Index
PART I THE CLASSICAL BASE
1.0 Gravitational and Centripetal Accelerations
2.0 Orbital Velocity Gradients
3.0 Light Refraction at the Mass Event Horizon
4.0 Space Curvature and the Deflection of Light
5.0 Light Deflection Angle
6.0 Effect of Mass on Light Deflection
7.0 Relation between Gravitational Acceleration and Deflection Angle
PART II THE RELATIVISTIC BASE
8.0 The Energy Aspect
9.0 Total Energy, Kinetic Energy, and Potential Energy
10.0 Gravitational Field Potential
11.0 Space-Time and Gravitational Field Potential
12.0 Effect of Mass on Space-Time
13.0 Relation between Gravity and Space-Time Warp Angle
14.0 The Reach of Gravity
PART III CONCLUDING OBSERVATIONS
15.0 Mass Event Horizon
16.0 Cosmological Implications
PART IV APPENDIX
Solar System Data
PART I THE CLASSICAL BASE
1.0 Gravitational and Centripetal Accelerations
2.0 Orbital Velocity Gradients
3.0 Light Refraction at the Mass Event Horizon
4.0 Space Curvature and the Deflection of Light
5.0 Light Deflection Angle
6.0 Effect of Mass on Light Deflection
7.0 Relation between Gravitational Acceleration and Deflection Angle
PART II THE RELATIVISTIC BASE
8.0 The Energy Aspect
9.0 Total Energy, Kinetic Energy, and Potential Energy
10.0 Gravitational Field Potential
11.0 Space-Time and Gravitational Field Potential
12.0 Effect of Mass on Space-Time
13.0 Relation between Gravity and Space-Time Warp Angle
14.0 The Reach of Gravity
PART III CONCLUDING OBSERVATIONS
15.0 Mass Event Horizon
16.0 Cosmological Implications
PART IV APPENDIX
Solar System Data
PART I THE CLASSICAL BASE
1.0 Gravitational and Centripetal Accelerations
The subject of this discussion is gravity. Our interest is to define, or redefine, what gravity does, not what
gravity is. Therefore, our point of departure must be based on the classical definition of gravity. Note that all
values are expressed in the exponential or scientific format, and their units are in the CGS system (centimeter,
gram, second) unless otherwise specified.
We will use two elementary equations for our discussion. The first one is Newton’s classical inverse square
relation:
g = M G / d^2 (1.1)
where: g is the acceleration due to gravity
M is the mass of the body to which gravity is attributed
G is the universal gravitational constant 6.67E-8 [cm3 g-1sec-2]
d is the distance from the body
The second one is the equation relating the centripetal acceleration of a body rotating at a constant velocity
around a center:
a = V^2 / d (1.2)
where: a is the centripetal acceleration directed toward the center
V is the linear velocity of the rotating body
d is the distance of the body from the center
Two essential aspects of these relations should be considered.
In Newton’s equation, an infinitesimally small mass generates an infinitely large gravity at an infinitely close
distance. This is due to the inverse square relation to distance and to the fact that Newton’s law is based on the
required assumption that mass is concentrated at the center of mass.
In the other equation, the velocity of a rotating mass cannot be greater than the speed of light.
2.0 Orbital Velocity Gradient
We will consider velocities of bodies orbiting the Sun. All data related to the major planets, the Moon, and the
Sun is given in the Appendix.
The velocity of a body orbiting another body of mass M at a distance d is given by equating gravitational force
to centripetal force, or g = a from equations (1.1) and (1.2) (the affected mass is the same in both equations):
V = sqr (M G / d) (2.1)
Velocity V is directly proportional to the square root of mass M and inversely proportional to the square root of
the distance from the center of mass M, and will be equal to the speed of light at a distance Rs defined as the
critical radius past which orbital velocity gradients would be greater than the speed of light. This critical radius
defines the event horizon of mass contracted into a spherical volume of that radius:
Rs = M G / C^2 (2.2)
The mass event horizon Rs is the locus at which the orbital velocity gradient is equal to the
speed of light.
Mass can not be contracted beyond the Rs radius, or mass event horizon: this is a world where the laws of
classical physics no longer apply. (Note that the Schwarzschild radius is twice the Rs radius defined by
equation (2.2).)
Multiplying and dividing equation (2.1) by Rs, and substituting equation (2.2) for Rs in the denominator, we
have:
V^2 = C^2 (Rs / d) = C^2 / (nRs) (2.3)
where (nRs) is the distance from the center of mass M expressed in Rs units.
When using Rs as unit of distance, orbital velocity gradients are independent of mass.
Using equations (1.1) and (2.2), the gravitational acceleration of a mass M contracted to its Rs volume (that is, a
volume of radius Rs) is:
go = C^4 / (M G) = 1.21E49 / M (2.4)
That is, gravitational acceleration at the mass event horizon is inversely proportional to mass
and, therefore, inversely proportional to the Rs radius.
We can therefore stipulate that the orbital velocity gradient equal to the speed of light at a mass
event horizon is independent of gravitational acceleration at that horizon; and that an orbital
velocity gradient equal to the speed of light is a property of the mass event horizon.
Table 2.1 below shows orbital velocities from actual data compared to the computed velocities from equation
(2.1) for the major bodies of the solar system. The gravitational and centripetal accelerations are those induced
by the Sun on the bodies.
Table 2.1
1.0 Gravitational and Centripetal Accelerations
The subject of this discussion is gravity. Our interest is to define, or redefine, what gravity does, not what
gravity is. Therefore, our point of departure must be based on the classical definition of gravity. Note that all
values are expressed in the exponential or scientific format, and their units are in the CGS system (centimeter,
gram, second) unless otherwise specified.
We will use two elementary equations for our discussion. The first one is Newton’s classical inverse square
relation:
g = M G / d^2 (1.1)
where: g is the acceleration due to gravity
M is the mass of the body to which gravity is attributed
G is the universal gravitational constant 6.67E-8 [cm3 g-1sec-2]
d is the distance from the body
The second one is the equation relating the centripetal acceleration of a body rotating at a constant velocity
around a center:
a = V^2 / d (1.2)
where: a is the centripetal acceleration directed toward the center
V is the linear velocity of the rotating body
d is the distance of the body from the center
Two essential aspects of these relations should be considered.
In Newton’s equation, an infinitesimally small mass generates an infinitely large gravity at an infinitely close
distance. This is due to the inverse square relation to distance and to the fact that Newton’s law is based on the
required assumption that mass is concentrated at the center of mass.
In the other equation, the velocity of a rotating mass cannot be greater than the speed of light.
2.0 Orbital Velocity Gradient
We will consider velocities of bodies orbiting the Sun. All data related to the major planets, the Moon, and the
Sun is given in the Appendix.
The velocity of a body orbiting another body of mass M at a distance d is given by equating gravitational force
to centripetal force, or g = a from equations (1.1) and (1.2) (the affected mass is the same in both equations):
V = sqr (M G / d) (2.1)
Velocity V is directly proportional to the square root of mass M and inversely proportional to the square root of
the distance from the center of mass M, and will be equal to the speed of light at a distance Rs defined as the
critical radius past which orbital velocity gradients would be greater than the speed of light. This critical radius
defines the event horizon of mass contracted into a spherical volume of that radius:
Rs = M G / C^2 (2.2)
The mass event horizon Rs is the locus at which the orbital velocity gradient is equal to the
speed of light.
Mass can not be contracted beyond the Rs radius, or mass event horizon: this is a world where the laws of
classical physics no longer apply. (Note that the Schwarzschild radius is twice the Rs radius defined by
equation (2.2).)
