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TOC
Supernova
The Cosmological Constant
The Evolution of the
Universe
Dark
Energy
Anti-Matter
Rotation
2-D Rotation
The Biggest Black Hole
References
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We know the universe is expanding in that matter on the large scale is spreading apart. Now the gravitational attraction of all matter acts to slow this expansion. The crucial question at the end of the 20th century was, is there enough matter to pull the universe back together in a big crunch or will it go on expanding forever and suffer an infinitely prolonged heat death?
For the universe to have lasted this long yet still remain relatively compact meant that it appears to be balanced on a knife-edge between these two states. A little greater average density and the universe would have collapsed long ago. A little more intrinsic kinetic energy and the galaxies would not have formed. Only the most accurate measurements then could determine on which side of this fine balance our ultimate fate lies.
The red shift gives us an extremely accurate measure of the rate of recession of a distant object but determining exactly how far away that object is, is not an easy matter. Obviously, the more distant the dimmer the object appears to be but stellar objects vary tremendously in brightness. Fortunately, there is one type of object whose intensity is remarkably consistent. This is a Type 1a supernova.
Hubble Space Telescope composite image of the Type 1a supernova SN 2014J in the galaxy M82, some 11.5 million light-years distant.
Supernova.
A Type 1a supernova occurs in binary systems containing a white dwarf. At some point as these slowly spiral into each other the disruption caused by their mutual gravitational attraction causes a tremendous explosion. Consequently, these supernovas occur under very specific conditions of stellar size, mass and composition resulting in an energy output and resulting brightness that is remarkably consistent. At its peak such supernova can outshine their entire host galaxy. These then are a light source of specific brightness and this can be used to accurately estimate their distance. Unfortunately, with this level of output the star quickly burns up or throws off the necessary mass-energy required to sustain its supernova stage and the phenomenon lasts only a matter of days. Also, because of the specificity of the conditions for the formation of a supernova they occur only at the rate of one per galaxy per year.
Fortunately, the universe is vast and the number of galaxies it contains immense making the occurrence of such supernova fairly common. Conversely, finding them during their short lifetime amongst the myriad of observable galaxies presents some difficulty. It was only with the advent of digital photography and computerised analysis in the late 1990’s that these could be detected in sufficient numbers. Combined with their red shifts and that of their host galaxy this gives us our best measure of the relationship between distance and recession velocity.
Now distance in space also represents distance in time. For example, Type 1a supernova SN 2014J above occurred in M82, 11.5 million light-years and consequently it also occurred million years ago; the time it has taken the light from it to reach us. So by measuring this supernova we get not only a picture of the relationship between distance and recession rate but also of how this has changed over time.
Now this experiment was to determine if the rate at which the expansion was slowing were sufficient to make all the matter collapse back or not. In one of those wonderful moments in science when something completely unexpected happens the researchers discovered that not only will the universe continue to expand indefinitely but this occurring at an expanding rate. That distant objects are actually moving away at a rate that increases with distance and hence the universal expansion is accelerating [9].
There has been one complication to these results. Subsequent observations have shown that there are actually two varieties of type 1A supernova, with different energies and consequent brightnesses. As a result the original estimate of the value of the rate of expansion will have to be revised downwards. The universe is expanding at an increasing rate, just not as fast as originally thought.
The Cosmological
Constant.
It should come as no surprise that Einstein’s General Theory of Relativity (GR) actually encompasses the possibility of such an increasing rate of expansion. Indeed, it was quite prevalent in his earliest papers. At that time the universe was believed to be eternal static and essentially flat. This was a problem for Einstein in that gravity is a result of a positive curvature.
Now solving the GR equations involves a process of integration, which invariably gives rise to a constant, in this case, the cosmological constant. For largely aesthetic reasons Einstein would have liked to have set this to zero. In order to produce a static universe he was forced to assigned this a very small positive value.
Later, when it was discovered that the universe is expanding, the cosmological constant was assumed to be zero and Einstein referred to originally retaining it as his ‘greatest mistake’. It is somewhat ironic then that half a century after his death that it has again emerged to describe the nature of an accelerating universe.
A positive cosmological constant produces a repulsive force that counters the attractive one of gravity. Unlike gravity, whose force diminishes inversely with the square of distance this one increases linearly with distance. . As a consequence, the attractive force of gravity will always win out over the repulsive force of the cosmological constant over the short range. In this case the short range being local galactic distances. Yet because the force due to the cosmological constant increases with distance there is comes a point at which these forces are equal and beyond which the repulsion of the cosmological constant is greater than the attractive one of gravity. With increasing distance beyond this critical one the force exerted by a positive cosmological constant continues to increase even as that of gravity diminishes still further.
