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| THE NEW LAWS OF _________ Paul Bird |
Gravity, in many ways, is the simplest of all known forces. Acting on virtually everything, it pulls things together in a slow and predictable way. So why is it that theories of gravity are usually the most difficult to understand and get to grips with? It is precisely because gravity is a universal force. It is difficult to stand back and observe the force in action because we, as observers, are always a part of the action. The second reason is that gravity is incredibly weak. For example a small magnet can pick a paper-clip up even though the entire mass of the Earth is trying to pull it back down. Nonetheless it is an essential force of nature. Without it, there would be no stars or planets or people. It is also, surprisingly, one of the most interesting. One of the reasons is that we, ourselves, are affected by gravity and so we can justify asking questions about what would happen to us in certain situations.
Unfortunately, due to its complexity, gravity is also the most mis- understood of the forces. A great deal of mysticism has grown up around gravitational theory, particularly contributed to by science fiction. Hence, in a lot of people's minds, gravity is connected with time-travel, intergalactic space-time wormholes and so forth. It is worth pointing out, that Albert Einstein, himself, was never awarded the Nobel Prize for his theory of gravity, the committee felt that his ideas were too metaphysical and not based on rigorous experiment. (He was however awarded it for his work on the photoelectric effect, after considerable pressure from his supporters!) The reason that his ideas became generally accepted can be traced to a variety of reasons. The most important could be Eddington's book on Einstein's theory of gravity and his subsequent much publicised trip to observe a solar eclipse and to 'prove' that light is affected by gravity. Although the results were never conclusive, through his convictions that Einstein's theory was correct, Eddignton managed to convince many fellow scientists, and the national press that the experiment was a success.
Another test of General Relativity is said to be the precession of Mercury's orbit. After taking into account all known gravitational excitations, there still seems to be no reason for that precession. The problem seemed to be solved when Einstein's theory did not predict perfectly elliptical orbits but slight precessions. The results agreed very well with the Mercury anomaly. [What is less well known, however, is that the theory does not agree so well with the observed precessions of all other planets in the solar system.]
Since gravity is a universal affect, it is important in discussing the evolution of the universe. This subject is cosmology. A good gravitational theory should predict things like the rate of expansion of the universe and also give us a good idea about the origins of the universe.
Many ancient Greek philosophers, such as Aristotle, believed that things fell because their 'natural place' is on the ground. Heavier things, obviously, fall faster than light things because a rock falls faster than a feather, for instance. They believed that the moon must be made of an incredibly light material for it not to fall to earth. Obviously, other Greeks had different ideas, but this was the accepted view at the time.
The fact that, in the absence of air, all objects would fall at the same rate is attributed to Galileo Galilei. He is said to have conducted an experiment from the Tower of Pisa by dropping two equally shaped objects of different weights from the top. The objects fell to the ground at more-or-less the same moment and so confirmed his theory.
Sir Isaac Newton's most notable achievement was to use mathematics and geometry to formalise the Laws of Motion. The mathematics he developed to explain motion is what we now call 'calculus'. His dot notation for change in respect to time is what we still use along with Leibniz's dx/dt notation which he had developed independently of Newton in France. Another important invention of Newton's was the idea of 'force' defined as mass times acceleration. The units of force still carry his name. Newton then went on to 'explain' gravity as a force which acted on all objects proportional to their masses and the inverse square of the distance between them. Some other people at that time also had the idea that gravity might be explained by an inverse square law of force, but it is thought to be Newton who first demonstrated that this law would produce elliptical orbits.
Albert Einstein's contribution was to develop the theory of (special) relativity. This theory predicts that nothing can travel faster than the speed of light. It also presented a unified picture of the 3 space and 1 time dimensions by interpreting acceleration as a 4-dimensional rotation in space-time. This means that an increase in an objects progression through space means a decrease in an objects progression through time. In other words everything moves more slowly inside an object as it approaches the speed of light. Also the mass of the object would increase hence the famous formula E=mc2. Again, other people also contributed to this theory, but it is fair to say that Einstein made the biggest contribution.
Later Einstein developed his theory of gravity, General Relativity. This described gravity as a consequence of 'curved space-time'. The main ingredients are the n-dimensional geometry of curved surfaces developed by Riemann, and his principle of equivalence which says that on a local scale (in minuscule portions of space-time) gravity is equivalent to acceleration. In other words, there should be no difference between an object in free-fall and an object in empty space. The main result of this principle is that it says that light must be affected by gravity. (Otherwise there would be a difference between a torch in empty space an a torch in free-fall.) It must be pointed out, however, that there was no evidence for the principle of equivalence. Einstein simply postulated it on aesthetic grounds. This, his opponents argue, is where he went wrong. The main predictions of General Relativity are the precession of Mercury, the bending of light and black holes.
Scalar Gravity, the subject of this book, can be interpreted as a theory of curved space (although it is not necessary). But it does not support the principle of equivalence. In other words, in this theory acceleration and gravity are two fundamentally different phenomena. Further, this theory states that light is not affected by gravity.
Richard Feynman was one of the first people to try to formulate a quantum theory of gravity. Having helped develop a theory of the electromagnetic force in terms of exchanges of photons, he turned his attention to the force of gravity. Naturally, he assumed that gravity could be thought of as exchanges of another particle called a 'graviton'. As General Relativity was the accepted theory of gravity at the time, he tried to write it as a theory of (spin-2) gravitons. What he, and other people, found was that this theory would not produce finite results. In other words, General Relativity could not be quantisized.
It is hoped that because Scalar Gravity is a simpler theory that it may be possible to quanitisize the theory. Although, this has not yet been investigated in any great detail. Scalar Gravity would be quantisized by spin-0 particles. Spin-0 gravitons were rejected by Feynman since they would not predict the bending of light.
Beyond theories of particles, it has been suggested that a theory of quantum gravity must incorporate the ideas of superstring theory. However since electrodynamics can be successfully quantisized in terms of particles it would be very strange if the same could not be done for gravity. Besides which, superstring theory in it's present formulation, only works in ten dimensions. In this book we shall be keeping to the classical solutions of Scalar Gravity and leave the theory of quantum gravity for another time.
In this book we shall be comparing three theories; Isaac Newton's inverse square law of gravity, Albert Einstein's General Relativity, and Scalar Gravity. The first, although not consistent with special relativity, is useful for comparison since it is by far the most useful theory and is the one used by NASA for instance when calculating the paths of satellites for instance. The second, although not proven by any means, is an interesting comparison seeing as it is definitely the most popular of all modern gravitational theories and is the one taught at undergraduate level at many universities. To get a good understanding of General Relativity you need know a bit about differential geometry and tensor theory. We shall resist, however, an explanation of Einstein's theory and try to keep the mathematics down to a minimum. Finally, Scalar Gravity is the theory most eluded to in this book, for the simple reason that little space has been given to it in other books. Scalar Gravity is the simplest possible relativistic theory of gravity.
Whether or not you believe that the evidence points to Einstein's General Relativity being correct, it is always useful to compare the standard theory with some other, possibly simpler, theory. It would then become clear which attributes of the theory are particular to itself and which are general attributes that all theories of gravity must have.
The Basics
Natural Units
In this book we shall use what are called 'natural units'. This is because they are formed from three of the natural constants. They are named after Max Planck who supplied the last of the natural constants and so made this system of units possible:
| value | ||
| speed of light | c | 299,800,000 m / s |
| gravitational constant | G | 6.6739 x 10-11 m3 / s2 kg |
| Planck's constant | ß | 6.626 x 10-34 kg m2 / s |
The gravitational constant gives the general strength of the gravitational force on any object. Planck's constant gives a relation between the energy of a wave or particle and its frequency.