Multiplying and dividing equation (2.1) by Rs, and substituting equation (2.2) for Rs in the denominator, we
have:
V^2 = C^2 (Rs / d) = C^2 / (nRs) (2.3)
where (nRs) is the distance from the center of mass M expressed in Rs units.
When using Rs as unit of distance, orbital velocity gradients are independent of mass.
Using equations (1.1) and (2.2), the gravitational acceleration of a mass M contracted to its Rs volume (that is, a
volume of radius Rs) is:
go = C^4 / (M G) = 1.21E49 / M (2.4)
That is, gravitational acceleration at the mass event horizon is inversely proportional to mass
and, therefore, inversely proportional to the Rs radius.
We can therefore stipulate that the orbital velocity gradient equal to the speed of light at a mass
event horizon is independent of gravitational acceleration at that horizon; and that an orbital
velocity gradient equal to the speed of light is a property of the mass event horizon.
Table 2.1 below shows orbital velocities from actual data compared to the computed velocities from equation
(2.1) for the major bodies of the solar system. The gravitational and centripetal accelerations are those induced
by the Sun on the bodies.
Table 2.1
|
If we assume that we can squeeze the mass of the Sun into a volume of smaller and smaller radiuses until we
reach the Rs radius of 1.47 kilometers, then a body orbiting this contracted Sun at a distance equal to its normal
radius of 696000 kilometers will have a velocity of 437 [km/sec]. The same body orbiting at the Rs radius would
have a velocity equal to the speed of light.
Table 2.2 shows gravitational accelerations and orbital velocity gradients for 1) a mass of one gram and 2) a
mass equal to that of the Sun, at distances from 1 to 20,000 Rs units.
Table 2.2
1) Mass = 1 [g] Rs = 7.41E-29 [cm]
2) Mass = 1.99E33 [g] Rs = 1.47E5 [cm]
reach the Rs radius of 1.47 kilometers, then a body orbiting this contracted Sun at a distance equal to its normal
radius of 696000 kilometers will have a velocity of 437 [km/sec]. The same body orbiting at the Rs radius would
have a velocity equal to the speed of light.
Table 2.2 shows gravitational accelerations and orbital velocity gradients for 1) a mass of one gram and 2) a
mass equal to that of the Sun, at distances from 1 to 20,000 Rs units.
Table 2.2
1) Mass = 1 [g] Rs = 7.41E-29 [cm]
2) Mass = 1.99E33 [g] Rs = 1.47E5 [cm]
|
3.0 Light Refraction at the Mass Event Horizon
The refraction of a beam of light going from a medium A to a medium B is given by Snell’s Law:
Cr sin (i) = Ci sin (r)
where: i is the angle of the incident light in medium A
r is the angle of the refracted light in medium B
Ci is the speed of the incident light
Cr is the speed of the refracted light
The angles are measured from the normal to the interface between the two media.
Let us assume that a beam of light with velocity C is directed to the mass event horizon. We then have:
Cr = C sin (r) / sin (i)
At the point of contact its velocity has to be reduced to zero for the beam to be absorbed into the world below
the mass event horizon. And so we have:
0 = C sin (r) / sin (i)
and sin (r) = 0 = sin (90 – a) = cos (a)
where "a" is the angle of deflection between the normal and the interface, or mass event horizon, and is equal to
90 degrees. This is independent of the angle of incidence i.
Therefore we can stipulate that a 90 degree deflection of light is a property of the mass event
horizon, and the deflection is independent of the angle of incidence.
The refraction of a beam of light going from a medium A to a medium B is given by Snell’s Law:
Cr sin (i) = Ci sin (r)
where: i is the angle of the incident light in medium A
r is the angle of the refracted light in medium B
Ci is the speed of the incident light
Cr is the speed of the refracted light
The angles are measured from the normal to the interface between the two media.
Let us assume that a beam of light with velocity C is directed to the mass event horizon. We then have:
Cr = C sin (r) / sin (i)
At the point of contact its velocity has to be reduced to zero for the beam to be absorbed into the world below
the mass event horizon. And so we have:
0 = C sin (r) / sin (i)
and sin (r) = 0 = sin (90 – a) = cos (a)
where "a" is the angle of deflection between the normal and the interface, or mass event horizon, and is equal to
90 degrees. This is independent of the angle of incidence i.
Therefore we can stipulate that a 90 degree deflection of light is a property of the mass event
horizon, and the deflection is independent of the angle of incidence.
4.0 Space Curvature and the Deflection of Light
One of the results of the general theory of relativity that has been proved correct by observation is the deflection
of a beam of light tangent to the Sun due to the curvature of space created by the presence of mass. We should
point out that the theory is concerned with the physical curvature of space generated by gravitational and
electromagnetic fields and not with the curvature or warping of space-time continuum as we will discuss in Part
II.
According to the theory, a beam of light that passes tangentially to the surface of the Sun should have a
deflection in degrees of arc from its direction of incidence given by:
alfa = 0.00047 rSun / d (4.1)
where rSun is the radius of the Sun and d is the distance from its center.
Half of this deflection is due to the Newtonian field of attraction of the Sun and the other half to the curvature of
space. We will then use:
alfa = 0.000236 rSun / d (4.2)
According to equation (4.2), the deflection at the Sun’s mass event horizon would be 111.4 degrees; at the Sun’
s surface 0.00024 degrees; at a distance equal to the Earth distance of 216 Sun’s radiuses 0.000001 degrees;
and at Jupiter 0.0000002 degrees.
5.0 Light Deflection Angle
It is intended to show that the deflection of light due to the curvature of space can be obtained independently
from the general theory of relativity by using only two premises:
1) light is deflected 90 degrees at the mass event horizon
2) equation (1.1): g = M G / d^2
From (1.1) at Rs we have: go = M G / Rs^2
at a distance > Rs: g = M G / d^2
and: (g / go) = (Rs / d)^2
The ratio (Rs / d) can be considered as the tangent of an angle A equal to:
A = Atan (Rs / d)
And at d = Rs:
Ao = Atan (Rs / Rs) = 45 [deg]
To satisfy the first premise, the angle of light deflection at Rs must be twice the angle Ao, and:
alfa = 2 Atan (Rs / d) (5.1)
where d is greater or equal to Rs.
It should be noted that, when d is less than Rs, the angle of deflection increases and reaches the value of 180
degrees at d = 0. We are below the mass event horizon and outside the scope of this discussion. However, the
point must be made that a curvature of space that generates a light deflection of 180 degrees can only be
achieved by a space structure folded onto itself. The spatial dimensions of such a structure are double the
ordinary space dimensions. Therefore, dimensions below the mass event horizon are fractal and vary from
three to six (time dimension excluded) depending on the closeness to the center of mass.
We must show that equation (5.1) is equivalent to equation (4.2):
2 Atan (Rs / d) <----> 0.000236 rSun / d
By equivalent we mean that both equations describe the light deflection phenomenon equally well.
From equation (4.2):
alfa = 0.000236 rSun / d
alfao = 0.000236 rSun / Rs
(alfa / alfao) = (Rs / d)
The ratio (Rs / d) can also be considered as the tangent of angle A giving the same result as in equation (5.1).
However, whereas equation (4.2) is based on the radius of the Sun and describes the specific case of light
deflection around the Sun, equation (5.1) instead is based on the Rs radius which is a property of mass and,
therefore, applies generally to all mass.
Table 5.1 shows the values of the two equations from Rs to 6 Rs, and at the Sun’s radius.
Table 5.1
One of the results of the general theory of relativity that has been proved correct by observation is the deflection
of a beam of light tangent to the Sun due to the curvature of space created by the presence of mass. We should
point out that the theory is concerned with the physical curvature of space generated by gravitational and
electromagnetic fields and not with the curvature or warping of space-time continuum as we will discuss in Part
II.
