MATHEMATICAL APPENDIX
The geometry of open and closed worlds
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- OPEN AND CLOSED WORLDS
- THE GEOMETRY OF A CLOSED WORLD
- THE GEOMETRY OF AN OPEN WORLD
- IMAGINARY TIME IN OPEN AND CLOSED WORLDS
- THE ORIGIN OF SPACE AND TIME
- THE ORIGIN OF MASS AND ENERGY
- TABLE OF CONTENTS
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We consider now the closed world of an isotropic Riemann space with constant positive curvature and the open world of a space with constant negative curvature. We call this last space also a Lobatjevsky space.
We shall see, that a three-dimensional Riemann space, with constant curvature, has the geometry of a three-dimensional hyper-spherecal surface in a fictitious euclidian 4-space, in analogy with a 2-dimensional spherical surface embedded in an euclidean 3-space. The curvature and radius of curvature of the space correspond to the curvature and radius of curvature of a spherical surface. A negative curvature will be presented by an imaginary radius.
We see in our thoughts a four dimensional euclidean space with coordinates x1, x2, x3 and x4. We let coincide the first three of these coordinates with our well-known three spatial coordinates. The fourth coordinate is a fictitious spatial coordinate. A hyper-sphere with radius a has in the this four-space the mathematical form x1²+x2²+x3²+x4²=a².
The distance between two points on the hyper-sphere is:
dL²=dx1²+dx2²+dx3²+dx4². We eliminate the fourth coordinate and find for the formula for the spatial distance:
dL²=dx1²+dx2²+dx3²+(x1.dx1+x2.dx2+x3.dx3)²/(a²-x1²-x2²-x3²).
We proceed to ordinary spherical coordinates r, X and Ø, defined by x1=r.cosØ.sinX, x2=r.sinØ.sinX and x3=r.cosX. We find then for the geometrical equation:
dL²=(dr²+r²dX²+r²sin²Xdز)+(r²dr²)/(a²-r²).
We can simplify this to:
dL²=(dr²)/(1-r²/a²)+r²dO² with dO²=(dX²+sin²Xdز).
The radius of curvature a we call also world radius a and the curvature k of the space has been defined by k=1/a². For k positive, negative or zero we talk of a three dimensional space, which is spherical, pseudo-spherical or flat. If the radius becomes very large, the curvature approaches to zero. In this case approach the spherical and pseudo-spherical space to the flat space with the pythagorean expression for the distance dL: dL²=dx1²+dx2²+dx3². A growing radius in the four-space corresponds to an expanding three dimensional space.
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b. The geometry of a closed world.
We use the ordinary spherical coordinates r, X and Ø, defined in x1=r.cosØ.sinX, x2=r.sinØ.sinX and x3=r.cosX. The infinitesimal distance dL between two points for a space with constant positieve curvature (K=1/a²), has then the form:
dL²=(dr²)/(1-r²/a²)+r²dO² met dO²=(dX²+sin²X.dز).
The first term in this geometric equation is connected with the infinitesimal radial distance Lr=dr/Ö(1-r²/a²). The second term is connected with the infinitesimal transversal distance dLO=r.dO.
The circumference of a circle is as usually 2.PI.r. De radius Lr of this circle follows from: Lr=ódr/Ö(1-r²/a²)=a.arcsin(r/a), whereby r can vary from 0 to a. Because r=a.sin(Lr/a) we can write the circumference of a circle with radius r as: circumference=2.PI.r=2.PI.a.sin(Lr/a). When we move away from the origin, we see at always larger growing radial distance, that the circumference of the circle reaches his maximum value 2.PI.a for Lr=PI.a/2 and after that decreases to zero for Lr=PI.a.
The two dimensional analogon of this is a surface of a sphere with radius a. Suppose that we are situated on the North Pole of our earth with world radius a and that Lr is the radial distance from the North Pole to a parallel-circle, measured along a meridian on the spherical surface. The circumference of the parallel-circle lies in a cross-section of a surface perpendicular to our direction of looking and also in the surface of the sphere. We measure the radius r in the cross-section. The equator lies on a radial distance Lr=PI.a/2 with maximum circumference 2.PI.a for a parallel-circle and on a distance Lr=PI.a lies the South Pole with circumference zero for the parallel-circle. All points of this spherical surface are equivalent, so that the North Pole, the point where we are stand now, has no favored position and could be chosen also elsewhere.