All values are now written in terms of the following natural units:
| value | relation | |
| Planck Mass | 5.46 x 10-8 kg | (ß c/G )½ |
| Planck Energy | 1.10 x 1029 J | (ß c5/G)½ |
| Planck Length | 4.05 x 10-35 m | (ß G/c3)½ |
| Planck Time | 1.35 x 10-43 s | (ß G/c5)½ |
The advantage of natural units is that we can dispense with most of the constants in our equations and hence concentrate on the essential mathematics. For example we have, in natural units, c=G=ß=1. The least 'natural' of all our constants might be the gravitational constant G. This is because it is referring to a particular force, i.e. gravity why the others are more general. So for instance, if we were talking about theories of magnetism, we might find different units more applicable. It is often remarked upon that each natural constant is used in one of the three great physical theories. The gravitational constant comes from Newton's Laws of Gravity. The speed of light is used in Einstein's Special Theory of Relativity and Plank's constant is a corner stone of the Quantum Theory (various artists). So it is assumed that the unit values of the Planck system represent important values in a forthcoming quantum theory of gravity which should combine all three of these theories.
Natural units are not very useful in everyday life, however, for example the Plank length is billions of times smaller than the size of an atom and the Plank energy is billions of times bigger than any power station could ever produce. The only reasonable quantity is the mysterious Planck Mass. It is the mass of a speck of dust. (About a hundredth of a milligram.) In terms of atoms and such like, this mass is enormous, but it just so happens that this mass is just about quantifiable in human terms. Some people have gone so far as to say that this is no accident!
Notation
We shall use the letter r with a line underneath to represent the position vector of an object. An r with no bar is the modulus of this vector, that is the radius or distance to the origin. A letter with a bar above it represents a unit vector in the relevant direction. Thus we have the relationship:
A dashed letter such as r' represents the change in r with respect to time. This is, more-or-less, the notation that Isaac Newton invented.
Laws of Motion
Before we look at gravity we should consider the Laws of Motion since the two things can not easily be separated. Newton's Laws of Motion are written in terms of the calculus also developed by Isaac Newton.
If we know the position and velocity for a particle at time T=0 and the acceleration is constant then we can work out the position of the particle at time T=t using the Taylor series expansion:
By differentiating the series again we can find the velocity at time t.
This breaks down of course if the acceleration is dependent on position. But they are useful near the surface of the Earth where the acceleration is approximately constant. With these laws Newton worked out that the path of a cannon ball shot into the air is a parabola. This can easily be seen simply from the fact that for the constant acceleration by gravity, the height is given by a quadratic in t.
Newton's first law states that an object's velocity will remain the same unless acted on by a force. This can be neatly expressed as:
force = mass x acceleration
This begs the questions, 'What is mass?' and 'What is force?' Mass is often defined as 'the property of an object which resists acceleration'. Whereas force is 'that which causes acceleration'. We should note that when we usually exert a force on an object, the force in question is the electromagnetic force. For example by pushing something with your hand you are causing the atoms in the object to repel the atoms in atoms in your hand via the electromagnetic force. Hence the first definition really only applies to non-gravitational forces.
The most useful thing about forces is that the total force on a body can be found by simply adding together all the different forces on that body. The acceleration caused by all these forces can then by found by dividing by the mass of the object. Forces can be added according to the parallelogram rule:
Another of Newton's laws says that for every action there is an equal and opposite reaction. Again, this is a consequence of the electric repulsion between charged particles which is given by,
which is symmetric between the particles. We shall see later that this law is not quite true for the gravitational force. Indeed the whole notion of 'forces' is probably not the best language to use when talking about gravity. It is sometimes best simply to talk of the acceleration caused by gravity since this is independent of the mass of the object anyway.
Thus we have seen that when using the language of classical mechanics we must not take for granted that the same language can be used for each different force of nature.
Gravitational Force
Now we shall address the question of the gravitational force. First we should think about what effects a theory of gravity should predict. Firstly it should predict that in the absence of air, an apple would fall from a tree with almost precisely constant acceleration until it hit the ground. That would go, of course, for any other object near the surface of the Earth. Experiments have shown that this acceleration is about 9.8m/s2. The acceleration is slightly lower for higher altitudes. Secondly the theory should predict that the planets follow approximately elliptical orbits around the sun. Keppler also found some relations regarding the speed of the planets and their position on their elliptical orbits which the theory should predict. We won't go into them however since they are superfluous to simple observations of the planets position and speed which can be used to test a theory of planetary motion.
When astronomers make observations of the planets' motion through the night sky, it is complicated by the fact that the Earth is also moving around the sun. So to determine the orbit of a planet we must first determine the orbit of the Earth. How is this done? Luckily, because the Earth is tilted on its axis, it is relatively easy to determine the period of Earth's orbit. We simple count the number of days between two Summer Solstices, these are the times when the sun sets furthest in the East(?). Next we need the distance to the sun. Once we have the orbit of the Earth it is then possible to calculate what the mass of the sun is according to our theory.
The distance from the Earth to the planets can be determined by observing them from two different points on the Earth's surface and noting the different position of the planet in relation to the 'fixed' stars. To find the distance to the sun we can measure the shadows that it makes at various points on Earth at the same time.
The reason for this short digression into astronomical observations is to show that gravitational theory is not simply an abstract subject but can actually be tested against observations of the movements of the planets in the night sky. Whereas if we start to talk about black holes, event horizons and so forth, we should be wary that we may be stepping beyond what our theory is required to explain. (Remember that Albert Einstein himself was not awarded the Nobel Prize for his gravitational theory for the very reason that his theory was considered too 'metaphysical'.)
The other side of the argument is that although Isaac Newton's theory is the most practical and reliable gravitational theory to date, it is nonetheless inconsistent with experiments that we might one day in the future be able to do. In other words, they are mathematically inconsistent.
Astronomical Values
Since gravitational theory is mostly concerned with movements of the sun, moon and the planets, it would be useful to know some details about them. We shall use some of these results later in the book. Beware however, because a value such as the mass of the moon is calculated using a gravitational theory and so would be different depending on the theory we use. However, this difference is too small to be concerned with.
| Body | Distance to the sun / 109m | Mass / 1021kg | Radius / 106m |
| Sun (Sol) | 0 | 1,988,430,000 | 695 |
| Earth | 150 | 5,980 | 6.4 |
| Mercury | 58 | 33 | 2.4 |
| Mars | 228 | 642 | 3.4 |
| Moon | 0.38 (to Earth) | 73 | 1.7 |
So you don't need any particularly complicated equipment to check your theories of gravity. Many planets are visible with the naked eye or through binoculars and their paths are traced out in the night sky in relation to the stars. The only luxury that one might require is time - since they do tend to move extremely slowly.
It is amazing to think that many ancient civilisations could predict eclipses of the sun and moon even without the aid of a gravitational theory.
Finally, having grown up knowing all about the force of gravity, it is difficult to imagine a time when people did not know what gravity was and that it was responsible for the movements of the planets. Yet it is perhaps not so surprising that the concept of 'force at a distance' took so long to take hold. For example it took the discovery of the rock magnetite, and its ability to magnetise iron to create magnets which could attract other metal objects for 'force at a distance' to be made shown explicitly. In fact, it was suggested even before Newton's time that the moon might be held in orbit around the world by magnetic forces. The jump from this to the postulation that a separate new force of nature is responsible for gravity is quite a large one.
Gravity is a force which moves an object (A) towards another object (B) with an acceleration dependent on the mass of the object B and the position and velocity of the object A.