According to the theory, a beam of light that passes tangentially to the surface of the Sun should have a
deflection in degrees of arc from its direction of incidence given by:
alfa = 0.00047 rSun / d (4.1)
where rSun is the radius of the Sun and d is the distance from its center.
Half of this deflection is due to the Newtonian field of attraction of the Sun and the other half to the curvature of
space. We will then use:
alfa = 0.000236 rSun / d (4.2)
According to equation (4.2), the deflection at the Sun’s mass event horizon would be 111.4 degrees; at the Sun’
s surface 0.00024 degrees; at a distance equal to the Earth distance of 216 Sun’s radiuses 0.000001 degrees;
and at Jupiter 0.0000002 degrees.
5.0 Light Deflection Angle
It is intended to show that the deflection of light due to the curvature of space can be obtained independently
from the general theory of relativity by using only two premises:
1) light is deflected 90 degrees at the mass event horizon
2) equation (1.1): g = M G / d^2
From (1.1) at Rs we have: go = M G / Rs^2
at a distance > Rs: g = M G / d^2
and: (g / go) = (Rs / d)^2
The ratio (Rs / d) can be considered as the tangent of an angle A equal to:
A = Atan (Rs / d)
And at d = Rs:
Ao = Atan (Rs / Rs) = 45 [deg]
To satisfy the first premise, the angle of light deflection at Rs must be twice the angle Ao, and:
alfa = 2 Atan (Rs / d) (5.1)
where d is greater or equal to Rs.
It should be noted that, when d is less than Rs, the angle of deflection increases and reaches the value of 180
degrees at d = 0. We are below the mass event horizon and outside the scope of this discussion. However, the
point must be made that a curvature of space that generates a light deflection of 180 degrees can only be
achieved by a space structure folded onto itself. The spatial dimensions of such a structure are double the
ordinary space dimensions. Therefore, dimensions below the mass event horizon are fractal and vary from
three to six (time dimension excluded) depending on the closeness to the center of mass.
We must show that equation (5.1) is equivalent to equation (4.2):
2 Atan (Rs / d) <----> 0.000236 rSun / d
By equivalent we mean that both equations describe the light deflection phenomenon equally well.
From equation (4.2):
alfa = 0.000236 rSun / d
alfao = 0.000236 rSun / Rs
(alfa / alfao) = (Rs / d)
The ratio (Rs / d) can also be considered as the tangent of angle A giving the same result as in equation (5.1).
However, whereas equation (4.2) is based on the radius of the Sun and describes the specific case of light
deflection around the Sun, equation (5.1) instead is based on the Rs radius which is a property of mass and,
therefore, applies generally to all mass.
Table 5.1 shows the values of the two equations from Rs to 6 Rs, and at the Sun’s radius.
Table 5.1
|
Defining variance as the sum of the squared differences of the values of equations (4.2) and (5.1) taken at n
points divided by the number of points, and standard deviation as the square root of the variance, we have:
Rs to ½ Sun radius: sd = 0.0444 [deg] n = 236000
½ Sun radius to Sun radius: sd = 9.6E-06 [deg] n = 236000
Sun radius to Earth: sd = 4.6E-07 [deg] n = 430000
Earth to Jupiter: sd = 1.4E-08 [deg] n = 452000
6.0 Effect of Mass on Light Deflection
Based on the two properties of the mass event horizon previously discussed, we can use equation (5.1):
alfa = 2 Atan (Rs / d)
to calculate the deflection of a beam of light tangent to a body of mass M due to the curvature of space. All we
need is the Rs of mass M, given by equation (2.2), and the radius of the body.
Table 6.1 shows the deflection angles in degrees of a beam of light tangent to the major bodies of the solar
system due to the curvature of space created by the bodies. This deflection excludes the effect of the Newtonian
field attraction.
Table 6.1
points divided by the number of points, and standard deviation as the square root of the variance, we have:
Rs to ½ Sun radius: sd = 0.0444 [deg] n = 236000
½ Sun radius to Sun radius: sd = 9.6E-06 [deg] n = 236000
Sun radius to Earth: sd = 4.6E-07 [deg] n = 430000
Earth to Jupiter: sd = 1.4E-08 [deg] n = 452000
6.0 Effect of Mass on Light Deflection
Based on the two properties of the mass event horizon previously discussed, we can use equation (5.1):
alfa = 2 Atan (Rs / d)
to calculate the deflection of a beam of light tangent to a body of mass M due to the curvature of space. All we
need is the Rs of mass M, given by equation (2.2), and the radius of the body.
Table 6.1 shows the deflection angles in degrees of a beam of light tangent to the major bodies of the solar
system due to the curvature of space created by the bodies. This deflection excludes the effect of the Newtonian
field attraction.
Table 6.1
|
A light beam tangent to Jupiter will be deflected 0.0081 seconds of arc as compared to 0.87 seconds of arc for
the Sun. If we include the Newtonian attraction the deflections would be 0.016 seconds of arc for Jupiter, and
1.74 seconds of arc for the Sun.
From equation (5.1), solving for distance d:
d = Rs / tan (alfa / 2) (6.1)
tan (alfa / 2) = 1 / (nRs) (6.2)
where (nRs) is the distance from the center of mass M expressed in Rs units.
For any deflection angle, the distance at which the deflection occurs is directly proportional to
the mass of the body creating the deflection.
Deflection angles have the same values at the same distance (expressed in Rs units) from the
center of mass.
When using Rs as unit of distance, orbital velocity gradients (Paragraph 2.0) and light deflection
angles are independent of mass.
An average galaxy of a trillion Suns will curve space and deflect light by 0.000243 degrees (the deflection
occurring at the Sun’s surface) at a distance from its center of about 7E17 kilometers, or 74000 light years. The
same galaxy will curve space and deflect light by 0.000001 degrees (the deflection occurring at a distance from
the Sun equal to that of the Earth) at a distance from its center of about 1.7E20 kilometers, or 18,000,000 light
years. In contrast, if the mass of such a galaxy were to be contracted to its Rs radius, the resulting event would
have a diameter of 3 trillion kilometers, or about 0.3 light years, and would deflect light by 90 degrees at its
horizon.
7.0 Relation between Gravitational Acceleration and Deflection Angle
Using equations (1.1) and (2.2), and substituting for distance d from equation (6.1), we have:
g = (C^2 / Rs) [tan (alfa / 2)]^2 (7.1)
g = [C^4 / (M G)] [tan (alfa / 2)]^2 (7.2)
For any given deflection angle, gravitational acceleration is inversely proportional to mass.
For any given mass, gravitational acceleration is directly proportional to the square of the
tangent of the semi-angle of deflection.
From equations (1.1) and (5.1) we also have:
(g / go) = [tan (alfa / 2)]^2 (7.3)
where go is the gravitational acceleration of mass M at its event horizon, given by equation (2.4):
go = C^4 / (M G) = 1.21E49 / M
PART II THE RELATIVISTIC BASE
8.0 The Energy Aspect
All mass moves. When discussing energy of a moving body we must address the subject from a relativistic
point of view.