The surface of a sphere in the 4-space has as usually the value 4.PI.r² and we can write this also as 4.PI.sin²(Lr/a). When we move from the origin, we see at growing radial distance, that the surface of a sphere reaches a maximum value 4.PI.a² for Lr=PI.a/2 and there after decreases to zero for Lr=PI.a.
For the volume V of this sphere applies then: V=ó4.PI.r²dLr or V=ó4.PI.a3sin²(Lr/a).d(Lr/a) or V=óPI.a3[1-cos(2Lr/a)]d(2Lr/a) or V=PI.a3[(2Lr/a)-sin(2Lr/a). The volume of a sphere will increase at larger values for Lr until it reaches his half-value for Lr=PI.a/2 and his maximum value for Lr=PI.a. The maximum volume is 2.PI.²am3.
This world with a constant positive curvature is spatially seen closed in itself. She has a finite volume, but no boundaries. The proportion between circumference and radius of a circle (2PI.r/Lr) is 2PI.sin(Lr/a)/(Lr/a) and is always smaller than 2.PI. Only if r is small in respect to the world radius a, we can take for this proportion 2.PI.
The geometry of the positively curved space is equal to the geometry on an hyper-sphere with real radius a in a fictitious 4-dimensional space. We can take also spherical coordinates in this Euclidean 4-space. These spherical coordinates a,X,Ø,ß are associated with the Euclidean coordinates x1,x2,x3,x4:
x1=a.sinß.cosØ.sinX, x2=a.sinß.sinØ.sinX, x3=a.sinß.cosX en x4=a.cosß, whereby r=a.sinß. The coordinates r,X,Ø coincide with our ordinary spatial coordinates. We consider ß as a spatial angle in a 4-space, which runs from 0 to PI. There apply dr=a.cosß.dß and dr²/(1-a²/r²)=dr²/cos²ß. The geometric equation then gets the form:
dL²=a²[dß²-sin²ß(dX²+sin²X.dز].
The infinitesimal radial distance is dLr=a.dß, so that Lr=a.ß. The infinitesimal transversal distance is dLO=a.sinß.dO with dO²=(dX²+sin²X.dز), so that a circumference of a circle is 2.PI.a.sinß and a spherical surface is 4.PI.a²sin²ß. The volume of a sphere is V=ó4.PI.r²dLr or V=ó4.PI.a²sin²ß.a.dß=óPI.a3[1-cos(2ß)]d(2ß) or V=PI.a3[2ß-sin(2ß). The maximum values (for r=a) then are for the circumference 2.PI.a, for the surface 4.PI.a² and for the volume 2.PI.²am3.
For the volume V we are allowed to write:
V=Vm/2.PI.[2ß-sin(2ß)] with Vm=2.PI.²am3 when ß=PI.
Because of the isotropy we are allowed to set the angles Ø and X equal zero in the expression for the interval ds²=c²dt²-dL². The interval gets the form ds²=c²dt²-a²dß². When we take c.dt=a.dt, then this becomes ds²=a²dt²-a²dß². In this expression is ß a pure space coordinate and t a pure time coordinate. The world radius a only depends on the time. We call also t and ß comoving coordinates. Now was r=a.sinß or dr=a.cosß.dß and cdt=a.dt, so that a=c.dt/dt and also a=(1/cosß).dr/dß. We see here a striking symmetry in the space and time coordinates for the magnitude of the world radius.
In this case we can consider t as the cosmic eigen-time, which play the role of the measure for the physical age of the universe. We can predicate ß the designation eigen-distance.
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c. The geometry of an open world.
When we take the curvature K=1/a² negative, then the world radius a becomes imaginary. When we replace in the formulas for positive curvature replace a by ia everywhere, then we get analogous results.
We use again the ordinary spherical coordinates r, X en Ø, defined by x1=r.cosØ.sinX, x2=r.sinØ.sinX and x3=r.cosX. The infinitesimal distance dL between two points, in a space with constant negative curvature, then has the form:
dL²=(dr²)/(1+r²/a²)+r²dO² with dO²=(dX²+sin²X.dز).
The first term in this geometric equation is related with the infinitesimal radial distance dLr=dr/Ö(1+r²/a²) and the second term with the infinitesimal transversal distance dLO=r.dO.