Isaac Newton showed that an acceleration proportional to the inverse square of the distance between the objects would give elliptical orbits of the planets (which is what was observed at the time). His law can be expressed as:
The barred r is a unit vector in the direction of the object B. This can be solved in one dimension (for example for an object falling directly towards object B) to give:
Unfortunately, this law means that objects near to the speed of light will be accelerated faster than the speed of light which we know does not happen. Can we modify Newton's law so that it still predicts elliptical orbits but also satisfies the laws of Special Relativity? We can (for r>2m at least). For a particle attracted to a stationary body, the New Law of Gravity can be expressed as:
So now the acceleration is dependent on the velocity of the object (A) as well as its distance from object (B). If the velocity is 1 (the speed of light in our units) then the acceleration is zero. Notice that this equation approximates to Newton's equation for small velocities and large distances. The equation can be solved in one dimension to give:
At first our new Law seems to break down at |r|=2m but this is not the case, since the solution to this equation works for all r>0. This law only works for a stationary object (B) since, as the gravitational force travels at the speed of light, the acceleration is dependent on the position of the object (B) at a time in the past. Alternatively we can say that the object (B) makes changes to the gravitational 'field' and the acceleration of object (A) depends on changes in the gravitational field. Changes in the field itself travel at the speed of light. The field must satisfy the continuity equation (or 'wave equation'). As according to Special Relativity mass is equivalent to energy (E=mc2), the gravitational field must interact with the energy (T) of the object (A). All this information allows us to write the field equations for this system:
Which can be solved for the empty space (T=0) outside the objects A and B. Using spherical co-ordinates the above becomes:
The multiplicative constant, a, is irrelevant since it can be divided out of the wave equation. The New Laws of Gravity can then be found by the 'acceleration equation' which we shall derive later:
We should note here that all these accelerations are with respect to the normal standard time of classical mechanics of an observer at 'rest' in regards to the 'fixed' stars and not the apparent time observed by someone travelling with the object. Although, it has been shown that 'time is relative', it is particularly useful to select out one particular time frame as we have done. That way we have a closer match with classical mechanics and we won't get in such a muddle.
Kinetic and Potential Energy
Kinetic Energy is a value that is conserved in interactions. It is defined for a single particle of mass M and velocity v by:
In a gravitational field the velocity of a particle changes, but according to the rule that energy must be constant, we introduce a value called 'potential energy' which is defined using the relation:
constant
So potential energy is defined up to an additive constant. For Newton's law of gravity we have:
Where a, b and c are constants. For Scalar Gravity the above relationship no longer applies, but it turns out that we can use a different relation:
constant
Then we have:
Although now it is no longer possible to talk of 'gravitational potential energy'. We can easily show that the values of kinetic and potential energy for scalar gravity approximate to those in Newton's Laws. If the speed is small and the distance is large then we can neglect terms bigger than quadratic in u and linear in r.
Which gives the values for Newtonian kinetic and potential energy up to an additive constant which we can ignore.
Potential field
A more appropriate definition of the gravitational potential might not be the scalar field but the logarithm of the scalar field, h(x). This is because a potential is given up to an additive factor whereas our scalar field is given up to a multiplicative factor. But if we take the logarithm then the potential is given up to an additive constant:
This notation can be useful, for example we have the identity:
Which, when combined with our acceleration equation, tells us that the acceleration is proportional to the divergence of the gravitational potential, for a time invariant field, which is what we should expect. This is very similar to Isaac Newton's theory. We can identify the field h(r) with the graviton wave function. Writing this out in a series:
It shows that a quantum theory of gravity involves contributions from graviton interactions of an arbitrary number of graviton particles. For a static field around a spherically symmetric source we get:
Total Energy
The total energy for a static system is found as follows.
We see again that the energy is real so long as the velocity is greater than the speed of light as r<2m and smaller than the speed of light in normal circumstances (r>2m).
Gravity on Earth
Using the radius of the Earth, which we can measure quite easily by making a few voyages around the globe and the acceleration due to gravity which we have measured to be around 9.8m/s2 we shall now calculate, using Newton's Laws of Gravity, the mass of the Earth. We must include the gravitational constant, G, in our equation which is basically a conversion factor into our system of units. We have the equation:
This gives for the mass of the Earth:
Which fits with the result from our table. We can check that this gives a reasonable result by calculating the volume of the Earth and calculating it's average density to see if this gives a reasonable enough result - which it does seem to. Incidentally, the density of the Earth can also be probed by detecting the paths of Earthquakes as they ripple through the Earth. To this date, all these tests are in good agreement with each other.
The value of G, our conversion factor, is the most difficult to find. It is usually found using a 'torsion balance' and measuring the very small rotations that are caused by gravity.
We can derive the usual formula for the potential energy near the Earth's surface as a function of the height by subtracting the potential at the Earth's surface from that at the desired height h:
Because h is so small compared to the radius of the Earth the formula predicts that for small heights we should see approximately constant acceleration.
The reason that we have used Newton's Laws is that they give a good enough degree of accuracy for large distances and slow velocities. It is simply a waste of time using a more complicated theory.
Elliptical Orbits
The greatest achievement of Newton's Laws of gravity was probably the prediction of elliptical orbits from the inverse square law. Before that, the orbits of the planets were first thought to be circular - because that was the most perfect shape and then later elliptical for exactly the same reasons. However Newton had now shown that a simple law could explain the entire movement of the celestial bodies and so put an end to the speculation of the exact shape of the orbits of the planets. However when Albert Einstein proposed his theory of General Relativity, it seemed as if the orbits might not be so perfect after all. It is interesting that Scalar Gravity, though a relativistic theory like Einstein's, does in fact predict perfectly elliptical orbits like Newton's theory.
It is quite easy to show that the orbits predicted by Newton's Laws and Scalar Gravity are elliptical and that those predicted by General Relativity are not. Using polar co-ordinates (see Appendix) and the fact that the total energy is the same throughout the orbit we have for Newton's Laws:
Since the force only acts in the radial direction, the angular momentum must also be constant. This is written in polar co-ordinates,
which gives us:
Now we change the dependency on r to the angle h,
to get:
This is an ordinary differential equation and can be solved quite easily with the aid of the substitution u=1/r to get:
for certain constants a,b and c. This is the equation of an ellipse in polar co-ordinates.
For Scalar Gravity we start with the equation:
Which has exactly the same form as the equation for Newton's equation and it also leads to the equation of an ellipse for different constants a, b and c.
For General Relativity we start with the equation:
which, because it has a different form to the last two equations does not lead to an equation for an ellipse (it has higher powers of 1/r). It does, though, approximate to an ellipse but there is a precession of the perihelion - which means that the point where it is farthest away is slightly different after each full orbit. It is very difficult to observe this precession for several reasons. Firstly the planets are moving very slowly, secondly other affects contribute to the precession such as the gravitational attraction of other planets, interstellar dust and even the shape of the sun. Even accounting for all these affects, many scientists (though not all) believe that the evidence is in agreement with the precession predicted by Einstein's theory.
The above differential equation can be solved using elliptical functions. So we have the confusing situation in which the equation of the ellipse is written in terms of circular functions (sin, cos) whereas the equation which is not an ellipse is written in terms of elliptical functions (sn, cn, dn).
Deriving the Acceleration Equation
We have already shown how to derive the New Laws of Gravity from the scalar field using the 'acceleration equation', but we have not yet justified doing this. The standard derivation of this equation is usually done in terms of variations of the 'Action' so we shall follows suit. We use the variational principles to derive it, in the same way as is done in General Relativity. First we write the action in terms of the clock speed s:
We then make a variation of this action using the Euler-Lagrange equations:
This becomes the following:
After some rearranging and putting everything in terms of the 'standard' time t, we have:
We can simplify this further. By taking n=0 we get the equation involving the time variable which is:
Now we can substitute this back into our equation to get rid of any relation to the clock speed ds/dt:
In our usual notation this is:
which is exactly the acceleration equation that we were looking for. This is arguably the most important result of Scalar Gravity. Provided we know the potential field at any time, we can work out the acceleration of a particle given its position and velocity.
The reason that the Scalar acceleration equation is so simple is primarily because Scalar Gravity is a 'flat space' theory. This result shows that in Scalar Gravity it is sometimes simpler to use 'standard' time rather than the clock rate of the object itself. This makes the visualisation of things much simpler. In General Relativity it is mostly simpler to think in terms of the clock rate of the object in question.
Another of the significant affects of gravity is the movement of the tides. A few days of observation would tell us that the tides come in and out every 12 hours so and half these times coincide when the moon is highest in the sky. A few months observations at night would tell us that the tides are highest during a full or new moon and lowest during a half moon. If we made observations for a few years we might notice other coincidences. We could use these conclusions to make a very accurate model of the tides but what are the scientific explanations of these principles? Obviously the tides must be caused by a gravitational affect, primarily of the moon but also of the sun.
