to avoid printing problems. We get:
Since BF is constant, we get:
Since I increases linearly with time from zero to IF during the second, the average current I during the second is IF / 2. From Equation C12, we obtain:
We could have accelerated the plates to produce any value whatever for BF, so:
Equation C16 is well known in electromagnetic texts. It is used, for example, to calculate the energy stored in an inductor.
Calculate the Relativistic Coriolis Field Due to Moving Masses
In a manner similar to the derivation of the magnetic field due to moving charges, we plan to calculate the relativistic Coriolis field between a pair of parallel massy plates moving in opposite directions.
Figure C3 (below) illustrates our thought experiment for generating a Coriolis field. Two plane, parallel plates are considered that have equal and uniform surface mass densities. Our experimental space is defined therebetween. The plates move in opposite directions at equal speeds, U, as seen by an observer, whom we shall refer to as Observer 1. The plates are assumed to extend large distances to the left and right, and in the direction perpendicular to the plane of the figure, so that edge effects need not be considered.
The top plate moves toward the right, and the bottom plate moves toward the left, relative to Observer 1, at speed U. A test particle, P having a positive mass, moves toward the right at velocity V, as seen by Observer 1. Indicia deliniate equal lengths, L along the top and bottom plates, and the mass between indicia is denoted M.
To simplify the math, we consider a limiting case in which U and V are both much less than the speed of light, and the discussion is simplified by assuming that V is less than 2U. (Although we are looking for relativistic effects, velocities approaching the velocity of light are not required. We will be invoking the Lorentz contraction and the relativistic increase of mass with velocity which occur for any nonzero velocity.) Because of symmetry, and other considerations, Observer 1 sees no gravitational field in the experimental space between the plates due to the masses on the plates, since fields due to the massy plates cancel in this space.
Figure C4 illustrates the system from the viewpoint of a second observer, denoted Observer 2. Observer 2 moves toward the right, relative to Observer 1, at velocity V, which is also the velocity of the test particle, P. When Observer 2 views the bottom plate, he sees that plate to be moving toward the left at an increased velocity, U + V . Because of the increased speed of the bottom plate, as seen by Observer 2, that observer sees the indicia on the bottom plate to be more closely spaced due to the Lorentz contraction, than was seen by Observer 1. The distance between indicia on the bottom plate, as seen by Observer 2, is denoted LB. Observer 2 also sees the masss between indicia on the bottom plate to also be increased, due to the relativistic increase of mass with velocity. The mass between indicia on the bottom plate, as seen by Observer 2 is denoted MB
In like manner, Observer 2 sees the indicia on the top plate to be spaced farther apart due to a reversed Lorentz contraction caused by the reduced speed of the top plate, as seen by Observer 2. The mass between indicia on the top plate is also reduced because the velocity of the top plate is reduced, relative to Observer 2. Figure C4 reminds us of these changes by the spacing of the indicia on the top and bottom plates, and the heaviness with which the plates are drawn. The heaviness with which the plates are drawn is indicative of the mass between indicia. The distance between indicia on the top plate, as seen by Observer 2, is is denoted LT, and the mass between indicia on the top plate is denoted MT.
Observer 1 sees no gravitational force on particle P because that observer sees zero gravitational field in the space between the top and bottom plates. Observer 2 sees zero Coriolis force on particle P because the velocity of particle P, as seen by Observer 2, is zero. That observer, however, sees a gravitational field, g2 in the space between the plates because the bottom plate has more mass per unit area than the top plate, as seen by that observer.
The increased mass per unit area of the bottom plate is due in part to the reduced distance, LB between indicia on the bottom plate, as compared to the increased distance, LT between indicia on the top plate. It is also due in part to the increased mass between indicia on the bottom plate, MB as compared to the reduced mass, MT on the top plate. The gravitational field g2 exerts a downward force on particle P, which we have assumed to have a positive mass. That force is interpreted by Observer 1 to be a Coriolis force because it is proportional to and perpendicular to the velocity V of particle P. We obtain the Coriolis field in the experimental space between the plates by calculating g2 and dividing by (2V).