If dt is the change in the time interval for a moving body; dx, dy, dz the change in the space interval during time
dt; dS the length of the change in space-time interval during time dt; and C the speed of light; then, from
Minkowski, the relation between time, space, and space-time is:
dS^2 = C^2 dt^2 – (dx^2 + dy^2 + dz^2)
= C^2 dt^2 – ds^2
= dt^2 (C^2 – dV^2)
where ds is the length of the change in the space interval, which is equal to the change in the time interval dt
times the change in velocity dV. If velocity V is constant, then:
S^2 = t^2 (C^2 – V^2)
Dividing by C^2:
S^2 / C^2 = t^2 [1 – (V / C)^2]
but S^2 / C^2 is equal to the square of the time T of space-time:
T^2 = t^2 [1 – (V / C)^2]
T = t sqr [1 – (V / C)^2]
The above can be represented geometrically by a right triangle, which we will call the space-time triangle, of the
following form:
Hypotenuse: C t
Vertical side: V t = C t sin (theta)
Horizontal side: S = C t cos (theta) = C T = C t R
Angle: theta
Where:
R = sqr [1 – (V / C)^2] = cos (theta) (8.1)
For a body of mass m orbiting a body of mass M, the velocity terms in equation (8.1) can be substituted as
follows:
Using equation (2.1) V^2 = M G / d:
R = sqr [1 – M G / (C^2 d)] (8.2)
Using equation (2.3) V^2 = C^2 (Rs / d):
R = sqr (1 – Rs / d) (8.3)
From equations (8.1), (8.2), and (8.3), we have:
(V / C)^2 = M G / (C^2 d) = (Rs / d) (8.4)
The total energy (E) of a body in motion is equal to the sum of its energy at rest (Eo) plus kinetic energy (Ke) and
potential energy (Kp).
E = Eo + Ke + Kp (8.5)
The energy at rest is given by
Eo = mo C^2 (8.6)
The kinetic energy Ke is the difference in energy levels between the body in motion (mass m) and the body at
rest (mass mo), or:
Ke = m C^2 – mo C^2 (8.7)
Using R above and noting that from relativity m = mo / R:
Ke = mo C^2 (1 / R – 1) = Eo (1 / R – 1) (8.8)
The potential energy is given by:
Kp = – (mo / R) M G / d (8.9)
where d is the distance of the orbiting body to the center of the body of mass M (note the minus sign).
Substituting into equation (8.5) and using equation (8.3) for R we have:
E = (mo / R) C^2 (1 – Rs / d)
= mo C^2 R = Eo R
or the equivalent:
E = m (C^2 – V^2)
E = m (C^2 – M G / d)
For instance: Jupiter has a mass of 1.9E27 kilograms and orbits the Sun with a velocity of 13.1 kilometer/second
at a distance of about 7.8E8 kilometers. Its kinetic energy from equations (8.1) and (8.8) is about 1.6E42 ergs.
(Because of Jupiter’s slow velocity compared to that of light, the same result is obtained from the classical
kinetic energy equation Ke = (1/2) m V^2) The potential energy from equation (8.9) is about 3.2E42 ergs; and the
total energy is about 1.7E51 ergs, very slightly lower than its energy at rest. This is due to the negative potential
energy that is about twice the value of the positive kinetic energy.
9.0 Total Energy, Kinetic Energy, and Potential Energy
We will start with noting the relation between kinetic and potential energies. From equations (8.8) and (8.9),
using equation (2.1) V^2 = M G / d and equation (8.1):
[Kp] / Ke = (1 – R^2) / (1 – R) (9.1)
where [Kp] is the absolute value of the potential energy.
At the mass event horizon (R = 0) the kinetic and the potential energies are equal in value but of opposite signs.
At exactly R = 1 (at infinite distance from the center of mass M) the ratio [Kp]/Ke is undefined (a singularity); but
at any other values of R the rate of change d([Kp] / Ke) / dR is equal to one, and
[Kp] / Ke = 1 + R (9.2)
Since R is going from zero to one depending on the distance from the center of mass, the ratio [Kp]/Ke goes
asymptotically from 1 to 2 with the potential energy of opposite sign to the kinetic energy.
Substituting equation (9.2) into equation (8.5) we have:
E = mo C^2 R = Eo R (9.3)
At the mass event horizon (R = 0) the total energy is equal to zero. At R = 1 (at infinite distance from the center of
mass M) the total energy is equal to the rest energy.
The total energy of a body of mass m rotating around a body of mass M can never be greater
than its rest energy as long as the speed of light is the limiting velocity.
This statement can be generalized by saying that matter cannot acquire more energy than its rest
energy regardless of conditions to which it is subjected.
We can also say that the mass event horizon is the locus of total energy unavailability and,
therefore, it is the locus of maximum entropy.
It is also of interest to note the relations between potential and kinetic energies and the energy of mass at rest
Eo.
For Ke = Eo we use equations (8.6) and (8.7) and solve for R, giving:
R = .5
Rs / d = .75
In terms of Rs units:
(nRs) = d / Rs = 1 / .75 = 1.333….
For Kp = Eo we use equations (8.6) and (8.9):
mo M G / (R d) = mo C^2
Using equation (2.2) for Rs:
R = Rs / d = sqr [1 – (Rs / d)]
Squaring and solving the resulting quadratic equation, we have:
Rs / d = [ – 1 + sqr (5)] / 2 = .618033988
In terms of Rs units:
(nRs) = d / Rs = 1.618033988
We will next note the relations among all energies, displayed numerically in Table 9.1 and mathematically in
Table 9.2.
Table 9.1
the Sun. If we include the Newtonian attraction the deflections would be 0.016 seconds of arc for Jupiter, and
1.74 seconds of arc for the Sun.
From equation (5.1), solving for distance d:
d = Rs / tan (alfa / 2) (6.1)
tan (alfa / 2) = 1 / (nRs) (6.2)
where (nRs) is the distance from the center of mass M expressed in Rs units.
For any deflection angle, the distance at which the deflection occurs is directly proportional to
the mass of the body creating the deflection.
Deflection angles have the same values at the same distance (expressed in Rs units) from the
center of mass.
When using Rs as unit of distance, orbital velocity gradients (Paragraph 2.0) and light deflection
angles are independent of mass.
An average galaxy of a trillion Suns will curve space and deflect light by 0.000243 degrees (the deflection
occurring at the Sun’s surface) at a distance from its center of about 7E17 kilometers, or 74000 light years. The
same galaxy will curve space and deflect light by 0.000001 degrees (the deflection occurring at a distance from
the Sun equal to that of the Earth) at a distance from its center of about 1.7E20 kilometers, or 18,000,000 light
years. In contrast, if the mass of such a galaxy were to be contracted to its Rs radius, the resulting event would
have a diameter of 3 trillion kilometers, or about 0.3 light years, and would deflect light by 90 degrees at its
horizon.
7.0 Relation between Gravitational Acceleration and Deflection Angle
Using equations (1.1) and (2.2), and substituting for distance d from equation (6.1), we have:
g = (C^2 / Rs) [tan (alfa / 2)]^2 (7.1)
g = [C^4 / (M G)] [tan (alfa / 2)]^2 (7.2)
For any given deflection angle, gravitational acceleration is inversely proportional to mass.
For any given mass, gravitational acceleration is directly proportional to the square of the
tangent of the semi-angle of deflection.
From equations (1.1) and (5.1) we also have:
(g / go) = [tan (alfa / 2)]^2 (7.3)
where go is the gravitational acceleration of mass M at its event horizon, given by equation (2.4):
go = C^4 / (M G) = 1.21E49 / M
PART II THE RELATIVISTIC BASE
8.0 The Energy Aspect
All mass moves. When discussing energy of a moving body we must address the subject from a relativistic
point of view.