Combined Forces: of gravity and that resulting from a positive cosmological constant. Gravity wins over short distances but there comes a point, beyond which, objects are actively repulsed with increasing force.
The Evolution of the
Universe.
At the beginning of the universe, at least the universe as we know it, it was filled with energy. This naturally began to dissipate outwards in all directions and the universe expanded. The force of gravity retarded this motion and if the initial energy had been below a certain level the mutual attraction of all the mass energy could have brought the whole to a halt and then drawn it back together. Once matter reached the critical distance where the repulsive force of the positive cosmological constant exceeded that of the attractive force of gravity, or equivalently the net curvature became negative, the universe was destined to expand at an ever increasing rate forever.
A positive cosmological constant implies that, in the absence of matter the universe in negatively curved. As the presence of matter causes space-time to curve positively in on itself, is there a similar cause for the negative curvature? When an object accelerates energy has to be supplied by or to it. So where does the energy for the acceleration of distant objects come from.
Dark Energy.
We are familiar with energy in the form of electromagnetic photons and gravitational gravitons. Gravitons only provide for a force of attraction. Electromagnetic forces are commonly applied to add energy to a system. Yet we do not any source of these that could provide the vast energies required to accelerate whole galactic clusters away from each other.
In just the same way that we have named the vast source of gravitons, dark matter, we have applied the term dark energy to the invisible source of this force of repulsion.
There is another reason why em energy cannot be the driving force behind this repulsion. Em energy dissipates with the square of the distance from its source. Conversely, a form of energy that could produce a positive cosmological constant has to be increasing linearly with distance. The source would have to be distributed everywhere and yet remain constant as the space expanded. Certainly, at the quantum level space is full of spontaneous energy fluctuations. It may be that as more space is created the energy of the universe does actually increase. Yet this would contravene the law of conservation of mass-energy. A law that, as far as we have been able to determine, operates in all em and gravitational interactions. The latter for both for normal and dark matter. Consequently, we should require considerable counter evidence indeed before we reject this conservation law.
Anti-Matter.
Anti-matter is matter but not at we know it. Every fundamental particle has an anti-matter counterpart. These counterparts have the opposite properties, such as charge and spin to their matter ones. They are created in pairs in extremely high-energy interactions, the simplest of which is when an incredibly energetic photon spontaneously generates a matter, anti-matter pair. As they are oppositely charged they attract one another and usually impact a short time later to recreate a high-energy photon. When these are created in high-energy experiments, large electromagnetic fields are used to separate the component parts.
Spontaneous Matter Anti-matter Creation: In the absence of any other forces the two antiparticles almost always recombine to recreate the high energy photon that generated them in the first place.
It is believed that this process occurs in the vacuum of space at all energies, everywhere, all the time. The higher the energy the less the frequency and the shorter the time the particles phase persists.
It has been hypothesised that, anti-matter is repelled by the gravitational field of normal matter. This is a consequence of an old theory that anti-matter is matter moving backwards in time. An idea that was generated by the observation that the above creation and annihilation process is time reversible, in that if it was played backwards it would be indistinguishable from a photon traversing right to left instead of left to right as shown above. This is not argument is not particularly convincing as, has already been observed in the section on entropy, this is true for any two particle interaction; such as the mutual orbiting or simple passage of two particles. The reasoning goes, that if anti-matter were travelling backwards in time, then it would be expected to be repelled by a gravitating mass.
The reasoning goes, that, if indeed anti-matter were repulsed by normal matter, then when the pairs are created in the vacuum of space during their brief existence they would orientate themselves around a large mass radiating outwards with the antimatter further from the gravitating mass, as shown below.
Hypothetical Alignment of Matter and ‘Anti-Gravity’ Antimatter in a Gravitational Field.
This would create a net repulsion in the same way that electrons do on the surface of objects preventing them merging when they approach one another. This would make it a candidate for the force behind the positive cosmological constant.
Anti-gravity is one of the holy grails of physics. So much so that any possibility is worth investigating. Consequently, circa 2016, an experiment is in preparation at the Giant Hadron Collider in Switzerland to test this hypothesis by observing the motion of anti-matter created in the collider under the action of the earth’s gravitational field. Unfortunately, this is doomed to failure.
There are two reasons for this. Firstly, the form of the resulting anti-gravity force does not fit that of a positive cosmological constant. Thought there is enough leeway in the accuracy of the measurements that makes it possible that such a force could account for our observations. The second and far more relevant objection is as follows. In the case of the simple, transitory pair, the initial and final photons both provide a gravitational pull on surrounding matter. . During the matter anti-matter pair phase the total mass-energy is the same and must also produce the same gravitational pull. If the antimatter component were repulsed by the gravitating body then the net force of the system would be zero, but only for the duration of their existence. This would break several well-established laws of physics.