These effects are due to the fact that on a large body, the gravitational forces are different between the top and bottom of the body. This causes a stretching effect.
The amount of stretching is dependent on how rigid the body is. Thus these tidal forces have little affect on the solid body of the Earth but a small but noticeable affect on the oceans and hence we have tides. Unlike length contraction of special relativity, this gravitational effect is not independent on the type of matter involved.
Let us now consider the problem of the acceleration caused by two stationary bodies. The easiest method is to solve the field equations. We shall use a co-ordinate system given by the distances towards each stationary object. The field equations are solved easily to give:
Then we can use the acceleration equation to give the acceleration in terms of the gravitational potentials:
where the barred letters are unit vectors in the direction of the objects. This result easily generalises to cases with three or more stationary objects. When the objects are very close together, their light horizons join up to form a single shape given by the formulae:
Moving sources
The case of moving sources is more difficult to solve. In fact very few cases are known. If the source follows some known mathematical curve then we can calculate the field caused by the source fairly easily. On the other hand if we have two or more bodies both affecting and being affected by the field then the situation becomes more difficult and it becomes necessary to use computer simulations. These split the field into many small cells and then use the differential equations to determine what happens to this cell at time t+1 using the knowledge of the cells surrounding it. This is similar to the way fluid systems such as the weather are calculated at the meteorological office.
One important point about field theories is that to find the affect the field has on a particular particle, we must first subtract the affect that the particle had on the field. That is to say, a particle does not affect itself. (If it did, the field would be infinite at this point and so cause the theory to collapse.) The reasons for this require a knowledge of quantum field theory, which explains the field as the exchange of quanta such as photons or gravitons. Even then, the self absorption of exchange particles has caused a headaches to some of the best physicists last century. Even now the rules which are employed to get rid of these infinities, although they work, are on somewhat shaky mathematical grounds.
Inside the Light Horizon (r<2m)
As we have already seen, orbits exist which pass through the false singularity at r=2m and back out again. The speed of an object at the radius r=2m is always the speed of light. Further, the speed of an object inside the Light Horizon (r<2m) is always above the speed of light. But there is no contradiction with Special Relativity here since the clock speeds of the objects is still real.
The total energy for a particle is constant throughout the orbit. It is given by:
The total energy also tells us about the type of orbit by considering the limit as r tends to infinity and realising that for a free particle in empty space k=1 or k>1.
If k<1 then the particle is in an elliptical orbit around the object. (It does not have enough energy to escape the gravitational pull. In fact, its total energy is less than its rest energy so it cannot become a free particle!) If k=1 then the particle will follow a parabolic path and if k>1 then the particle will follow a hyperbolic path.
When the particle is at a distance r<2m then the particle will be travelling faster than the speed of light. The clock speed for a particle in free fall is:
We use the positive square roots in both cases which gives a modulus function. This is because clock speed must be positive.
So what would an observer see if they were in an orbit that passes through r=2m? A person observing from the outside would see some strange optical affects such as when the observer is accelerating towards the person at superluminal velocity, time would appear to be going backwards! But this is purely an optical affect. The observer would also see similar optical affects in regards to the person outside the Light Horizon.
[How would observers measure the speed of light? Due to tidal forces, the observer will be stretched in the direction of the massive object, and squashed in the direction perpendicular to the orbit so the speed of light may be measured the same for the entire orbit?]
Unfortunately, this result doesn't open up the possibility of faster than light travel since a particle only travels faster than the speed of light when close to a particle r<2m. Although an interesting question comes from this: Can an object A pass a signal to an object B faster than speed of light by sending it around an orbit through the light horizon? It is given that the objects A and B are both outside the light horizon. This is not so clear since although the signal has longer to go, it is also going more quickly. The solution to this problem may tell us something about where the laws of special relativity must hold.
Lastly, we have called the distance r=2m the 'light horizon' instead of the 'event horizon' as in General Relativity since objects can pass over this horizon in real time in Scalar Gravity. Further more at the distance r=2m they will all be travelling at exactly the speed of light hence the name 'light horizon'.
The Death of Stars
Another question is what happens to a star when it dies according to Scalar Gravity? Obviously it won't form a black hole, but is there something similar that Scalar Gravity predicts. In other words can an object be compressed below a radius of 2m? This is not as straight forward as it sounds since we have already seen that as objects pass the radius of 2m they must be travelling at the speed of light. When particles get this close, we must also consider other forces such as the electromagnetic force. Closer than this, we must consider the effects of the Pauli exclusion principle which acts like a force pushing Fermion particles further apart and Boson particles closer together. An interesting question here is if, taking all these forces into account, there is a maximum density that a star can be compressed into. According to standard General Relativity it is thought that there is no maximum density and a large star will collapse into a black hole and the particles inside the black hole will end their time in a singularity.
Gravitational Field of Charged Particles
Since the gravitational force is caused by all forms of energy, for a stationary particle it is not only it's rest mass which causes a gravitational field but also it's charge. This is due to the electromagnetic field contributing to the energy. Some interesting consequences stem from this. The consequences can be split into two cases. The first is when the charge is substantially greater than the rest mass of the object such as an elementary particles like electrons or charged atoms, ions. The second case deals with objects such as stars (or neutral atoms) in which the mass is much greater than the charge due to the positive and negative charges making the whole object more or less neutral.
In both cases the gravitational potential is given by the equation:
Which gives:
We can write this in a more useful way as:
Which if the charge is much greater than the mass is always positive and so the clock speed for a particle orbiting the object is always positive. If we imposed this condition on all elementary particles we should expect that neutral particles like the photon and the neutrinos should also be massless. Unfortunately this condition breaks down for the case of the Zo particles and of course for composite particles such as the neutral pion.
The velocity corresponding to this potential is:
Which means that beyond a certain radius, depending on the initial speed of the test particle, the gravitational force becomes repulsive. This is not just for charged particles but all particles feel this repulsive force caused by gravity.
General Relativity vs. Scalar Gravity
Scalar Gravity and General Relativity do actually have much in common. For example they both can be interpreted in terms of space-time metrics. The difference is that Scalar Gravity is restricted to the class of flat-space-time metrics. We can write Scalar Gravity as a metric theory by defining a metric as follows:
So we have the metric equation:
We can derive the clock-rate equation from this:
Some of the differences between the two theories are discussed below.
Scalar gravity predicts that any object travelling at the speed of light is not affected by gravity. On the other hand, General Relativity predicts that because space-time is interpreted as a curved manifold, light will follow curved paths and hence will appear to be affected by gravity. It is near impossible to measure the bending of light by gravity. Light is too fast and gravity is too weak. Most experiments rely on astrological observations such as observing stars close to the sun during a solar eclipse. Unfortunately this is not as simple as it sounds and due to effects such as the refraction of light by the earths atmosphere, the sun's corona and interstellar dust, the evidence so far is inconclusive.
Other 'evidence' for the bending of light is the observation of so-called gravitational lenses in which light from a far off nebulae or star cluster is bent around an intervening galaxy. The result of which rings are seen around the galaxy. The interpretation of these rings as the result of gravitational lensing is controversial. Other explanations for these phenomena such as the remnants of super novae explosions and so forth have been put forward. It is even possible that light could be refracted but normal interstellar dust. Besides which, statistically less gravitational lenses are observed than are predicted. In short, the jury is still out.
Scalar gravity predicts precise elliptical orbits for an object orbiting a stationary body in otherwise empty space. Unfortunately situations are never as simple as this and when comparing theories to observation we must include effects from all the planets, asteroids, interstellar dust and so on. General Relativity predicts that if all these things were accounted for, the orbits of the planets would still not be perfectly elliptical and there would be a precession of the orbit. The result predicted for Mercury matches quite well while it doesn't match so well for the other planets. Many other effects that might be responsible for the shift in Mercury's orbit have been dismissed such as the shape of the sun and other unknown planets. In short General Relativity is just one of many theories to explain Mercury's strange orbit.
