**Fig. 1**- Newton's geometric proof that an impulsive pushing graviton mechanism is consistent with Kepler's 3rd law of planetary motion, because equal areas will be swept out in equal times (the three triangles of equal area,

*SAB, SBC*and

*SBD,*all have an equal base of length

*SB,*and they all have altitudes of equal length), together with a diagram we will use for a more modern analysis. Newton's geometric proof of centripetal acceleration, from his book

*Principia,*applies to any elliptical orbit, not just circular orbits as Hooke's easier inverse-square law derivation did. (Newton didn't include the graviton arrow, of course.) By Pythagoras' theorem

*x*

^{2}=

*r*

^{2}+

*v*

^{2}

*t*

^{2}, hence

*x*= (

*r*

^{2}+

*v*

^{2}

*t*

^{2})

^{1/2}. Inward motion,

*y*=

*x - r*= (

*r*

^{2}+

*v*

^{2}

*t*

^{2})

^{1/2}-

*r*=

*r*[(1 +

*v*

^{2}

*t*

^{2}/

*r*

^{2})

^{1/2}- 1], which upon expanding with the binomial theorem to the first two terms, yields:

*y*~

*r*[(1 + (1/2)

*v*

^{2}

*t*

^{2}/

*r*

^{2}) - 1] = (1/2)

*v*

^{2}

*t*

^{2}/

*r*. Since this result is accurate for infidesimally small steps (the first two terms of the binomial become increasingly accurate as the steps get smaller, as does the approximation of treating the triangles as right-angled triangles so Pythagoras' theorem can be used), we can accurately differentiate this result for

*y*with respect to

*t*to give the inward velocity,

*u = v*

^{2}

*t/r*. Inward acceleration is the derivative of

*u*with respect to

*t*, giving

*a = v*

^{2}/

*r*. This is the centripetal force formula which is required to obtain the inverse square law of gravity from Kepler's third law: Hooke could only derive it for circular orbits, but Newton's geometric derivation (above, using modern notation and algebra) applies to elliptical orbits as well. This was the major selling point for the inverse square law of gravity in Newton's

*Principia*over Hooke's argument.

See Newton's

*Principia,*Book I, The Motion of Bodies, Section II: Determination of Centripetal Forces, Proposition 1, Theorem 1:

*'The areas which revolving bodies describe by radii drawn to an immovable centre of force ... are proportional to the times on which they are described.*For suppose the time to be divided into equal parts ... suppose that a centripetal [inward directed] force acts at once with a great impulse [like a graviton], and, turning aside the body from the right line ... in equal times, equal areas are described ... Now let the number of those triangles be augmented, and their breadth diminished

*in infinitum*... QED.'

This result, in combination with Kepler's third law, gives the inverse-square law of gravity, although Newton's argument is using geometry plus hand-waving so it is actually

*far less rigorous than my rigorous algebraic version above.*Newton failed to employ calculus and the binomial theorem to make his proof more rigorous, because he was the inventor of them, and most readers wouldn't be familiar with those methods. (It doesn't do to be so inventive as to both invent a new proof and also invent a new mathematics to use in making that proof, because readers will be completely unable to understand it without a large investment of time and effort; so Newton found that it payed to keep things simple and to use old-fashioned mathematical tools which were widely understood.)

Newton in addition worked out an ingeniously simple proof, again geometrically, to demonstrate that a solid sphere of uniform density (or radially symmetric density) has the same net gravity on the surface and at any distance, for all of its atoms in their three dimensional distribution, as would be the case if all the mass was concentrated in a point in the middle of the Earth. The proof for that is very simple: consider the sphere to be made up of a lot of concentric shells, each of small thickness. For any given shell, the geometry is such as that a person on the surface experiences small gravity effects from small quantities of mass nearby on the shell, while most of the mass of the shell is located at large distances. The inverse square effect, which means that for equal quantities of mass, the most nearby mass creates the strongest gravitational field, is thereby offset by the actual locations of the masses: only small amounts are nearby, and most of the mass of the shell is at a great distance. The overall effect is that the effective location for the entire mass of the shell is in the middle of the shell, which implies that the effective location of the mass of a solid sphere seen from a distance is in the middle of the sphere (if the density of each of the little shells, considered to be parts of the sphere, is uniform).