If dt is the change in the time interval for a moving body; dx, dy, dz the change in the space interval during time
dt; dS the length of the change in space-time interval during time dt; and C the speed of light; then, from
Minkowski, the relation between time, space, and space-time is:
dS^2 = C^2 dt^2 – (dx^2 + dy^2 + dz^2)
= C^2 dt^2 – ds^2
= dt^2 (C^2 – dV^2)
where ds is the length of the change in the space interval, which is equal to the change in the time interval dt
times the change in velocity dV. If velocity V is constant, then:
S^2 = t^2 (C^2 – V^2)
Dividing by C^2:
S^2 / C^2 = t^2 [1 – (V / C)^2]
but S^2 / C^2 is equal to the square of the time T of space-time:
T^2 = t^2 [1 – (V / C)^2]
T = t sqr [1 – (V / C)^2]
The above can be represented geometrically by a right triangle, which we will call the space-time triangle, of the
following form:
Hypotenuse: C t
Vertical side: V t = C t sin (theta)
Horizontal side: S = C t cos (theta) = C T = C t R
Angle: theta
Where:
R = sqr [1 – (V / C)^2] = cos (theta) (8.1)
For a body of mass m orbiting a body of mass M, the velocity terms in equation (8.1) can be substituted as
follows:
Using equation (2.1) V^2 = M G / d:
R = sqr [1 – M G / (C^2 d)] (8.2)
Using equation (2.3) V^2 = C^2 (Rs / d):
R = sqr (1 – Rs / d) (8.3)
From equations (8.1), (8.2), and (8.3), we have:
(V / C)^2 = M G / (C^2 d) = (Rs / d) (8.4)
The total energy (E) of a body in motion is equal to the sum of its energy at rest (Eo) plus kinetic energy (Ke) and
potential energy (Kp).
E = Eo + Ke + Kp (8.5)
The energy at rest is given by
Eo = mo C^2 (8.6)
The kinetic energy Ke is the difference in energy levels between the body in motion (mass m) and the body at
rest (mass mo), or:
Ke = m C^2 – mo C^2 (8.7)
Using R above and noting that from relativity m = mo / R:
Ke = mo C^2 (1 / R – 1) = Eo (1 / R – 1) (8.8)
The potential energy is given by:
Kp = – (mo / R) M G / d (8.9)
where d is the distance of the orbiting body to the center of the body of mass M (note the minus sign).
Substituting into equation (8.5) and using equation (8.3) for R we have:
E = (mo / R) C^2 (1 – Rs / d)
= mo C^2 R = Eo R
or the equivalent:
E = m (C^2 – V^2)
E = m (C^2 – M G / d)
For instance: Jupiter has a mass of 1.9E27 kilograms and orbits the Sun with a velocity of 13.1 kilometer/second
at a distance of about 7.8E8 kilometers. Its kinetic energy from equations (8.1) and (8.8) is about 1.6E42 ergs.
(Because of Jupiter’s slow velocity compared to that of light, the same result is obtained from the classical
kinetic energy equation Ke = (1/2) m V^2) The potential energy from equation (8.9) is about 3.2E42 ergs; and the
total energy is about 1.7E51 ergs, very slightly lower than its energy at rest. This is due to the negative potential
energy that is about twice the value of the positive kinetic energy.
9.0 Total Energy, Kinetic Energy, and Potential Energy
We will start with noting the relation between kinetic and potential energies. From equations (8.8) and (8.9),
using equation (2.1) V^2 = M G / d and equation (8.1):
[Kp] / Ke = (1 – R^2) / (1 – R) (9.1)
where [Kp] is the absolute value of the potential energy.
At the mass event horizon (R = 0) the kinetic and the potential energies are equal in value but of opposite signs.
At exactly R = 1 (at infinite distance from the center of mass M) the ratio [Kp]/Ke is undefined (a singularity); but
at any other values of R the rate of change d([Kp] / Ke) / dR is equal to one, and
[Kp] / Ke = 1 + R (9.2)
Since R is going from zero to one depending on the distance from the center of mass, the ratio [Kp]/Ke goes
asymptotically from 1 to 2 with the potential energy of opposite sign to the kinetic energy.
Substituting equation (9.2) into equation (8.5) we have:
E = mo C^2 R = Eo R (9.3)
At the mass event horizon (R = 0) the total energy is equal to zero. At R = 1 (at infinite distance from the center of
mass M) the total energy is equal to the rest energy.
The total energy of a body of mass m rotating around a body of mass M can never be greater
than its rest energy as long as the speed of light is the limiting velocity.
This statement can be generalized by saying that matter cannot acquire more energy than its rest
energy regardless of conditions to which it is subjected.
We can also say that the mass event horizon is the locus of total energy unavailability and,
therefore, it is the locus of maximum entropy.
It is also of interest to note the relations between potential and kinetic energies and the energy of mass at rest
Eo.
For Ke = Eo we use equations (8.6) and (8.7) and solve for R, giving:
R = .5
Rs / d = .75
In terms of Rs units:
(nRs) = d / Rs = 1 / .75 = 1.333….
For Kp = Eo we use equations (8.6) and (8.9):
mo M G / (R d) = mo C^2
Using equation (2.2) for Rs:
R = Rs / d = sqr [1 – (Rs / d)]
Squaring and solving the resulting quadratic equation, we have:
Rs / d = [ – 1 + sqr (5)] / 2 = .618033988
In terms of Rs units:
(nRs) = d / Rs = 1.618033988
We will next note the relations among all energies, displayed numerically in Table 9.1 and mathematically in
Table 9.2.
Table 9.1
Table 9.1 above shows the relations among the variables discussed in this paragraph at three distances from
the mass event horizon: 1) at a distance from Rs where Kp = Eo; 2) at a distance equal to the Sun’s radius; and
3) at a distance equal to Jupiter’s distance from the Sun. The Table is in two parts: the top part is for a body of
mass mo equal to Jupiter’s orbiting a body of mass M equal to the Sun’s, both masses assumed concentrated at
the center of each respective bodies. The bottom part is for the opposite condition where the body of mass M is
now orbiting the body of mass mo.
Energy values (in black) are dependent on mass and distance while ratios between rest, kinetic, potential, and
total energy (in red) are constant for the same distance in Rs units. When using those units, the ratios are
independent of the mass of either body.
When using Rs as unit of distance, orbital velocity gradients (Paragraph 2.0), light deflection
angles (Paragraph 6.0), and energy gradients ratios are independent of mass.
Table 9.2 Energy ratios of an orbiting body.
the mass event horizon: 1) at a distance from Rs where Kp = Eo; 2) at a distance equal to the Sun’s radius; and
3) at a distance equal to Jupiter’s distance from the Sun. The Table is in two parts: the top part is for a body of
mass mo equal to Jupiter’s orbiting a body of mass M equal to the Sun’s, both masses assumed concentrated at
the center of each respective bodies. The bottom part is for the opposite condition where the body of mass M is
now orbiting the body of mass mo.
Energy values (in black) are dependent on mass and distance while ratios between rest, kinetic, potential, and
total energy (in red) are constant for the same distance in Rs units. When using those units, the ratios are
independent of the mass of either body.
When using Rs as unit of distance, orbital velocity gradients (Paragraph 2.0), light deflection
angles (Paragraph 6.0), and energy gradients ratios are independent of mass.
Table 9.2 Energy ratios of an orbiting body.
E / Eo = R
Eo / E1 = R
E / E1 = R^2
E / Ke = R^2 / (1 – R)
E / Kp = R^2 (1 – R^2)
E / (Ke + Kp) = R^2 / (R^2 – R)
Kp / Ke = (1 – R^2) / (1 – R) = (1 + R)
R = sqr [1 – (V / C)^2] (8.1)
= sqr [1 – M G / (C^2 d)] (8.2)
= sqr (1 – Rs / d) (8.3)
Eo / E1 = R
E / E1 = R^2
E / Ke = R^2 / (1 – R)
E / Kp = R^2 (1 – R^2)
E / (Ke + Kp) = R^2 / (R^2 – R)
Kp / Ke = (1 – R^2) / (1 – R) = (1 + R)
R = sqr [1 – (V / C)^2] (8.1)
= sqr [1 – M G / (C^2 d)] (8.2)
= sqr (1 – Rs / d) (8.3)
10.0 Gravitational Field Potential
The gravitational field potential is defined as P = – M G / d. The relation among gravitational field potential, orbital
velocity, and distance is obtained directly from equation (8.4):
[P] = (M G / d) = V^2 = C^2 (Rs / d) = C^2 / (nRs) (10.1)
where [P] is the absolute value of the gravitational field potential.