This is probably the most famous of all the predictions of General Relativity and the one that Albert Einstein was most against. It is a result of the fact that light is affected by gravity in this theory. The result of this is that there should exist objects for which the gravitational force is so strong that light cannot escape from them. Hence making them invisible. Further if an object enters this 'black hole' it spirals towards a central 'singularity' and is destroyed. Scalar Gravity also predicts strange affects close to dense massive objects but the theory does not have singularities or black holes.
Einstein's first theory of gravity did not in fact predict the big bang. However, after conversations with Edwin Hubble who had measured the speed with which galaxies were flying apart, Einstein changed his equations so that it they would predict a big bang. He later called the term he had removed the 'greatest mistake of my life,' which as many of today's scientists would agree was a bit dramatic. Nowadays many people use Einstein's equations to investigate the big bang. They do this by finding solutions to his equations in which the universe is expanding and see what was happening when you put t=0. The big bang is very similar to the time-reverse of a collapsing star except that a star is finite while the universe is (as far as we know) infinite. Thus we can say that if a star can get big enough to collapse into a black hole, then the universe itself must have started out like this.
Scalar Gravity also predicts the expansion of the universe, as does Isaac Newton's theory. The differences lie in that General Relativity requires that the universe started in a singularity whereas the case is not so clear for Scalar Gravity.
Conclusion
Although General Relativity is by far the more complicated of the two theories, and predicts such strange objects as black holes and singularities, it does seem to fit the experimental data better than Scalar Gravity. However, most tests of General Relativity are designed to compare it with Isaac Newton's theory of gravity to which General Relativity always comes out on top. The tests to compare General Relativity with Scalar Gravity are more subtle and so the experimental data is far less accurate. A definitive answer could be given if the affects of gravity on light were measured by an Earth based experiment which due to the speed of light and the weakness of gravity would be very difficult and expensive with today's technology. So for now we must simply rely on astronomical observations as evidence, some of which are open to interpretation.
An interesting question is what would have happened if Scalar Gravity was found before General Relativity? In many ways Scalar Gravity is the natural extension of Newton's Laws to incorporate Special Relativity. Would scientists have been so ready to give that theory up in favour of a theory of curved space-time? On the other hand, it is possible that the formulation of Scalar Gravity would not have been at all obvious until after a theory of curved space-time had been formulated. All we can do here is put forward both theories on an equal footing and hope that one day a combination of indisputable experimental data and mathematical consistency will separate the two.
On The Experimental Differences
Let us try to design an experiment which will separate out General Relativity from Scalar Gravity. Despite the many differences between the theories, we shall see that it is not so easy to separate them experimentally.
Let us design a box in which the experiment is to take place. The box is of width d and is resting horizontally on the ground. A beam of particles of velocity u are shot from the top left hand corner of the box. The beam of particles hit the right side of the box and we are to measure the distance from the top right hand corner of the box to the target.
According to our theories, the beam will end up lower down if General Relativity (G.R) is true than if Scalar Gravity (S.G) is true.
For example if we shone a laser from the top of Ben Nevis in Scotland to the top of Canary Warf in London, England, the light should be drawn towards the downards towards Earth by about a thenth of a millimetre.
We can work out the absolute differnece as follows.
on the Earth's surface given that:
The distance down according to Newton's formula would be:
the Earth surface approximation gives (because the second term is constant):
We find that the distance downwards according to S.G is
The Earth surface approximation gives:
We find that the distance downwards according to G.R is
The difference between these is then:
Which shows that the difference is directly proportional to the time and that the difference is greatest for
which is as expected.
One of the more interesting predictions of scalar gravity is that a particle travelling at high speeds will experience less gravitational force than a particle at rest. In particular, as it reaches the speed of light, the gravitational force becomes zero. One might ask if this is of any practical use in building the mythical antigravity device. Unfortunately, the answer seems to be no. Let us see what it takes to construct a device which decreases the gravitational force on itself by 1%. We shall start with a ring of metal with a radius of one metre. We shall now, but some means (probably using magnets) being to rotate this ring
through the axis of symmetry. According to our laws of gravity the acceleration of the ring by gravity will decrease by:
Where n is the number of rotations per second. For a decrease of 1%, this means u must be 10% the speed of light:
So would have to spin the ring at approximately half a million times a second - and the ring would be by no means levitating this point the gravity only being reduced by 1%. For example if the experiment was done on Earth, the acceleration due to gravity would still be around 9.7m/s2.
How would this spinning ring be used to slow down the acceleration of a non-spinning object? Lets imagine an object of mass m attached to the spinning ring in some way (magnets again?) then the acceleration of the combined system would be:
And so, if the mass of the object is not too heavy, the combined system will still have less gravitational acceleration.
Another way to look at this is that by spinning the ring, we are adding energy to the system, and the more energy it has, the more freedom it has and the less it is 'trapped' within the gravitational system of the Earth. In a similar way, that adding energy in terms of photons to electrons in an atom, the further away the electron becomes from the central nucleus.
So apart from the obvious practical difficulties, it is theoretically possible to construct an antigravity machine providing the laws of Scalar Gravity are correct. Such a machine would look like this:
However, the energy required to reduce the gravity by 1% is:
That comes to 500 million megajules - which is quite a bit more than we could get from a gallon of petrol! For higher speeds we use the relativistic equation for increase in energy and we find that a 50% reduction in gravitational acceleration needs 37 billion megajules and to decrease the gravity to 1% needs over 809 million billion megajules.
Lets put aside antigravity machines for now and ask if there are any other tests of this theory. For example, what experiments have been done on the gravitational forces on high speed particles in particle accelerators such as CERN? Unfortunately the question is not so simple as that, since in order to keep the particles going around the ring at CERN, powerful magnets are used which totally overpower any gravitational affects. (It has not even been experimentally proven that antiparticles aren't repelled by the gravitational field - although consistency requires that they aren't.) What of other high speed particles such as the electrons in atoms? Would this not alter the affect of gravity on different substances with different arrangements of electrons? Indeed this would be the case if Scalar Gravity is correct and thus the equivalence principle is wrong. Again, this is very difficult to test as an electron is but 1/1836th the mass of a proton and the predicted differences in the affect of gravity are negligible. There are experiments planned by the French to measure the difference in gravitational acceleration between two different substances in the vacuum of outer space. Any difference in gravitational acceleration will disprove the equivalence principle - although I'm sure a variety of other explanations will be put forward for the discrepancy.
Because the universe is so 'smooth' it is believed that there must have been a time when all the particles of the universe were very close together and then they all moved apart at speeds greater than the speed of light. This inflationary period is predicted by Scalar Gravity since when the particles were below a radius of 2m from each other they must necessarily be moving at speeds beyond the speed of light.
Many models have been suggested for the universe. Either the universe starts from a big bang singularity. Or it has collapsed from a previous universe and 'bounces' back into the universe we know today. This scenario is the one supported by Scalar Gravity. Although when particles get this close together we must also consider other quantum effects such as the creation and annihilation of particle antiparticle pairs. Either way, Scalar Gravity does not have singularities and so the only way the universe could have a beginning is if all information was somehow lost from the collapsing universe before it bounced back to form ours. The only way this could happen is if the universe reached a state of minimum entropy. That is the point where the particles are closest together must be the most ordered state possible. The problem of this initial state is often referred to as the 'initial conditions'. Often people have tried to get around the problem of finding initial conditions by proposing that there aren't any. For instance if the universe started in a singularity. Other imaginative solutions have been proposed which we shan't go into.
Stress-Energy
In General Relativity, this quantity is a matrix (technically a 'tensor') quantity. However in Scalar Gravity it is much simpler. It represents the energy of a distribution of matter or radiation and is usually written in terms of density and pressure as follows:
If we were to be more precise then we would write this Stress-Energy function in terms of all the different fields of all the different elementary particles and forces. However we are not interested in them here, only their overall density and pressure.