The source of the positive cosmological constant, and unfortunately one of anti-gravity, will have to be sought elsewhere.
There is one, well known, phenomenon that not only counters the force of gravity but also increases in direct proportion to distance.
Rotation.
Rotation produces an effect that can counter the force of gravity. This is most easily seen in orbiting spacecraft and space stations. It is for this reason that orbiting astronauts experience weightlessness. Though, strictly there is only a surface within an orbiting satellite where the gravity is exactly zero and elsewhere conditions of micro-gravity exists which acts towards the earth at all points between that surface and the earth and away at all points outside.
This counter action of rotational force is also apparent in the motions of the planets and their satellites and indeed any orbital motion. The moon does not crash to the earth because the speed of its orbit exactly counters the force of gravity in the same way that that of the earth counters the pull of the sun.
If you are not convinced by these arguments you can test this for yourself. Get hold of bucket with a strong handle. The reason for this last you will discover soon enough. Now partly fill in with water. Preferably outdoors, rapidly swing this in a large arc up and above you and down again. If you did it sufficiently fast, and the handle held, the water should have remained in the bottom of the bucket throughout.
Rotation then, like a positive cosmological constant, provides an apparent counter for the force of gravity. In addition, and very significantly, with a constant rate of rotation this force increases linearly with distance.
Unfortunately, the simple, flat, two-dimensional rotation with which we are familiar has a definite direction, an axis of rotation, while the cosmological constant has appears to have none, operating, as it does, equally in all directions. This is something we will come back to shortly for now we will concentrate on the similarities.
For a cosmological constant of 2.036 x 10-52 (the value determined circa 2010) this is equivalent an angular velocity of space of 2.47 x
10-18 radians/sec. In addition, rotation as we know it involves discrete objects linked to a common centre, while the cosmological constant relates to a property of space itself. For it to be related to rotation space itself would have to be rotating simultaneously in two orthogonal directions. In a simply connected space this results in a new common axis of rotation; as shown in the figure below.
Rotation of a Spherical Surface: Any attempt so make an object such as a ball rotate in two different directions results in a one, single-axis, net rotation as shown in the figure above.
As we have seen, space itself is infinitely flexible and that it could stretch and expand in all directions simultaneously. This leaves the question of what could cause this rotation. There are a couple of obvious candidates. The simplest one is rotating matter.
As we saw in another section, the natural state of matter is rotation, from tiny particles such as electrons to galactic clusters. Even photons have an intrinsic spin. One consequence of GR is that rotating matter produces a small rotational drag on the surrounding space. Normally, this is far to imperceptible small to have any measurable effect. Except that is in very extreme cases such as rotating black holes.
These rotations act in every direction and the cumulative effect does apply a net repulsive force. Whether this is sufficient to account for the value of the cosmological constant has not been investigated. In addition, this drag does not increase linearly with increasing distance. Though it could account for the observations within the degree of accuracy.
We are thrown back and requiring a uniform two-dimensional rotation. Fortunately, there is a simple, very common, three dimensional object whose surface can simultaneously rotate in two perpendicular directions, while largely retaining its shape and resulting in a force that acts almost uniformly over the whole surface.
If you would like to figure this out for yourselves then pause here. If you need a hint, try watching a few episodes of the Simpsons.
2-D Rotation.
For those who have realised what the shape is, or are too impatient to try, the answer is a torus. Examples of which are, Homer Simpson’s favourite food, the donut and inner tubes of bicycle tyres. The later you can rotate about the major axis, indeed that is what happens when a wheel is turning. Take it off the wheel and it can also be rotated about a circular axis around the length of the tube. These rotations can be applied simultaneously and as they are orthogonal result in forces that act at right angles to each other in exactly the form we need to produce force that would be equivalent to a positive cosmological constant on this two-dimensional surface.
Two-Dimensional Rotation on the Surface of a Torus.
Any surface with a hole in it, such as a torus, is said to be not simply connected. This idea can be immediately extended to higher dimensions. So is our three dimensional space simply connected or not? In fact, there is strong evidence that it is not simply connected and that is does indeed possess a hole. In fact it has many, to the point where it is positively riddled with them; black ones. Massive black holes of a vast range of sizes exist at the heart of almost, if not, every galaxy. A number of much smaller stellar sized ones have also been detected by their effects on matter in the local neighbourhood. Add to that there are most probably vast numbers of smaller ones that we cannot detect. There may even be much bigger black holes too, the existence of which we may already be observing. Recent fine measurements of the background radiation appear to show an area of terrific space drift, the dark flow towards some ‘great attractor’. Could this great attractor be a super-colossal black hole? Possibly, but as yet, due to the vast distance involved and the correspondingly enormous time in the past from which this observation originates, these measurements are highly tentative.