When using Rs as unit of distance, orbital velocity gradients (Paragraph 2.0), light deflection
angles (Paragraph 6.0), energy gradients ratios (Paragraph 9.0), and gravitational potential
gradients are independent of mass.
From the space-time triangle of Paragraph 8.0 we have:
V^2 = C^2 (1 – R^2)
Equation (10.1) then becomes:
[P] = (M G / d) = V^2 = C^2 (Rs / d) = C^2 (1 – R^2) (10.2)
11.0 Space-Time and Gravitational Field Potential
From the space-time triangle of Paragraph 8.0 we also have:
tan (theta) = (V t) / (C T) = V / (R C) (11.1)
theta = Atan [V / (R C)] (11.2)
We define the angle theta of the space-time triangle as the warp angle of the space-time
continuum.
Using equation (2.3) for V / C:
warp = Atan [sqr (Rs / d) / R] (11.3)
From equation (10.2), using equation (8.1) for R:
[P] = [C sin (warp)]^2 (11.4)
The gravitational field potential is proportional to the square of the sine of the space-time warp
angle, the proportionality constant being the square of the speed of light. [P] is equal to C^2
when warp angle is 90 degrees at the mass event horizon; and equal to zero when warp angle is
zero degrees at infinite distance from the center of mass generating the field.
The relation between warp angle and the angle of light deflection (Paragraph 6.0) is given by:
tan (alfa / 2) = [sin (warp)]^2 (11.5)
which is obtained from equations (5.1), (10.2), and (11.4).
Table 11.1 shows the values of the two angles for a distance from 1 to 6 Rs units, and for a distance equal to the
Sun’s radius.
Figure 11.1 shows the deflection angle (in red) and the warp angle (in black) from one Sun radius to two Sun
radiuses as well as the angles that would occur if the Sun were contracted below its radius.
Table 11.1
The gravitational field potential is defined as P = – M G / d. The relation among gravitational field potential, orbital
velocity, and distance is obtained directly from equation (8.4):
[P] = (M G / d) = V^2 = C^2 (Rs / d) = C^2 / (nRs) (10.1)
where [P] is the absolute value of the gravitational field potential.
When using Rs as unit of distance, orbital velocity gradients (Paragraph 2.0), light deflection
angles (Paragraph 6.0), energy gradients ratios (Paragraph 9.0), and gravitational potential
gradients are independent of mass.
From the space-time triangle of Paragraph 8.0 we have:
V^2 = C^2 (1 – R^2)
Equation (10.1) then becomes:
[P] = (M G / d) = V^2 = C^2 (Rs / d) = C^2 (1 – R^2) (10.2)
11.0 Space-Time and Gravitational Field Potential
From the space-time triangle of Paragraph 8.0 we also have:
tan (theta) = (V t) / (C T) = V / (R C) (11.1)
theta = Atan [V / (R C)] (11.2)
We define the angle theta of the space-time triangle as the warp angle of the space-time
continuum.
Using equation (2.3) for V / C:
warp = Atan [sqr (Rs / d) / R] (11.3)
From equation (10.2), using equation (8.1) for R:
[P] = [C sin (warp)]^2 (11.4)
The gravitational field potential is proportional to the square of the sine of the space-time warp
angle, the proportionality constant being the square of the speed of light. [P] is equal to C^2
when warp angle is 90 degrees at the mass event horizon; and equal to zero when warp angle is
zero degrees at infinite distance from the center of mass generating the field.
The relation between warp angle and the angle of light deflection (Paragraph 6.0) is given by:
tan (alfa / 2) = [sin (warp)]^2 (11.5)
which is obtained from equations (5.1), (10.2), and (11.4).
Table 11.1 shows the values of the two angles for a distance from 1 to 6 Rs units, and for a distance equal to the
Sun’s radius.
Figure 11.1 shows the deflection angle (in red) and the warp angle (in black) from one Sun radius to two Sun
radiuses as well as the angles that would occur if the Sun were contracted below its radius.
Table 11.1
|
Figure 11.1
12.0 Effect of Mass on Space-Time
From equation (11.3), solving for distance d:
d = Rs {1 + 1 / [tan (warp)]^2}
= Rs / [sin (warp)]^2 (12.1)
warp = Asin [sqr (Rs / d)]
= Asin {sqr [1 / (nRs)]} (12.2)
where (nRs) is the distance from the center of mass M expressed in Rs units.
For any space-time warp angle, the distance at which the warp occurs is directly proportional to
the mass of the body creating the warp.
Space-time warp angles have the same values at the same distance (expressed in Rs units) from
the center of mass.
When using Rs as unit of distance, orbital velocity gradients (Paragraph 2.0), light deflection
angles (Paragraph 6.0), energy gradients ratios (Paragraph 9.0), gravitational potential gradients
(Paragraph 10.0), and space-time warp angles are independent of mass.
We can use equation (12.2) to calculate the warp angle generated by a body of mass M. All we need is the Rs of
mass M from equation (2.2) and the distance from the center of mass M.
Table 12.1 shows the space-time warp angles in degrees generated by the Sun at the major bodies of the solar
system and the warp angles generated by the major bodies at their surface.
Table 12.1
From equation (11.3), solving for distance d:
d = Rs {1 + 1 / [tan (warp)]^2}
= Rs / [sin (warp)]^2 (12.1)
warp = Asin [sqr (Rs / d)]
= Asin {sqr [1 / (nRs)]} (12.2)
where (nRs) is the distance from the center of mass M expressed in Rs units.
For any space-time warp angle, the distance at which the warp occurs is directly proportional to
the mass of the body creating the warp.
Space-time warp angles have the same values at the same distance (expressed in Rs units) from
the center of mass.
When using Rs as unit of distance, orbital velocity gradients (Paragraph 2.0), light deflection
angles (Paragraph 6.0), energy gradients ratios (Paragraph 9.0), gravitational potential gradients
(Paragraph 10.0), and space-time warp angles are independent of mass.
We can use equation (12.2) to calculate the warp angle generated by a body of mass M. All we need is the Rs of
mass M from equation (2.2) and the distance from the center of mass M.
Table 12.1 shows the space-time warp angles in degrees generated by the Sun at the major bodies of the solar
system and the warp angles generated by the major bodies at their surface.
Table 12.1
|
The space-time warp angle generated by the Sun at Pluto is 3.3 seconds of arc; at Jupiter is 9 seconds of arc; at
the Earth is 20.5 seconds of arc.
The space-time warp angle generated by Pluto at its surface is 0.6 seconds of arc; by Jupiter at its surface is
28.9 seconds of arc; by the Earth at its surface is 5.4 seconds of arc; and by the Sun at its surface is 300.2
seconds of arc.
Figure 12.1 shows the relation between distance and mass for three warp angles: 90 degrees, 0.0834 degrees,
and 0.000739 degrees.
Figure 12.1
the Earth is 20.5 seconds of arc.
The space-time warp angle generated by Pluto at its surface is 0.6 seconds of arc; by Jupiter at its surface is
28.9 seconds of arc; by the Earth at its surface is 5.4 seconds of arc; and by the Sun at its surface is 300.2
seconds of arc.
Figure 12.1 shows the relation between distance and mass for three warp angles: 90 degrees, 0.0834 degrees,
and 0.000739 degrees.