In the following, we use the fact that the average gravitational potential is inversely proportional to the density. This is required for conservation of energy. The equation we have to solve for cosmology is the following:
Now since the density is inversely proportional to the 'volume of expansion' we get,
So we have the simple result that the volume of the universe increases as the square of the time since the big bang. This is the same result that we get from both General Relativity and Newtonian gravity. The difference here is that in Scalar Gravity, the universe can start off from a finite size, d. In fact this model is one in which the universe before the big bang collapses to a finite size and then re-expands. We might ask, what force is it that stops the universe collapsing to a point-like singularity? But this is the wrong question to ask since the expansion of the universe is not related to forces.
Also, unlike General Relativity, the conservation of energy holds throughout. This being due to the gravitational 'potential' cancelling out the kinetic movement of the expansion.
Another interesting situation occurs if we set d<0. Then we get a universe which starts from a big bang, expands and then re-collapses. Continuity arguments would suggest that the universe which starts out at a finite size is more likely since this does not involve any 'singularities'. Although, we should note that for the situations near the beginning (or indeed end) of the universe we need to include other physical effects such as the electromagnetic forces between particles and other quantum effects.
Hubble Constant
The Hubble constant defined as the speed of expansion divided by the size of the universe is, in most theories, proportional to the age of the universe. This is assuming that the expansion follows approximately a power law:
Thus from the above we should expect the age of the universe to be around 2/3 smaller than the Hubble time.
We can measure the Hubble constant for observations of the red-shift of far away galaxies and by guessing the distance to them by observing the brightness of certain familiar stars. The Hubble constant is approximately:
1/H = 1010years
This gives the age of the universe around 12-15 billion (109) years old. Other measures of the universe such as calculations star formation and observations of the composition of stars in distant galaxies, confirms that this value is about right. At this time it is not wise to be any more accurate than this considering that the official age of the universe given by astronomers and cosmologists changes every few years. But it does confirm that the expansion of the universe follows approximately a power law.
Just as electromagnetism explains the propagation of electromagnetic waves such as radio waves, light waves, gamma rays, X-rays and so on, a theory of gravity should explain the propagation of a new kind of radiation called gravitational radiation. In terms of quantum theory, this is a natural consequence of thinking of gravity as the exchange of 'gravitons'. Unlike electromagnetic waves, however, gravitational waves do not have a polarisation - that is if you take the Scalar Gravity model. If you take Einstein's General Relativity then gravitational waves must have two polarisation's! In Scalar Gravity, gravitational waves arise naturally out of a simple time dependent solution of the wave equation. For instance for waves emanating from a spherical source we have:
A solution to this is:
There is no analogue of gravitational radiation in Isaac Newton's theory since, in his theory, gravity acts instantaneously between two points in space.
The most important outstanding problem in modern physics, it could be argued, is of finding a quantum theory of gravity. It is known, for instance, that Albert Einstein's General Relativity, when quantisized into a theory of interacting spin-2 gravitons is not finite. That is, it gives infinite probabilities. Extensions of this theory to incorporate particles of lower spin - the 'supergravities' also failed to be finite. (It was hoped that the sum of the integer spin loops would cancel with the half-integer spin loops). What's more, the largest of the supergravities 'N=8 Supergravity' does not even predict enough particles to include all those that we know of. 'N=8' is thought to be the largest since higher theories must incorporate particles of higher spins which was thought to be inconsistent.
Scalar Gravity, however, can be quantisized into a theory of spin-0 gravitons. Theories of spin-0 gravitons are usually rejected because they do not predict that gravity affects light. However, this is not a problem for our purposes. Further, since we no longer require that all spins be below 2, we can find larger scalar supergravity theories. The most promising is N=12 Scalar Supergravity. It contains the following sets of fields:
| Field |  |
 |
 |
 |
 |
 |
| Name | Graviton | Gravitino | Boson | Fermion | Higgs | Higgsino? |
| Predicted Particles | 1 | 6 | 16½ | 27½ | 30 . 95 | 24 . 76 |
| Known Particles | 1 | 0 | 12 | 24 | ? | 0 |
*Note: I now believe that indeed N=8 is the correct way to go and that the hadrons such as the protons and neutrons are not made of quarks, but of leptons held together by a weak-like force so that the charges of the leptons contintually change and it is ownly their average charge which is fractional. N=8 says that there are 7-leptons (1 more than we know of) and 7 bosons (1 photon 3 weak force and 3 more weak-type forces) and 1 gravitino (with 8-components). This is in contrast with N=8 supergravity which has 28 bosons and 56 fermions! If we accept N=8 then we don't have to add higher dimensions to make the theory work. The rest of this chapter is still relevent if one subsitutes 8 for 12.
The number of particles are calculated using the fact that the fields are anti- symmetric in all the indices and that a particle of spin-N is described by 2N real variables. What we find in this theory is that the spins of the particles (equal to half the number of indices) is two minus the accepted value. But there is nothing fundamentally wrong with this, since integer particles remain integer particles, and importantly, the bosons remain vector (spin 1) particles.
We shan't go any further into this subject since this requires knowledge of quantum field theory and superspace. But we shall just note that besides N=12 giving the best fit for the number of particles, it also suggests some interesting geometric interpretations. For instance for spin-N particles, a maximum of 6/N can meet in an interaction (less at lower energies). Now, for N= ½, 1 and 1½, this gives exactly the number of vertices of the triangular faced Platonic solids hence the paths of the particles to or from the interaction could be equally spaced in 3 dimensions. [Also, the 8 components of a fermion can be related nicely to the 8 vertices of a cube, in which the indices of the field are given by the 3 edges coming into the vertex numbered 1 to 12.]
If this theory fails to be finite (which is quite likely considering the history of particle field theories), the next step would be a theory of strings or membranes. There doesn't seem to be any theory beyond membranes since N=12 can have a maximum of 6 dimensions (3+1 space-time and 2 membrane-surface co-ordinates) while still having the possibility of giving finite answers. String theory cannot predict any more particles than a particle theory since one must be an approximation of the other but the method of calculation might lead to finite results. The current difficulty with string and membrane theory (apart from being in 10 and 11 dimensions) is that while all string diagrams can be classified, the problem of classifying 3 dimensional 'surfaces' has not been solved. In fact, it has been shown, that 4 dimensional 'surfaces' and beyond cannot be classified since it can be shown that it is impossible to construct an algorithm to determine whether any two surfaces (given in an appropriate coded form) are the same i.e. can be transformed into one another by successive manipulations. However a theory of strings would probably only be of mathematical interest since most results can be determined through the approximation of point particles.
The diagrams for a theory of Scalar Supergravity are very simple. They simply involve graphs in which the lines are given up to N numbers from 1 to N. A line with no numbers is interpreted as a graviton, those with one number are components of a 'gravitino', those with two numbers are components of a boson and those with three are components of a fermion and so it goes on.
The only condition is that no lines with the same digits on them can meet at a vertex. This restricts, somewhat, the number of interactions possible. Each diagram is given a number associated with the probability of it occurring. Swapping any pair of numbers in a diagram gives the negative of that number. This anti-symmetric nature of the diagrams is responsible for the Pauli exclusion principle which says that identical fermions cannot occupy the same space. (More precisely, the probability that two fermions are close together becomes lower than normal while the probability that two fermions are close together becomes higher than normal.) This principle is the reason why the electrons in atoms form distinct shells and hence why we have so many distinct chemical elements. It is also the mechanism through which photons in a laser stay together for long distances.
As we have seen, a graviton which is represented by a line without any numbers can occur an arbitrary amount of times in an interaction. This means that it can affect all particles and hence gives a reason for the universal nature of gravity.
For those that know about such things, the Feynman rules for the graphs of N=12 Scalar Supergravity include the following:
Vertex
In the vertex below the bold letters refer to different momentums and the light letters are the indices of the various particles.
