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The space-time continuum
- THE THREE DIMENSIONS OF SPACE
- THE FOURTH DIMENSION OF THE TIME
- THE INTERVAL OF SPACE-TIME
- THE AREAS OF SPACE-TIME
- THE VIEWS OF SPACE AND TIME
- THE SPACE-TIME CONTINUUM
- THE EXPANDING UNIVERSE AND THE SPACE-TIME CONTINUUM
- EINSTEIN. THE GENERAL FIELD EQUATION
- THE CURVATURE OF SPACE-TIME
- DIMENSION AND LIFETIME OF OUR UNIVERSE
- TABLE OF CONTENTS
The three dimensions of the space.
We can describe mathematically the trajectories of planets and stars in space by means of a cartesian system of three mutual perpendicular lines. The length x, width y and height z play the role of the three spatial coordinates. We call them also the x, y and z dimensions. For the distance r of a point P in space (with coordinates x,y,z) from the origin O of our system of coordinates (with coordinates 0,0,0) applies the geometric relation of Pythagoras. This formula has the form: r²=x²+y²+z².
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The fourth dimension of the time.
In many cases we like to show the role of the time in a mathematical expression of the movement of a body. We choose then the time as the fourth dimension. Hereby we wish to give the time a similar role as the spatial coordinates x, y and z. In mathematical diagrams the time is always delineated perpendicular on the other coordinates.
We can catch the trajectory of a ray of light, departing from the origin O and arriving in point P, in the formula r=ct, whereby c is the speed of light. We are allowed to write this relation as c²t²=x²+y²+z² and also as x²+y²+z²+i²c²t²=0, whereby i²=-1 and i=Ö-1.
In place of t we take ict as the fourth coordinate. The fact that the ict-coordinate represents a complex number, is no objection for a mathematician, because a solid theory already exists for these numbers. In fact the complex number i=Ö-1 here has the formal function as informer of the fact, that the line of time stands perpendicular on the spatial lines. A space-time, wherein a moving body can be described with the coordinates x, y, z and ict, we call a Minkowsky space-time.
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The interval s is defined in the relation: s²=r²+i²c²t² of s²=x²+y²+z²+i²c²t².
In this case we talk of the interval between the space-time point P (with coordinates x,y,z,ict) and the space-time origin O (with coordinates 0,0,0,0). In place of space-time point we use in the future the name event. An event is then an occurrence, that takes place on a certain place and on a certain moment. The relation for the interval, as formulated above, can be considered as the space-time distance of event P to the origin O of space and time in four dimensions. The formula for that interval can be considered as a generalization of the law of Pythagoras for four dimensions. Here is formulated mathematically that the dimensions x, y, z and ict are mutually perpendicular.When a particle P is moving, then the world points of P form an world line in the Minkovsky space-time. A particle that is moving with a constant velocity v depicts a straight world line. Like that is OP the world line of particle P, that departed from the origin with a constant velocity v. The tangent of the angle Ø, formed by OP and the ict-coordinate, is proportional with v/c.
The ict-coordinate itself is the world line of a not moving particle (our origin), of course with velocity zero. A particle that is moving from the origin with the speed of light has the bissectrice between the r-coordinate and the ict-coordinate as world line with angle Ø = 45 degrees.
Because no velocities are allowed larger than the speed of light, there exist no world lines with an angle Ø > 45 degrees. This means that it is not allowed to delineated world lines originating from the origin with an angle Ø > 45 degrees.
We shall see later, that a rotation of the r-coordinate and ict-coordinate over an angle Ø in the Minkovsky space-time corresponds to a Lorentz-transformation.
The rotation of a world line of an electron, that is emitting a photon, changes by rotation over 90 degrees in this Minkovsky space-time in the creation or annihilation of an electron-positron pair. For this see the space-time diagrams for elementary particles in our microcosm. Velocities larger then the speed of light are here surely allowed.
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Dutch => English ----------------------------
Toekomst => Future
Heden => Present
Verleden => Past
Verboden => Forbidden
This diagram is called a space-time diagram, this is a system of coordinates with vertically the time t and horizontally the distance r.
Our space-time consists of events. These events can be connected as cause and result or occur separately. Two events P and O can only be connected as cause and result, if there can be exchanged signals between P and O.
For events in the present, that occur on the point of time now, we choose the point of time t=0. For events in the future applies then the relation t>0 and for events in the past applies the relation t<0.
Because the velocity of light is the maximum velocity for a signal, aplies for all events P with the relation r²>c²t² or r²-c²t²=s²>0, that these events can not be connected with O and are situated in a forbidden area for an observer in O.
All events P with r²=c²t² or r²-c²t²=s²=0 are connected with event O as cause and result. They are connected via light or other electromagnetic signals.
For all events with the relation r²-c²t²=s²<0 applies r²<c²t². These events can be connected via signals below the velocity of light, or when r<ct as events in the future (with t>0) or when r<-ct as events in the past (with t<0) for an observer in O.
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The views of space and time, which I wish to lay before you, have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.
H. Minkowski, 1908.
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The formulas of Lorentz apply for uniform movements of bodies or spatial volumes filled with mass and energy. An uniform movement is a movement with invariable velocity in an invariable direction.
When there is a change of direction at high speeds, then applies the invariance of the interval only local. We can divide the space-time, in which the variable movement takes place, in very small pieces along the trajectory. Each piece of space-time is so small, that the change in speed or direction stays negiglible small. In this very small piece of space-time, also called infinitesimal area, applies local the invariance of the space-time interval. For this interval applies in infinitesimal form the relation ds²=dx²+dy²+dz²-c²dt².