Figure 12.1
The space-time warp angle generated by the presence of mass is uniform throughout the universe. It only
differs in values that are directly proportional to the mass that is generating the warp and to the distance from
its center.
An average galaxy of a trillion Suns will warp space-time to a warp angle of 0.0834 degrees (the warp occurring
at the Sun’s surface) at a distance from its center of about 7E17 kilometers, or 74000 light years. The same
galaxy will warp space-time to a warp angle of 0.00569 degrees (the warp occurring at a distance from the Sun
equal to that of the Earth) at a distance from its center of about 1.5E20 kilometers, or 16,000,000 light years. If the
mass of such a galaxy were to be contracted to its Rs radius, the resulting event would have a diameter of 3
trillion kilometers, or about 0.3 light years, and would have a warp angle of 90 degrees at its horizon.
13.0 Relation between Gravitational Acceleration and Space-Time Warp Angle
Using equations (1.1) and (2.2), and substituting for distance d from equation (12.1), we have:
g = (C^2 / Rs) / {1 + 1 / [tan (warp)]^2}^2
= (C^2 / Rs) [sin (warp)] ^ 4 (13.1)
g = [C^4 / (M G)] / {1 + 1 / [tan (warp)]^2}^2
= [C^4 / (M G)] [sin (warp)]^4 (13.2)
For any given warp angle, gravitational acceleration is inversely proportional to mass.
For any given mass, gravitational acceleration is directly proportional to the fourth power of the
sine of the warp angle.
From equations (13.1) or (13.2) we also have:
g / go = [sin (warp)]^4 (13.3)
where go is the gravitational acceleration of mass M at its mass event horizon, given by equation (2.4):
go = C^4 / (M G) = 1.21E49 / M
Figure 13.1 shows the relation between gravitational acceleration and warp angle for Jupiter, the Sun, and for
the average galaxy of one trillion Suns.
Figure 13.1
differs in values that are directly proportional to the mass that is generating the warp and to the distance from
its center.
An average galaxy of a trillion Suns will warp space-time to a warp angle of 0.0834 degrees (the warp occurring
at the Sun’s surface) at a distance from its center of about 7E17 kilometers, or 74000 light years. The same
galaxy will warp space-time to a warp angle of 0.00569 degrees (the warp occurring at a distance from the Sun
equal to that of the Earth) at a distance from its center of about 1.5E20 kilometers, or 16,000,000 light years. If the
mass of such a galaxy were to be contracted to its Rs radius, the resulting event would have a diameter of 3
trillion kilometers, or about 0.3 light years, and would have a warp angle of 90 degrees at its horizon.
13.0 Relation between Gravitational Acceleration and Space-Time Warp Angle
Using equations (1.1) and (2.2), and substituting for distance d from equation (12.1), we have:
g = (C^2 / Rs) / {1 + 1 / [tan (warp)]^2}^2
= (C^2 / Rs) [sin (warp)] ^ 4 (13.1)
g = [C^4 / (M G)] / {1 + 1 / [tan (warp)]^2}^2
= [C^4 / (M G)] [sin (warp)]^4 (13.2)
For any given warp angle, gravitational acceleration is inversely proportional to mass.
For any given mass, gravitational acceleration is directly proportional to the fourth power of the
sine of the warp angle.
From equations (13.1) or (13.2) we also have:
g / go = [sin (warp)]^4 (13.3)
where go is the gravitational acceleration of mass M at its mass event horizon, given by equation (2.4):
go = C^4 / (M G) = 1.21E49 / M
Figure 13.1 shows the relation between gravitational acceleration and warp angle for Jupiter, the Sun, and for
the average galaxy of one trillion Suns.
Figure 13.1
14.0 The Reach of Gravity
The space-time warp angle can be used to compare the effect of mass on space by defining a warp W at which
the effect of gravity is considered nil, and call the distance from that point to the center of mass “the gravity
reach”. We will choose the conservative warp at 1.5 times the distance of Pluto from the Sun as the angle at
which the effect of the Sun’s gravity on space is considered nil:
W = 7.39E-04 [deg] (14.1)
We can then calculate the gravity reach distance Rc as function of mass using equation (12.1):
Rc = Rs / [sin (W)]^2 (14.2)
Table 14.1 shows the results for Jupiter, the Sun, and for our average galaxy of one trillion Suns (see also
Figure 12.1):
Table 14.1
The space-time warp angle can be used to compare the effect of mass on space by defining a warp W at which
the effect of gravity is considered nil, and call the distance from that point to the center of mass “the gravity
reach”. We will choose the conservative warp at 1.5 times the distance of Pluto from the Sun as the angle at
which the effect of the Sun’s gravity on space is considered nil:
W = 7.39E-04 [deg] (14.1)
We can then calculate the gravity reach distance Rc as function of mass using equation (12.1):
Rc = Rs / [sin (W)]^2 (14.2)
Table 14.1 shows the results for Jupiter, the Sun, and for our average galaxy of one trillion Suns (see also
Figure 12.1):
Table 14.1
|
It is very clear from the above that the effect of gravity is felt at a considerable distance from the center of mass.
A planet the size of Jupiter is affecting space for at least 8.5 million kilometers around it. The Sun’s reach is
conservatively estimated at 9 billion kilometers. The reach of an average galaxy is immense: almost one billion
light years! The inside of a galaxy, especially close to its center, is a cauldron of roiling space-time due to the
motion of bodies within that space. The halo of a galaxy is a more tranquil zone, but even there, because of the
rotation of the galaxy and of the bodies within the galaxy, space-time is continuously “moving”, and this
movement is propagated for millions, even billions, of light years, affecting other galaxies and the whole
structure of the universe. This structure is as pockmarked as the lunar surface: a beam of light traveling
through it is bounced around as a ball in a pinball machine with black holes as pockets.
The propagation of the gravitational field across immense distances of space and time poses questions. How
can a galaxy affect space-time at such immense distances if the only medium available to convey information is
the cosmologically slow speed of light? It is inconceivable that the gravity of an average galaxy takes one billion
years to make itself felt at a conservative distance that is comparable to the 1.5 times the distance of Pluto from
the Sun. An observer at that distance would not only see the galaxy but experience its gravity pull for one billion
years after the galaxy’s demise. A condition must exist in the space-time fabric that allows the gravitational field
to be conveyed much faster than the speed of light, if not instantly.
We must ask ourselves: is mass creating the turbulence in the surrounding space-time; or is the turbulence of
space-time creating the concentration of energy that is called mass? If the latter is true, then gravity may not
be a property of mass but, instead, it may be a property of the condition of space-time. In this
case matter is no longer the cause of the gravitational field potential and there is no need for a medium to
convey information to the space-time.
PART III CONCLUDING OBSERVATIONS
15.0 Mass Event Horizon
The mass event horizon is defined in equation (2.2) as the radial distance from the center of a body of mass M at
which the orbiting velocity gradient generated by the body is equal to the speed of light:
Rs = M G / C^2
This equation is obtained directly from the two classical equations defining gravitational and centripetal
accelerations. The mass event horizon distance Rs is not the Schwarzschild radius which is twice the Rs
distance: the event horizon is based on orbital velocity while the Schwarzschild radius is based on escape
velocity.
If a body of mass M is contracted into a volume of radius Rs (the event) then the horizon of that event has the
following properties: orbital velocity gradient equal to the speed of light; light deflection angle of 90 degrees;
space-time warp angle of 90 degrees; and total energy content of mass orbiting at the horizon equal to zero.
The horizon defines the critical distance below which the laws of classical physics may not apply. This would
include: orbital velocity gradients greater than the speed of light which will make the relativity term
R = sqr [1 – (V / C)^2] imaginary; and light deflection angles greater than 90 degrees which will increase the
number of spatial and temporal dimensions and make them fractal.