This vertex contributes the following term:
Propagator
The line below,
contributes the following term:
One of the basic elements of these graphs is the totally anti-symmetric tensor which has values one of the values {-1,0,1}:
If any two indices are swapped this becomes the negative:
The letters a,b,c and so on range from 1 to 12. This embodies the Pauli exclusion principle and Bose/Fermi statistics. The second element is the 12x12 momentum matrix. The matrix must satisfy the condition:
This restricts the matrix to be a 4n x 4n matrix since the smallest linear matrix which satisfies an equivalent condition is:
Which is a 2-spinor matrix (due to the fact that it can be written as a 2x2 complex matrix). Our fundamental 12x12 momentum matrix is a 3x3 block matrix with the above matrix in the diagonals. This 3x3 block is likely to correspond to the SU(3) representation of the strong-force.
The various terms are multiplied together and the integrals are taken over the free momentums. This gives, after further manipulation, a value of the graph in question. There are further rules which we shan't go into but it gives a fair idea of the basic ideas.
One thing you may have noticed is that all the graphs all look more-or-less alike. How then can we have seemingly different particles such as electrons, neutrinos, and quarks? And equally, why are the forces such as electro- magnetism, the weak and strong forces so different? The answer seem to lie in the process of 'spontaneous symmetry breaking'. An example of this is crystal formation. Above a certain temperature a liquid has a general symmetry in that each bit looks like any other bit. But as it freezes, the symmetry breaks and it forms crystals whose edges pick out certain directions as special. Similarly with particles in the universe, as the universe cools the background fields are expected to take configurations which specify certain particles as special. The fields responsible for this are called the Higgs fields (after the Scottish physicist who thought of a similar process in connection with superconductors). The particular configurations of the Higgs fields pick out certain particles and give them mass. It is thought that above a certain temperature, as the background Higgs fields break out of their configurations and all fermions and bosons have the same masses, all particles would be indistinguishable from one another. This is also known as grand-unification.
We might finally just note that the number 12 (and the number 24) are very important numbers in mathematics. For instance, 12 is the solution to the question: In what number of dimensions N, can you pack (N-1) dimensional spheres with the least space between them? So there are intriguing hints at the links between the theory of Supergravity and abstract number theory. Should we be surprised at this? Probably not both are in essence dealing with complex logical structures. And obviously any bit of mathematics can be linked to any other bit in a finite number of logical steps. However an interpretation of Scalar Supergravity in terms of number theoretic ideas would certainly be very interesting.
So in conclusion we might argue that the current difficulty in supergravity theory is to somehow merge the 4 dimensional 'N=8 Scalar Supergravity' theory which has the wrong number of particles with the 6 dimensional 'N=12 Scalar Supergravity' theory which has the wrong number of dimensions. To explain the discrepancy we might have to resort to string or membrane theory to somehow incorporate the extra two dimensions or some other trick yet to be invented. It is interesting to note that the 2-spinor formalism can incorporate another pair of energy-like momentums (and the corresponding pair of time-like dimensions), although it is then impossible to write it as a 2x2 complex matrix. Whether these extra 2 time-like dimensions are significant, whether they correspond to string parameters, or whether they are simply unobservable remains to be seen.
It is interesting to speculate that time is infact a 3-vector just like space and that not only does time have a magnitude but also an angular component (the big bang being the origin). Although it turns out that these angular co-ordinates corresponding to compact dimensions are unobservable.
Position Representation
Although field theory and Feynman diagrams are often framed in terms of the momentum representation, we should strictly frame it in terms of position, since this is in many respects more fundamental whereas 'momentum' is merely a description of the movement of waves through space. The momentum wavefuntion was created by decomposing the normal wave function into waves as follows:
From this defintion of momentum it allows us to make the following substitutions:
This substitution is implied from our defintition. The propergators in the momentum representation can be thought of coming from more fundamental propagators in the position representation as follows:
In usual quantum electrodynamics in 4 dimensions we make use of the following propergator for a massless boson:
Notice that the propagator for a scalar gravition is ln|x-y| which at first doesn't seem to make sense since this does not tend to zero in any limit. But when we remember that there can be an infinite series of graviton contributions forming an exponential series, we get:
Which does tend to zero in the limit and is what we would expect from a graviton propagator. In second quantisation the fields are operators acting on the state-space. In the position representation we have, for example:
Where this state-vector here representing fermions is a totally anti-symmetric function.
It is easiest, with special relativity, to start with the fundamental equation and then to justify it, rather than to start with classical theory and derive the equation.
The validity of the equation is justified experimentally everyday in particle accelerators such as CERN when scientists observe particles travelling at speed close to the speed of light. The fundamental equation of relativity can be written:
If s represents the time experience by the object then this equation can be read: The 'clock-rate' of an object squared is one minus the speed of the object squared. Hence we get the equation,
It is sometimes more useful to write the equation in a metric form. We do this by interpreting dx as a minuscule length and then multiplying the equation by dt2.
Thus special relativity is simply Pythagoras' theorem in 4 dimensions with a negative signature for time. If we form the vector:
we can write this more simply as,
where
Length Contraction
As we have seen, as an object moves faster its clock-rate slows down. Physically speaking, this is due to the fact that photons which are responsible for the electromagnetic force, have further to go and so all mechanical movements (including clocks, air particles and brain signals) all slow down.
Another affect is length contraction. This means everything gets squashed in the direction of motion. If someone is inside the object, their eyes will also be squashed in this direction and so they won't notice anything strange. However, if they looked out of a window, for instance, they would notice some strange visual affects due to their squashed eyes. Physically speaking, length contraction is due to the fact that a fast moving object does not emit a spherical wave, but a wave squashed in the direction of motion. Again this affects everything held together by the electromagnetic force, in particular, atoms. All these affects result in the fact that an observer inside a fast moving object will measure the speed of light to be exactly the same as for an observer at 'rest'. Let us write the velocity as:
Acceleration can now be interpreted as a rotation in 3+1 dimensional space-time. The rotation involving the time dimension is called a Lorenz rotation. For example:
Lorenz rotations keep the 4 dimensional length given by the (3+1)D Pythagoras' theorem the same just as a rotation of a rod in 3 dimensions keeps the length of the rod the same. Now it is easy to show that lengths in the direction of motion are contracted:
General Relativity
In General Relativity we make the metric into a general symmetric matrix dependent on the co-ordinates of space-time,
this does not give us enough information however so Einstein postulated the Principle of Equivalence. The result being that the metric takes the following form for a static spherically symmetric source:
while in Scalar Gravity we simply make the metric proportional to a scalar field.
We can see straight away that this will give a different result for a static spherically symmetric source. However, it turns out that it is the dt term that gives the greatest effect and in fact we find that the two theories agree in this term.
This must be the case in order for the two theories to approximate to Newton's inverse square law of force. This result, for General Relativity, was found by Schwarzschield even before Einstein found any exact solutions to his equations.
In General Relativity the clock rate is given by:
Or for an object in free fall this becomes:
Which shows that when r=2m, according to this theory, the velocity, u, must be zero. Otherwise the clock speed would not be real. Thus everything (including light) must slow down to zero velocity as it moves towards the 'event horizon' of the black hole. In fact further investigation will show that the object never reaches the event horizon. Although since this slowing down is accompanied by a slowing of clock-rates, someone falling into a black hole will not think they are slowing down. This poses an interesting paradox, since the person will believe he will enter the black hole in a finite time.
Also, notice that inside the distance r=2m, the clock rate is negative which means that clocks should be going backwards!
In General Relativity we find that for a particle in free fall straight towards the source:
If we put the metric for both these theories in this form:
Then the results can be summarised in this table:
| General Relativity | Scalar Gravity | |
| A(r) | 1-2m/r | 1-2m/r |
| B(r) | 1/(1-2m/r) | 1-2m/r |
| C(r) | 1 | 1-2m/r |
| D(r) | 1 | 1-2m/r |
| (clock rate)2 | (1-2m/r) - r'2/(1-2m/r) - (u2-r'2) | (1-u2)(1-2m/r) |
| speed at r=2m | O | 1 |
The clock speed in free fall is simply the A(r) term of the metric which is the same in both cases. In both cases the clock rate for a stationary object at a distance of r from the source is:
This has an imaginary value when r<2m but this simply tells us that an object cannot be stationary in the region r<2m due to the fact that objects are superluminal in this region.