This 2D rotation has nothing to do with the familiar axis of rotation of the black hole above. Rather it is a result of the way it ‘swallows’ the surrounding space-time. This is a form of flow into the black hole from the two directions perpendicular to the surface simultaneously. While the forces produced by orthogonal rotations on non-simply connected surface are not absolutely isotropic the differences in the on the large scale are relatively small. Such variations would be lost in those due to local gravitational effects.
The Biggest Black Hole.
The quantum nature of the universe places is a fundamental limit to how small a black hole can be. Similarly, a positive cosmological constant places an upper limit. As we have seen, the surface of a black hole, it event horizon, is a highly significant part of a black hole. It is Lorentz invariant, in that it unaffected by the space and time distortions of the gravitating mass. Thermodynamically, it represents the minimum size for matter in a state of maximum entropy. Essentially, it represents a stable quantity in an otherwise highly undefined system. It is fundamental and its size is governed only by the amount of mass-energy contained within it.
Conversely the curvature of the surrounding space in the two other orthogonal directions is determined by both the mass of the black hole and the cosmological constant. As we saw earlier for every object there is a distance at which the effect of cosmological constant equals that of the contained mass. At this point the space is flat and beyond curves increasingly negatively.
The condition for the existence of a black hole is that the space is acutely positively curved. Obviously, when the space becomes negatively curved it can no longer sustain a black hole. Consequently, with the current estimate of the value of the cosmological constant, a black hole of with a mass of 5.78 x
1052 kg with a radius of 8.58 x 1025 m would undergo such spontaneous disintegration.
This is the limit that the current value of the cosmological constant places on the size of a super-colossal black hole. At this point the mass-energy within, most of which, as we have seen, is trapped on the region of the black holes surface, will begin accelerating away from the black hole.
Consider, a black hole just within the limit imposed by the cosmological constant. At this point the mass-energy within is highly concentrated, extremely energetic, and consequently hot, and in a state of maximum entropy, maximum uniformity. Now add just a little more mass-energy to take it beyond the critical size and this extremely hot uniform mass-energy would dissipate rapidly in all directions uniformly. That is until, quantum fluctuations introduces minor variations that grow greater in time as the expansion continues.
You may find this highly suggestive. If so the following will also be of interest. Present estimates of the mass of the universe range from
1051 to 1055 kg.
References.
[1] Jacob D. Bekenstein, "Universal upper bound on the entropy-to-energy ratio for bounded systems", Physical Review D, Vol. 23, No. 2, (January 15, 1981), pp. 287-298.
[2] Bardeen, J. M.; Carter, B.; Hawking, S. W. (1973). "The four laws of black hole mechanics". Communications in Mathematical Physics 31 (2): 161–170.
[3] Bekenstein, Jacob D. (April 1973). "Black holes and entropy". Physical Review D 7 (8): 2333–2346.
[4] K. Schwarzschild, "Uber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie", Sitzungsberichte der Deutschen Akademie der Wissenschaften zu Berlin, Klasse fur Mathematik, Physik, und Technik (1916) pp 189.
[5] K. Schwarzschild, "Uber das Gravitationsfeld einer Kugel aus inkompressibler Flussigkeit nach der Einsteinschen Theorie", Sitzungsberichte der Deutschen Akademie der Wissenschaften zu Berlin, Klasse fur Mathematik, Physik, und Technik (1916) pp 424.
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[8] Holba, Ágnes; Horváth, I.; Lukács, B.; Paál, G. (1994). "Once more on quasar periodicities". Astrophysics and Space Science 222: 65.
[9] S. Perlmutter et al. (The Supernova Cosmology Project) (1999). "Measurements of Omega and Lambda from 42 high redshift supernovae". Astrophysical J. 517 (2): 565–86.
[10]. Adam G. Riess et al. (Supernova Search Team) (1998). "Observational evidence from supernovae for an accelerating universe and a cosmological constant". Astronomical J. 116 (3): 1009–38.
[11] Hunter G., Kowalski M., Mani R., Wadlinger L. P., Engler F. and Richardson T.. Photon Diameter Measurements. Gravitation and Cosmology: From the Hubble Radius to the Planck Scale (eds. R.L.Amoroso et al. 2002 Kluwer Academic Publishers. The Netherlands.)
[12] G. G. Hall and S. B. Jones, Information and Entropy for a Planar Box, Amer. J. Phys. 41, 213 (1973).
[13] N. Jarosik et al (2011). "Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Sky Maps, Systematic Errors, and Basic Results". "The Astrophysical Journal Supplement Series" 192: 14.
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