The horizon is the locus where the total energy of an orbiting body is zero. A body entering the sub-event world
does not increase the total energy of that world. Sub-event worlds are closed systems where energy can neither
get in nor get out: they are invariant.
The concept of mass event horizon provides a means of unifying aspects that may appear separate: orbital
velocity gradients, gravitational potential gradients, energy ratios, light deflection angles, and space-time warp
angles are all independent of mass when the radius of the horizon is used as unit of distance. The mass event
horizon makes those aspects uniform throughout the universe: each aspect has equal values at the same
distance measured in Rs units.
16.0 Cosmological Implications
If we define the universe as an event then the event horizon of the universe is given by equation (2.2)
Rs = M G / C^2, where M is the mass of the universe. This event horizon has the same properties as all mass
event horizons. From equations (8.5), (8.6), (8.7), and (8.9) we have:
E = mo C^2 + (m C^2 – mo C^2) – m M G / Rs = 0
And
m C^2 = m M G / Rs (16.1)
At the mass event horizon, the kinetic potential C^2 is equal to the gravitational potential M G / Rs. This equality
of potentials has to be true for the life of the universe. We can then use equation (16.1) as a cosmological model:
M C^2 – M^2 G / r = 0 (16.2)
where M is the mass of the universe and r is its “radius”.
Equation (16.2) is identical to equation (2.2) that defines the radius Rs at the event horizon of mass M. But the Rs
definition in equation (2.2) is based on a point mass orbiting at the horizon with a velocity equal to the speed of
light. Equation (16.2), however, is for a universe that is expanding at the speed of light. This introduces a time
parameter into the equation that requires mass and/or the speed of light to be variable. Furthermore, equation
(2.2) defines the limiting values of a system when viewed from above its mass event horizon; equation (16.2),
instead, must define a system when viewed from below the event horizon.
This cosmological model is in contrast to all current models even though it is grounded on solid classical and
relativistic principles. This new cosmology is discussed in “The Symmetrical Universe: a Cosmological Model”
PART IV APPENDIX
Solar System Data
A planet the size of Jupiter is affecting space for at least 8.5 million kilometers around it. The Sun’s reach is
conservatively estimated at 9 billion kilometers. The reach of an average galaxy is immense: almost one billion
light years! The inside of a galaxy, especially close to its center, is a cauldron of roiling space-time due to the
motion of bodies within that space. The halo of a galaxy is a more tranquil zone, but even there, because of the
rotation of the galaxy and of the bodies within the galaxy, space-time is continuously “moving”, and this
movement is propagated for millions, even billions, of light years, affecting other galaxies and the whole
structure of the universe. This structure is as pockmarked as the lunar surface: a beam of light traveling
through it is bounced around as a ball in a pinball machine with black holes as pockets.
The propagation of the gravitational field across immense distances of space and time poses questions. How
can a galaxy affect space-time at such immense distances if the only medium available to convey information is
the cosmologically slow speed of light? It is inconceivable that the gravity of an average galaxy takes one billion
years to make itself felt at a conservative distance that is comparable to the 1.5 times the distance of Pluto from
the Sun. An observer at that distance would not only see the galaxy but experience its gravity pull for one billion
years after the galaxy’s demise. A condition must exist in the space-time fabric that allows the gravitational field
to be conveyed much faster than the speed of light, if not instantly.
We must ask ourselves: is mass creating the turbulence in the surrounding space-time; or is the turbulence of
space-time creating the concentration of energy that is called mass? If the latter is true, then gravity may not
be a property of mass but, instead, it may be a property of the condition of space-time. In this
case matter is no longer the cause of the gravitational field potential and there is no need for a medium to
convey information to the space-time.
PART III CONCLUDING OBSERVATIONS
15.0 Mass Event Horizon
The mass event horizon is defined in equation (2.2) as the radial distance from the center of a body of mass M at
which the orbiting velocity gradient generated by the body is equal to the speed of light:
Rs = M G / C^2
This equation is obtained directly from the two classical equations defining gravitational and centripetal
accelerations. The mass event horizon distance Rs is not the Schwarzschild radius which is twice the Rs
distance: the event horizon is based on orbital velocity while the Schwarzschild radius is based on escape
velocity.
If a body of mass M is contracted into a volume of radius Rs (the event) then the horizon of that event has the
following properties: orbital velocity gradient equal to the speed of light; light deflection angle of 90 degrees;
space-time warp angle of 90 degrees; and total energy content of mass orbiting at the horizon equal to zero.
The horizon defines the critical distance below which the laws of classical physics may not apply. This would
include: orbital velocity gradients greater than the speed of light which will make the relativity term
R = sqr [1 – (V / C)^2] imaginary; and light deflection angles greater than 90 degrees which will increase the
number of spatial and temporal dimensions and make them fractal.
The horizon is the locus where the total energy of an orbiting body is zero. A body entering the sub-event world
does not increase the total energy of that world. Sub-event worlds are closed systems where energy can neither
get in nor get out: they are invariant.
The concept of mass event horizon provides a means of unifying aspects that may appear separate: orbital
velocity gradients, gravitational potential gradients, energy ratios, light deflection angles, and space-time warp
angles are all independent of mass when the radius of the horizon is used as unit of distance. The mass event
horizon makes those aspects uniform throughout the universe: each aspect has equal values at the same
distance measured in Rs units.
16.0 Cosmological Implications
If we define the universe as an event then the event horizon of the universe is given by equation (2.2)
Rs = M G / C^2, where M is the mass of the universe. This event horizon has the same properties as all mass
event horizons. From equations (8.5), (8.6), (8.7), and (8.9) we have:
E = mo C^2 + (m C^2 – mo C^2) – m M G / Rs = 0
And
m C^2 = m M G / Rs (16.1)
At the mass event horizon, the kinetic potential C^2 is equal to the gravitational potential M G / Rs. This equality
of potentials has to be true for the life of the universe. We can then use equation (16.1) as a cosmological model:
M C^2 – M^2 G / r = 0 (16.2)
where M is the mass of the universe and r is its “radius”.
Equation (16.2) is identical to equation (2.2) that defines the radius Rs at the event horizon of mass M. But the Rs
definition in equation (2.2) is based on a point mass orbiting at the horizon with a velocity equal to the speed of
light. Equation (16.2), however, is for a universe that is expanding at the speed of light. This introduces a time
parameter into the equation that requires mass and/or the speed of light to be variable. Furthermore, equation
(2.2) defines the limiting values of a system when viewed from above its mass event horizon; equation (16.2),
instead, must define a system when viewed from below the event horizon.
This cosmological model is in contrast to all current models even though it is grounded on solid classical and
relativistic principles. This new cosmology is discussed in “The Symmetrical Universe: a Cosmological Model”
PART IV APPENDIX
Solar System Data
|
The gravity is measured at the surface of the solar system body.
The mass event horizon radius Rs is obtained from equation (2.2): Rs = M G / C^2
The gravity reach Rc is based on the space-time warp angle of 0.000739 degrees which is the warp angle of the
Sun at 1.5 times Pluto’s distance (Paragraph 14.0).
The mass event horizon radius Rs is obtained from equation (2.2): Rs = M G / C^2
The gravity reach Rc is based on the space-time warp angle of 0.000739 degrees which is the warp angle of the
Sun at 1.5 times Pluto’s distance (Paragraph 14.0).
===================================================================================================
Copyright © 2005 Giulio. C. Cima All Rights Reserved
===================================================================================================
Copyright © 2005 Giulio. C. Cima All Rights Reserved
===================================================================================================
|
| Gravity and Space-Time by Giulio. C. Cima (Copyrigth © 2005 G. C. Cima All rights reserved) |