From this table, the most noticeable difference is the speed in which particles enter the horizon at r=2m. For General Relativity the speed decreases to zero and the particle never crosses the boundary while in Scalar Gravity the particle crosses the boundary at exactly the speed of light.
Finally we can see that Scalar Gravity is considerably simpler than General Relativity.
Here we shall outline the basics of Albert Einstein's Theory of Gravity. We start with the metric define by:
Einstein's main concept was the principle of General Covariance. This means that all field are invariant under general co-ordinate transformations. This is another way of saying that the affects of gravity in infinitesimal portions of space-time are equivalent to any other kind of acceleration. (Also called the 'principle of equivalence'). Since the metric is essentially a co-ordinate transformation from flat to curved space, this means that fields must be invariant under transformations given by the metric. For instance:
Let us find an expression involving the derivative that is invariant under this transformation. We might start with the general expression:
We can quite easily show that this expression (called the covariant derivative) is only invariant under the transformation if we set:
(This is called a Christoffel symbol.) Now we wish to find an invariant quantity involving two derivatives. We can take try taking the commutor of a pair of covariant derivatives of the field.
to get a symmetric quantity with two derivatives. This, incidentally, is called the 'curvature tensor'. We find that if we contract the indices, then we get an invariant quantity R. In General Relativity this quantity replaces the wave equation in Newtonian mechanics which also has two derivatives. Hence the action for General Relativity is:
Paths of particles follows the shortest curved line between two points on this 4 dimensional curved surface given by the metric.
These is not the only field equations based on the principle of equivalence however they are merely the simplest. We might want to look for a field theory with up to fourth derivatives. First we find a symmetric quantity made from the curvature tensor called the Weyl tensor:
where the brackets [,] stand for commutation of the indices. This tensor has the property that if we contract any two indices, the tensor vanishes (try it). Form this we can form a new action:
This theory, called 'Conformal Gravity' gives some interesting results although it is precisely the same as general Relativity in areas of empty space since the contracted curvature tensor becomes zero and so does not alter most of our discussion.
In terms of quantum gravity, it was once thought that Conformal Gravity would produce a theory that gave finite answers since its four derivatives caused the terms to converge more rapidly - however it was found that the theory had problems with causality and so had to be rejected.
Spherical Co-ordinates
As many situations in this book are concerned with objects moving around a stationary spherically symmetric source, the most useful co-ordinate system involves the distance to the object and two angles determining latitude and longitude. We shall list some results, without proof, from spherical geometry:
The wave equation is given in Cartesian and spherical co-ordinates by the following operator:
For working out elliptical orbits we need to know the velocities in polar co-ordinates. These are given, again without proof:
where, again, the barred letters are unit vectors in the various directions. Remember the unit vectors in polar co-ordinates are not constant but dependent on the position.
All these formulae can be found in various books on mechanics or can be worked out quite easily.
The Action Principle
The action principle states that any physical theory can be summarised by a function of certain variables and their derivatives. In classical mechanics these variables may be the positions and velocities of the particles and the function could represent the energy of the system. In quantum mechanics the variables may be fields.
Further, at least in the semi-classical limit, the equations of motion of the objects in question can be described by minimising this function. This is the 'principle of least action'. (For the case of quantum interactions this function becomes an operator and individual equations of motion don't exist). To minimise a function containing derivatives requires the Euler-Lagrange formula:
A small change in the Lagrangian function should not affect the action:
Which means:
Usually only the first derivative is required. This is a very important formula for instance, in deriving the geodesic equation in special relativity or the path of light through glass of varying density. It can be derived as follows. First using the Taylor series:
The last step uses the chain rule on the last term. Now since the third term is a total derivative, and using the condition that all the values tend to zero towards outer space, this means that the last term can be ignored and we are left with the Euler Lagrange equations.
This method is a result of the broader topic, 'The calculus of variations.'
The following is a sample of code from a simulation program for a particle with co-ordinates (x,y) moving around a stationary object of mass m at the origin. The accuracy depends on the variable, s, which can be set to 0.01 for fast computers.
Newton's Law of gravity is given by:
r=sqrt(x*x+y*y);
vx=vx-m*x/(r*r*r)*s;
vy=vy-m*y/(r*r*r)*s;
x=x+vx*s;
y=y+vy*s;
plot(200+x,200+y);
For Scalar Gravity make the following changes:
uu=vx*vx+vy*vy;
vx=vx-m*(x*(1-uu))/(r*r*(r-2*m))*s;
vy=vy-m*(y*(1-uu))/(r*r*(r-2*m))*s;
You may like to change the code a bit. For instance changing 2*m to 3*m. You will find that only 2*m gives orbits which continue past r=2*m and have no precessions. You can change the program to show the total energy:
E=(1-2*m/r)/(1-uu);
For General Relativity we have the much more complicated expressions:
R=(x/r)*vx*vx+(y/r)*vy*vy;
[[vx=vx-(x/r/r)*( (r*vx*vx-y*R)/(2*r)
+ m*(x*R)/(r-2*m) - m*x*(1-2*m/r)/r/r )*s;
vy=vy-(y/r/r)*( (r*vy*vy-y*R)/(2*r)
+ m*(x*R)/(r-2*m) - m*y*(1-2*m/r)/r/r )*s;]] WRONG!!!
The energy is:
E=(1-2*m/r)*(1-2*m/r)/((1-2m/r)-R/(1-2m/r)+uu-R);
Some good values to start with are:
x=50, y=50;
vx=0.2, vy=0;
m=30;
For a simulation of non-static forces, we would have to model the entire gravitational field by splitting it up into small squares or 'cells'. Not unlike the way fluids are modelled or indeed the weather.
These exercises are useful to get a better feel of what is going on. There is no set method of attack - indeed it is often useful to approach some exercises from several different directions. Those marked with an asterisk are at a slightly more advanced level.
1. Elliptical Orbits
Using the substitution u=1/r show that the following differential equation,
gives the elliptical formula:
and find the constants a, b and c. Also show that the above equation is an ellipse by writing it in Cartesian co-ordinates. This exercise is recommended to all readers, since one cannot claim to understand Newton's Laws of gravity if one has never derived the equation for an ellipse from the inverse square law.
2. Lorenz Rotation
Show that the following Lorenz rotation keeps the 4-dimensional length the same. Explain why u must represent velocity here.
The Lorenz rotation lies at the heart of special relativity. It is interesting, however, that it was not Einstein but a mathematician friend of his that first pointed out that relativity theory could be unified by considering time as the 4th dimension.
3. Geodesic Equation
Show that a variation of the following action,
gives rise to the geodesic equation which lies at the heart of Einstein's General Relativity which is written in a particularly simple form as:
or more usually as:
This equation is usually kept in this form in terms of the variable, s, since this is more useful for certain calculations and follows the curved space-time philosophy, but it can just as easily be written in terms of the classical 'standard' time. Show that this equation can be written in terms of 'standard' time as:
This formulation is useful for creating computer simulations for instance.
3. General Relativity Acceleration Equation
Using the Schwarzshield metric for a spherically symmetric source and the acceleration equation just found, show that for a particle in free fall straight towards the object, this gives a radial acceleration of:
.........???
4. General Covariance
All the fields in General Relativity are invariant under local co-ordinate transformations. This is because the gravitational metric acts as a rotation matrix which compensates for any co-ordinate transformation in the fields. However this is provided we transform the metric in the opposite direction to all the other fields so the local co-ordinate transformations cancel out. Scalar Gravity, like Newton's Law of Gravity, is not invariant under local co-ordinate transformations. This means that in certain senses Scalar Gravity has less 'symmetry' than General Relativity. On the other hand Scalar Gravity does not treat gravity any differently to any other field. Does this make Scalar Gravity a poorer description of gravity than General Relativity? Discuss.
Now that you have learned the simple field theory of Scalar Gravity you may now like learn Albert Einstein's Theory of curved space-time to compare the two. Some good books are listed here.
Here are some books about James Maxwell's field theory of electromagnetism.