To obtain g2, we require the difference in surface mass density of the two plates, as seen by Observer 2. To simplify the calculation, we utilize the requirement that U and V are both much less than the velocity of light, c. This is the same assumption which is used when we approximate the kinetic energy of a massy body as being equal to its mass divided by 2 and multiplied by the square of its velocity. We ignore terms in (U/c)4,(V/c)4, U2V2/c4, as well as higher terms. We also ignore corrections of the order of U2/c2 in comparison with unity, which would be applied equally to both the top and bottom plates.
In the following, we calculate LT and LB from the Lorentz contraction, and we calculate MT and MB from the relativistic increase of mass with velocity . To further simplify the calculations, we define L0 to be the distance between indicia on either plate as seen by an observer travelling at the same velocity as the plate. Employing L0 rather than L causes equal errors in both the top and bottom plates, and the errors are of the order of U2/c2 in comparison with unity. For a similar simplification, we define M0 to be the mass between indicia on either plate as seen by an observer travelling at the same velocity as the plate. Employing M0 rather than M causes equal errors in both the top and bottom plates, and the errors are of the order of U2/c2 in comparison with unity.
We obtain for the mass between indicia for the bottom plate, using the formula for mass versus velocity:
For the distance between indicia on the bottom plate, we obtain, applying the Lorentz contraction:
The ratio of these yields the mass per unit area on the bottom plate:
To expand the quantity in brackets, we employ the binomial theorem. In general:
For the case at hand, a = 1 and N = -1, so the binomial theorem becomes:
Applying the binomial theorem to equation C19, we note that for this case, b = - (U+V)2/c2. As for the magnetic case discussed above, we only require the first two terms because we are neglecting (U/c)4,(V/c)4, U2V2/c4 and higher terms, in comparison to unity. Thus, we obtain for the mass density on the bottom plate:
For the mass per unit area on the top plate, we again employ the mass between indicia, MT divided by the length between indicia, LT. Hence:
LT = L0[1 - (U-V)2/c2] 1/2
Hence, the mass per unit area on the top plate is:
Again, we expand the quantity in square brackets by the binomial theorem. For the case at hand, a = 1 and N = -1, so the binomial theorem becomes:
Applying the binomial theorem to equation C21, we note that for this case, b = - (U-V)2/c2.
As for the bottom plate discussed above, we only require the first two terms because we are neglecting (U/c)4,(V/c)4, U2V2/c4 and higher terms, in comparison to unity. Thus, we obtain for the mass density on the uppper plate:
The gravitational field in the space between the bottom and top plates is not affected by identical terms in the expressions for the mass densities on the plates. It only depends on terms which are different. From Equations C20 and C22, we see an excess positive mass density on the bottom plate of
The excess mass density on the bottom plate is matched by an equal mass decrement on the top plate.
We seek the gravitational field g2 due to the excess mass density on the bottom plate. We employ the law of gravity in vector form, which was presented above in Equation C1. That equation is repeated here:
By applying the divergence theorem and Equation C1 to the excess positive mass density on the bottom plate, we obtain for g2 the following value.
To obtain the Coriolis field, we employ the following equation, which expresses the force generated by a Coriolis field Ω operating on a mass m travelling at velocity v relative to the observer.
Hence, to obtain the magnitude Ω of the Coriolis field vector Ω, we divide g2 by (2V).
We obtain for the Coriolis field Ω between plates having a surface mass density of M0/L0 moving at velocity U in opposite directions:
Since we have assumed that U is much less than c, we get:
We now note that M/L U is a planar mass current, which we denote J. It expresses the mass for each meter perpendicular to the plane of Figure C3 or C4 that is carried past a fixed point per second, as seen by Observer 1. Its magnitude is the same for both the top and bottom plates. Hence, we obtain the following equation for the Coriolis field generated in the space between the plates by planar mass currents of magnitude J in the top and bottom plates.
To obtain the direction of the coriolis field vector Ω, we employ the vector equation C25. We thus find it to be orthogonal to the plane of Figures C3 and C4. It is directed from the figures toward the reader. Note that its direction is opposite to that of the magnetic field B generated by the moving charged plates in the electromagnetic case, just as the gravitational field g2 shown in Figure C4 is in a direction opposite to that of the electric field E2 shown in Figure C2.