Feynman discusses the Newton proof in his November 1964 Cornell lecture on 'The Law of Gravitation, an Example of Physical Law', which was filmed for a BBC2 transmission in 1965 and can viewed on google video here (55 minutes). Feynman in his second filmed November 1964 lecture, 'The Relation of Mathematics to Physics', also on google video (55 minutes), stated:

'People are often unsatisfied without a mechanism, and I would like to describe one theory which has been invented of the type you might want, that this is a result of large numbers, and that's why it's mathematical. Suppose in the world everywhere, there are flying through us at very high speed a lot of particles ... we and the sun are practically transparent to them, but not quite transparent, so some hit. ... the number coming [from the sun's direction] towards the earth is less than the number coming from the other sides, because they meet an obstacle, the sun. It is easy to see, after some mental effort, that the farther the sun is away, the less in proportion of the particles are being taken out of the possible directions in which particles can come. So there is therefore an impulse towards the sun on the earth that is inversely as square of the distance, and is the result of large numbers of very simple operations, just hits one after the other. And therefore, the strangeness of the mathematical operation will be very much reduced the fundamental operation is very much simpler; this machine does the calculation, the particles bounce. The only problem is, it doesn't work. .... If the earth is moving it is running into the particles .... so there is a sideways force on the sun would slow the earth up in the orbit and it would not have lasted for the four billions of years it has been going around the sun. So that's the end of that theory. ...

'It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of spacetime is going to do? So I have often made the hypothesis that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed, and the laws will turn out to be simple, like the chequer board with all its apparent complexities.’

The error Feynman makes here is that quantum field theory tells us that there are particles of exchange radiation mediating forces normally, without slowing down the planets: this exchange radiation causes the FitzGerald-Lorentz contraction and inertial resistance to accelerations (gravity has the same mechanism as inertial resistance, by Einstein's equivalence principle in general relativity). So the particles do have an effect, but only as a once-off resistance due to the compressive length change, not continuous drag. Continuous drag requires a net power drain of energy to the surrounding medium, which can't occur with gauge boson exchange radiation unless acceleration is involved, i.e., uniform motion

*doen't*involve acceleration of charges in such a way that there is a continuous loss of energy, so uniform motion doesn't involve continuous drag in the sea of gauge boson exchange radiation which mediates forces! The net energy loss or gain during acceleration occurs due to the acceleration of charges, and in the case of masses (gravitational charges), this effect is experienced by us all the time as inertia and momentum; the resistance to acceleration and to deceleration. The physical manifestation of these energy changes occurs in the FitzGerald-Lorentz transformation; contractions of the matter in the length parallel to the direction of motion, accompanied by related relativistic effects on local time measurements and upon the momentum and thus inertial mass of the matter in motion. This effect is due to the

*contraction*of the earth in the direction of its motion. Feynman misses this entirely. The contraction of the earth's radius by this mechanism of exchange radiation (gravitons) bouncing off the particles, gives rise to the empirically confirmed general relativity law due to conservation of mass-energy for a contracted volume of spacetime, as proved in an earlier post. So it is

*two for the price of one: the mechanism predicts gravity but also forces you to accept that the Earth's radius shrinks, which forces you to accept general relativity, as well.*Additionally, it predicts a lot of empirically confirmed facts about particle masses and cosmology, which are being better confirmed by experiments and observations as more experiments and observations are done.

As pointed out in a previous post giving solid checkable predictions for the strength of quantum gravity and observable cosmological quantities, etc., due to the equivalence of space and time, there are 6 effective dimensions: three expanding time-like dimensions and three contractable material dimensions. Whereas the

*universe as a whole is continuously expanding in size and age,*gravitation

*contracts*matter by a small amount locally, for example the Earth's radius is contracted by the amount 1.5 mm as Feynman emphasized in his famous

*Lectures on Physics.*This physical contraction, due to exchange radiation pressure in the vacuum, is not only a contraction of matter as an effect due to gravity (gravitational mass), but it is also a contraction of

*moving matter*(i.e., inertial mass)

*in the direction of motion*(the Lorentz-FitzGerald contraction).