The circumference of a circle is as usually 2PI.r. The radius Lr of this circle follows from: Lr=ódr/Ö(1+r²/a²)=a.arcsinh(r/a), whereby r can vary from 0 to infinite. Because r=a.sinh(Lr/a) we can also write the circumference of a circle with a radius Lr as: circumference=2.PI.r=2.PI.a.sinh(Lr/a). When we move from the origin, we see at growing radial distance, that the circumference of the circle becomes larger and larger.
The two dimensional analogon of this is the curved surface of the ending of a trumpet, where the curvature of the ending extends to infinity. We look at the curvature in the length direction of the trumpet. The negative curvature K correspondents with a circle of radius a, which touches a point of our trumpet surface, whereby the radius lays not inside but outside the trumpet. We are situated in a point where the curvature starts and measure the radial distance along the surface in the direction of the ending. The circumference of a circle is situated in a cross-section of a surface perpendicular to our direction of looking and the trumpet. We measure the radius in the area of the cross-section.
The surface of a sphere has as usually the value 4.PI.r². For the volume V of this sphere then applies: V=ó4.PI.r²dLr. The surface and the volume become larger and larger when the distance Lr becomes larger and finally become infinite large.
This world with constant negative curvature is spatially seen open and has an infinitely large volume and is without borders
The proportion between circumference and radius of a circle is 2.PI.sinh(Lr/a)/(Lr/a) and is always larger than 2.PI.. Only when r is small with respect to the world radius a, we are allowed to take for this proportion 2.PI.
The geometry of the negatively curved space is equal to the geometry on a pseudo hyper-sphere with an imaginary radius in a fictitious 4-dimensional space.
We can take also spherical coordinates in this Euclidean 4-space. These spherical coordinates a,X,Ø,ß are associated with the Euclidean coordinates x1,x2,x3,x4:
x1=a.sinhß.cosØ.sinX, x2=a.sinhß.sinØ.sinX, x3=a.sinhß.cosX and x4=a.cosß, whereby r=a.sinhß.
We can find the analogon for the space-time geometry on the pseudo hyper-sphere which imaginary radius, compared with the geometry on the hyper-sphere with real radius, by replacing everywhere in the formulas s,ß,a by is,iß,ia.
The space-time interval gets the form:
ds²=c²dt²-a²[dß²+sinh²ß(dX²+sin²X.dز].
By using c.dt=a.dt, we get the form:
ds²=a²dt²-a²[dß²+sinh²ß(dX²+sin²X.dز].
The infinitesimal radial distance is dLr=a.dß, so that Lr=aß.
The infinitesimal transversal distance is dLO=a.sinhß.dO with dO²=(dX²+sinh²X.dز), so that a circumference of a circle is 2.PI.a.sinhß and a surface of a sphere is 4.PI.a²sinh²ß.
The volume of a sphere is V=ó4.PI.r²dLr or V=ó4.PI.a²sinh²ß.a.dß or V=óPI.a3[cosh(2ß)-1]d(2ß) or V=PI.a3[sinh(2ß)-2ß].Because of the isotropy we are allowed to set the angles Ø and X equal zero in the expression for the interval ds²=c²dt²-dL². The interval gets the form ds²=c²dt²-a²dß². When we take c.dt=a.dt, then we have ds²=a²dt²-a²dß². In this expression is ß a pure spatial coordinate and t a pure time coordinate. The world radius only depends on the time. We call t and ß also comoving) coordinates. Now was r=a.sinhß and cdt=a.dt, so that a=c.dt/dt and also a=(1/coshß).dr/dß.
Also in this case we can consider t as the cosmic eigen-time, which play the role of the measure for the physical age of the universe and ß as eigen-distance. Also here we see a striking symmetry in the space and time coordinates.
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d. The imaginary time in open and closed worlds.
Prof. Stephen Hawking got the idea to take the time (lifetime of the universe) imaginary and to give the time the same role as the spatial coordinates. In the two dimensional analogon for a closed world, the North Pole of the earth is now the beginning of the time and the distance from the North Pole to a parallel-circle is the lifetime of the universe. The radius of a parallel-circle remains again a measure for the spatial dimension of the universe. This universe remains finite and unlimited in the space. The South Pole represents then the final contraction or collapse of the universe. In the real time we have a beginning and an ending (two singularities), whereas there happens nothing in the imaginary time in particular. If we walk from the North Pole to the South Pole on the real sphere, we can simply keep walking arriving on the South Pole and return eventually to the North pole. The poles have on our globe no meaning as singular points are to be considered as normal points. In a simultaneous manner the poles are no singular world points on our two dimensional analogon for this imaginary time-space.