Calculate the Energy in Coriolis Field Ω
We now seek the energy associated with the Coriolis field Ω caused by the planar mass currents, J. As we did for the magnetic case, to simplify the arithmetic, we assume that the plates are separated by a unit distance (1 meter) and we assume that the plates are first at rest, and then accelerate uniformly for one second to reach a final speed, UF. The final speed, UF, corresponds to a planar mass current JF . We consider the energy in a cubic meter of space - one meter left to right, one meter between the plates, and one meter perpendicular to the figure. We employ the gravitational equivalent of the Faraday law of induction to calculate a gravitational field which opposes (or aids) acceleration of the plates. At each instant during the second, Equation C27 applies. At the end of the one second acceleration, the Coriolis field between the plates has a final value, ΩF , which is determined by substituting JF for J in Equation C27.
The gravitational equivalent of the Faraday law of induction is derived in appendix C1. See Appendix C1 The formula, Equation C1-1 is repeated here:
We find it convenient to define a quantity, the gmf, that is defined around any closed path. The gmf around a closed path is defined to be the integral of g dot multiplied with path length along the closed path. It is anologous to the emf of electromagnetism. We are viewing from the perspective of Observer 1, Figure C3. As we did earlier for the magnetic case, we consider a closed path surrounding the cubic meter of space mentioned above (not shown). Our path starts at a point on the top plate, goes one meter to the right on the top plate, drops straight down to the bottom plate, goes one meter to the left on the bottom plate, and then rises straight up to the original point on the top plate. Only the first and third portions of the closed path, which lie along the top and bottom plates, make any contribution to the energy demanded (or contributed) by the Coriolis field as it increases during the second.
For our case, we employ the Coriolis field from equation C27. Since we have assumed the mass current J to increase linearly during the second to its maximum value, JF, we have:
Thus, from Equations C28 and C1-1, the gmf around the closed path is:
Since the average value of J during the second is JF/2, the energy due to the gmf acting on J during the second is gmf * JF/2 Thus, the total energy is:
Rewriting Equation C27, we obtain: JF = (c2/4πG)ΩF We now employ this to replace JF with ΩF in C29 and we obtain:
Cancelling reciprocal terms, we obtain:
And for the mass density of the Coriolis field, we obtain:
We could have moved massy plates to obtain any value whatever for ΩF, so the mass density of a Coriolis field Ω is:
The minus sign in Equation C33 indicates that the Coriolis field has a negative mass density. To verify this, we note that the Coriolis field we have derived is directed from the plane of Figure C3 toward the reader. During the one second acceleration of the plates, the Coriolis field has a time derivative which is also a vector quantity, and it is likewise pointing from the plane of the figure toward the reader. By the gravitational equivalent of the Faraday law of induction, this causes a curl of the gravitational field vector that is directed from the observer toward the figure. This field exerts a force on the top plate toward the right and a force on the bottom plate toward the left. Thus, the Coriolis field, as it is being created, delivers energy to the two plates. By applying the conservation of energy, we infer that the Coriolis field has negative energy and hence negative mass.
We now compare the magnitude of the mass density calculated in Equation C32 with the mass density obtained earlier in Equation C3. Our result is smaller by a factor of 2 than the density in Equation C3. Resolving this discrepancy is left as a challenge to the reader. Is this a matter of bad algebra in the analysis above, or a profound point of physics?
One issue to consider is that, when we used Figures C3 and C4 to obtain the Coriolis field, we applied both the Lorentz contraction and also the relativistic mass increase to the moving plates. When we did a similar calculation to derive the magnetic flux density B based on Coulomb's law and the appropriate relativistic transformations, we only used the Lorentz contraction, applying it to the moving charges in the plates. We did not treat the charges on the plates as increasing with velocity, because electric charge does not increase with velocity.
Another potential source for a factor of 2 discrepancy is that the magnetic relation, Equation C9 differs by a factor of 2 from its Coriolis equivalent, Equation C25. Likewise, the differential form of the Faraday law of induction, C13, differs from its Coriolis equivalent, C1-1, by a factor of 2.
Thus, the theory, so far, has a discrepancy of a factor of 2. What is clear, however, is that the Coriolis field has a negative energy density. This can be seen most simply by the fact that the gmf caused by the increasing Coriolis field Ω in the Coriolis case helps the massy plates accelerate, whereas the emf caused by the increasing magnetic field
We note that time rate gradient effects have not been considered in the preceeding calculation. Could such effects be the answer? Click the link below for a discussion of time rate gradient effects.
News Flash: Coriolis Fields are generally non-uniform.