When we divide the whole space-time along the trajectory in overlapping infinitesimal areas, then it is theoretical possible to catch the movement in mathematical formulas. Hereby we use then the differential and integral calculus. When we divide the whole space-time in each other overlapping infinitesimal areas, then we call this totality also the space-time continuum.
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The expanding universe and the space-time continuum.
According to the cosmologists our universe is expanding under the influence of the laws of Newton for gravity. The totality of mass and energy in our universe is subjected to this gravity, whereby variable movements appear, that we can describe only by using the space-time continuum.
Only when the expansion velocities are very small compared with the velocity of light the classical formulas for distance, time, mass and energy apply under Galilean transformations. The geometry of the space can be described by the geometry of Euclides.
When the velocities are larger and the changes in velocity are negiglible, then we use the law of the invariance of the space-time interval under Lorentz transformations. The Lorentz formulas for the observed dilatation of time and shortening of length are a direct result of the invariance of the interval. We can describe the geometry of the space-time by using the laws of the special relativity theory of Einstein.
When the velocities are large compared with the speed of light and when the changes in velocities are larger, in the form of accelerations or decelerations, then we use the space-time continuum. In this last case only the law of invariance applies for the total mass-energy content of a body or spatial confined volume . We can describe the geometry of the space-time by using the general theory of relativity of Einstein.
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Einstein. The general field equation.
R[mn] - ½ R.g[mn] = k.T[mn]
This equation is composed of multi dimensional entities, called tensors. The indexes m and n of the tensors go through the values 0, 1, 2 and 3 corresponding with 4 dimensions, 3 for the space and 1 for the time.
On the left side of the equation we see a geometrical tensor, composed of two other tensors. These are the Ricci tensor R[mn] and the fundamental tensor g[mn]. These both tensors are determined by the geometry of the space-time. The scalar factor R is derived from the Ricci tensor and is also called the scalar curvature of space.
On the right side of the equation we see a ordinary constant k and the energy tensor T[mn]. The energy tensor is determined by all mass and energy present in space. The constant k is proportional with the gravitational constant in the law of Newton for the universal gravitation.
We can apply the field equation to a isolated spatial volume, such as our solar system, black holes or the space as a totality. To say it in words, this means that the geometry of the space corresponds with the total quantity of energy in this space, inclusive the energy, that correspondents with masses.
N.B. We can determine R, according to the tensor calculus, as trace of R[mn] and T as trace of T[mn]. Through contraction of all indexes we have then local (because trace g = 4) the relation for the scalar curvature R = - kT, whereby T is representing the total mass-energy of the space.
This scalar curvature of the space corresponds directly with the scale factor of our universe, whereof the magnitude is determined by the recent rate of expansion of our space-time. This applies only in the case that we apply the general formula to our entire observable universe.
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In the general field equations of Einstein the total mass-energy content of a spatial limited volume is expressed in the geometrical curvature of space-time in four dimensions. These field equations can be solved in specific simple cases. Par example there is a solution for these equations for a space-time with negligible small mass-energy content. The curvature is then zero and this space-time is called flat. When the totality of mass-energy content is positive or negative, then the curvature also is positive or negative. For the space-time interval we can find an expression, wherein this curvature plays an important role, provided that we assume diverse simple conditions. When the curvature of space-time for our whole universe is an invariable entity for our entire universe in the continuum of space-time as a whole, the expressions for the interval even take a simple mathematical form.
Dutch => English The distance r between systems of stars set vertically against the time t horizontally calibrated for the point of time now (t=0) for a curvature k larger then zero, equal to zero or smaller than zero.
In this case we can use the geometry of Riemann for the positive curved space-time and the geometry of Lobatsjevski for the negative curved space-time. The geometry of the flat space-time of Minkovski follows from both mentioned geometries, if we let approach the curvature of space-time to zero. The interval takes the form, that we found earlier for the local invariant.
Also in the curved space-time continuums the form of the interval approaches the local invariant, when the distance and time stay small in cosmic view for the infinitesimal area under contemplation. With other words, the curvature of space-time is only observable on cosmic large scale. Cosmic large scale means that distances are large in comparison with the dimension of the universe and that lapses of time are large in comparison with the lifetime of the universe.
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Dimension and lifetime of our universe.
Recent estimations give for the lifetime of our universe a number of 15 billion (15 thousand millions) years, with an uncertainty of plus or minus 5 billion years. This means that the dimension of our universe corresponds with 15 billion light years. We can say, that until a distance of three billion light years from us, the local invariance applies and space-time is flat, so that the geometry of Minkovski applies here.
Only for much larger distances the curvature of space-time shall become observable . Because we see over these large distances an universe out of a far past and therefore so much different from what we see not far from us, we are not able to calibrate measurements reliably for the moment being. The research for these far regions of our cosmos coincides with the question concerning the curvature of our space-time as a whole, upon it is positive, zero or negative. The same question applies for the total content of mass and energy of our cosmos.
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<< 1 TEXTS ABOUT OUR SPACE-TIME
== 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
>> 8 THE WINDOWS OF TIME FOR OUR SPACE
>> 9 TERMINOLOGY FOR COSMOLOGY
>> 10 MATHEMATICAL APPENDIX PART 1 | PART 2
>> 11 SUMMARY OF THE CHAPTERS
>> 12 REFERENCES AND LITERATURE
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