If the imaginary time is the real time, we can state that the universe is also unlimited in the time and nevertheless has a finite (or cyclic) existence. The laws of nature were and remain always valid also during the beginning and ending phases of the universe.
In the same way we can consider the narrow opening of a trumpet for the analogon for the open world as the beginning of the time and the radial distance along the trumpet in the direction of the end as the age of the universe.The radius of the parallel-circle (in a cross-section of the trumpet) remains also here a measure for the spatial dimension of the universe. This universe remains unlimited and infinite in the space. If we close the opening of the beginning of the trumpet with a spherical surface with a positive curvature, this world is also unlimited in the time. The laws of nature remain valid for the beginning of the universe. Because this universe continues to expand eternally, there is no end phase. Also the open universe is infinite in the time.
More about this: see the lecture of prof. Hawking about The Beginning of Time.
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e. The origin of space and time.
In the general field equations of Einstein the geometry of the space-time (determined in the geometrical tensor) is expressed in the energy contents, including all masses, of a space-time volume (in the energy tensor). Under the assumption of isotropy and homogeneity in our system of coordinates these field equations are to solve even without too much effort.
There exist three possible solutions, which correspond to a closed spherical, open flat or open pseudo-spherical universe with corresponding geometry such as Riemann, Minkovsky or Lobatsjevki space-time.
At local level, such as the area of our solar system, we can consider the space-time as almost flat, whereas at more precise calculations, in the neighborhood of our sun, the space-time is deformed to a Riemann space-time. The same applies to our galaxy core, because the presence of a giant black hole in the center.
For larger distances up to approximately three billion light years the space-time is also almost flat, except in the neighborhood of black holes in the cores of galaxies. For larger distances our knowledge is at present time such that this space-time area seems flat, except in the neighborhood of large concentrations of galaxies. The properties of these Riemann space-time areas are determined by the masses of our sun, black holes or large concentrations of galaxies with each their own specific solutions.
The solution of the general field equations of Einstein for our solar system exists for a long time and is known as the so-called Schwartzschild solution. Also for black holes there exist similar solutions. The solutions for the space-time as a whole like above outlined, exist already almost a century.
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f. The origin of mass and energy.
Such as space and time are aspects of the same entity in the space-time tensor, also mass and energy are aspects of the same entity, in the mass-energy tensor, such as is expressed in the general field equations. To our large surprise these tensors however are also equal to each other. The three simple solutions of the field equations can be set into a form, which has irrefutable agreements with three possible quantum situations.
This we can see also in the form of the formula for the scalar curvature R of the space-time, expressed in the energy contents T of a space-time volume: R = - k.T, where the constant k is dependent on the gravitation constant in the law of Newton for the universal gravitation and R respectively. T can be a zero, positive or negative quantity.
In which situation our current universe finds itself as a whole, is still an unsolved riddle for our scientists. Some recent discoveries plead for a flat universe (R is zero) and other for a hyperbolic or a pseudo-spherical universe (R is negative), whereas also a spherical universe (R is positive) seems not yet excluded.
According to the modern quantum mechanics, a certain probability is granted to each possible quantum situation, where also a combination of all possible situations with several probability factors is allowed. According to my conviction our current universe exists from such a combination of these quantum situations. It will last probably still a while, before our scientists can give more information about this.
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SEE ALSO: MATHEMATICAL APPENDIX PART 2
<< 1 TEXTS ABOUT OUR SPACE-TIME
<< 8 THE WINDOWS OF TIME FOR OUR SPACE
<< 2 THE SPACE-TIME CONTINUUM
<< 3 THE SPACE-TIME IN THE MACROCOSM
<< 4 THE SPACE-TIME IN THE MICROCOSM
<< 5 SPACE-TIME DIAGRAMS FOR PARTICLES AND PHOTONS
<< 6 TABLE FOR QUARKS AND HADRONS
<< 7 TABLE FOR QUARKS AND LEPTONS
<< 9 TERMINOLOGY FOR COSMOLOGY
=> 10 MATHEMATICAL APPENDIX PART 1 | PART 2
>> 11 SUMMARY OF THE CHAPTERS
>> 12 REFERENCES AND LITERATURE
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