When most of this Coriolis web site was written, time rate gradient effects were not considered. In particular, it was not realized that Coriolis fields are generally non-uniform. Our reasoning (by revelation from the Lord?) is as follows.
We consider the simple case of a Coriolis field that is seen by observers attached to a rotating coordinate system. They could, for example, be attached to a rotating solid structure such as a rotating space station. We assume that the rotating solid structure does not have sufficient mass to cause significant fields due to frame dragging. This requirement is easily met. In spite of the great mass of the Earth, the Coriolis field due to its frame-dragging effect has not yet been measured, even though the Coriolis field seen by observers on the Earth due to rotation of the coordinate frame is sufficient to cause cyclones, hurricanes, jet streams and the deflection of ocean currents.
It has been noted that if a plurality of clocks are attached to a rotating structure, clocks at great distances from the axis of rotation will indicate less passage of time than clocks located near the axis. The general relativity explanation is that clocks at great distances from the axis are at a lower gravitational potential than clocks near the axis. (This has been measured in the Harvard Tower experiment, and is discussed in the web site TimeRateGradient.com that is listed below.)
We consider observers attached to the rotating structure at various radial distances from the axis of the structure. Each observer has a clock, and each has a view in a radially outward direction. The observers see stars to be orbiting about the structure, in the same manner that stars seen by observers on the Earth appear to be orbiting about the Earth. Observers near the axis will see stars to orbit with the same period as the structure. However, observers at great radial distances will see the stars to be orbiting at greater rates. That is because the clocks of the observers at great radial distances run slower than clocks of observers near the axis. Any measurement of Coriolis field strength by an observer far from the axis will be increased in the same proportion as the reciprocal of the rate at which his clock runs. Aircraft turn signals, or freely-gimballed gyroscopes attached to the structure at great radial distances will show greater Coriolis fields than such instruments placed near the axis. As we place turn rate instruments farther and farther out, we obtain greater and greater values as we approach a radius at which the instruments would be approaching the speed of light, as seen by an observer on the axis.
We believe that it is a property of the Coriolis field itself that its field strength changes in a direction perpendicular to the field. The vector operator known as the curl, applied to the Coriolis field at a point, yields a nonzero vector. That vector is perpendicular to the radius of the point and perpendicular to the axis of rotation.
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Appendix C1: Derivation of the Gravitational Equivalent of the Faraday Law of Induction
Figure C6 illustrates the arrangement as seen by an observer who undergoes the same angular acceleration as the disk. That observer sees all of space to be filled with a Coriolis field which has a nonzero time derivative, dΩ/dt. Associated with that nonzero time derivative, dΩ/dt, is a gravitational field g which has a nonzero curl. (We refer to the vector operator, the curl, which appears in Maxwell's electromagnetic equations.) Figure C6 does not show a centrifugal acceleration field because that is irrelevant to our quest for the gravitational equivalent of the Faraday law of induction. Of course, at the instant when the angular velocity of the disk is zero, there is no centrifugal acceleration field. Also, if a centrifugal acceleration field is present, it has a divergence but not a curl.
To obtain the gravitational equivalent of the Faraday law of induction, we seek a relationship between the curl of g and dΩ/dt. For a simple (but not rigorous) derivation, we calculate the integral of g multiplied by an element of length around the circumference of the disk, and then divide by the area of the disk. We note that for a point on the rim of the disk, the magnitude of g is RdΩ/dt. Also, the circumference of the disk is 2πR and the area of the disk is πR2. Thus, we obtain:
We have inserted the minus sign into equation C1-1 because by inspecting Figure C6, we see that the direction of g is such that its curl is directed downward in the figure, opposite to the direction of dΩ/dt .
We have employed a very simple integration - around the circumference of the disk, to obtain Equation C1-1. Integration about any other loop would have yielded the same result.
It is of interest to note that the balance organs in our inner ears operate by the gravitational equivalent of the Faraday law of induction. In each ear, we have three fluid passages lying in substantially orthogonal planes. They are referred to as "semicircular canals", even though each canal forms a complete loop. When you begin to turn your head, or otherwise change the angular velocity of your head, the law behind Equation C1-1 causes a gravitational field, relative to your head, having nonzero curl. This causes fluid motion in the semicircular canals, and this motion is sensed by hair cells in the canals. Three canals are needed because Ω is a vector having three components. We stand in awe of our Creator.
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