This contraction necessitates the correction which Einstein and Hilbert discovered in November 1915 to be required for the conservation of mass-energy in the tensor form of the field equation. Hence, the contraction of matter from the physical mechanism of gravity automatically forces the incorporation of the vital correction of subtracting half product of the metric and the trace of the Ricci tensor, from the Ricci tensor of curvature. This correction factor is the difference between Newton's law of gravity

*merely expressed mathematically as 4 dimensional spacetime curvature with tensors*and the full Einstein-Hilbert field equation; as explained on an earlier post, Newton's law of gravitation when merely expressed in terms of 4-dimensional spacetime curvature gives the

*wrong*deflection of starlight and so on. It is absolutely essential to general relativity to have the correction factor for conservation of mass-energy which Newton's law (however expressed in mathematics) ignores. This correction factor

*doubles the amount of gravitational field curvature experienced by a particle going at light velocity,*as compared to the amount of curvature that a low-velocity particle experiences. The amazing thing about the gravitational mechanism is that it yields the full, complete form of general relativity in addition to making checkable predictions about quantum gravity effects and the strength of gravity (the effective gravitational coupling constant,

*G*). It has made falsifiable predictions about cosmology which have been spectacularly confirmed since first published in October 1996. The first major confirmation came in 1998 and this was the lack of long-range gravitational deceleration in the universe. It also resolves the flatness and horizon problems, and predicts observable particle masses and other force strengths, plus unifies gravity with the Standard Model. But perhaps the most amazing thing concerns our understanding of spacetime: the 3 dimensions describing

*contractable*matter are often asymmetric, but the 3 dimensions describing the

*expanding*spacetime universe around us look very symmetrical, i.e. isotropic. This is why the age of the universe as indicated by the Hubble parameter looks the same in all directions: if the expansion rate were different in different directions (i.e., if the expansion of the universe was

*not*isotropic) then the age of the universe would appear different in different directions. This is not so. The expansion does appear isotropic, because those time-like dimensions are all expanding at a similar rate, regardless of the direction in which we look. So the

*effective*number of dimensions is 4,

*not 6*. The three extra time-like dimensions are observed to be identical (the Hubble constant is isotropic), so they can all be most conveniently represented by one 'effective' time dimension.

Only one example of a very minor asymmetry in the graviton pressure from different directions, resulting from tiny asymmetries in the expansion rate and/or effective density of the universe in different directions, has been discovered and is called the Pioneer Anomaly, an otherwise unaccounted-for tiny acceleration in the general direction toward the sun (although the exact direction of the force cannot be precisely determined from the data) of (8.74 ± 1.33) × 10

^{−10}m/s

^{2}for long-range space probes, Pioneer-10 and Pioneer-11. However these accelerations are very small, and to a very good approximation, the three time-like dimensions -

*corresponding to the age of the universe calculated from the Hubble expansion rates in three orthagonal spatial dimensions*- are very similar.

Therefore, the full 6-dimensional theory (3 spatial and 3 time dimensions) gives the unification of fundamental forces; Riemann’s suggestion of summing dimensions using the Pythagorean sum

*ds*= å (

^{2}*dx*

^{2}) could obviously include time (if we live in a single velocity universe) because the product of velocity,

*c,*and time,

*t,*is a distance, so an additional term

*d(ct)*can be included with the other dimensions

^{2}*dx*

^{2}

*, dy*

^{2}

*,*and

*dz*

^{2}. There is then the question as to whether the term

*d(ct)*will be added or subtracted from the other dimensions. It is clearly negative, because it is, in the absence of acceleration, a simple resultant, i.e.,

^{2}*dx*

^{2 }

*+ dy*

^{2}

*+ dz*

^{2}=

*d(ct)*, which implies that

^{2}*d(ct)*changes sign when passed across the equality sign to the other dimensions:

^{2}*ds*= å (

^{2}*dx*

^{2}) =

*dx*

^{2 }

*+ dy*

^{2}

*+ dz*

^{2}–

*d(ct)*= 0 (for the absence of acceleration, therefore ignoring gravity, and also ignoring the contraction/time-dilation in inertial motion); This formula,

^{2}*ds*= å (

^{2}*dx*

^{2}) =

*dx*

^{2 }

*+ dy*

^{2}

*+ dz*

^{2}–

*d(ct)*is known as the ‘Riemann metric’ of Minkowski spacetime. It is important to note that it is

^{2},*not*the correct spacetime metric, which is precisely why Riemann did not discover general relativity back in 1854.

Professor Georg Riemann (1826-66) stated in his 10 June 1854 lecture at Gottingen University,

*On the hypotheses which lie at the foundations of geometry:*‘If the fixing of the location is referred to determinations of magnitudes, that is, if the location of a point in the

*n-*dimensional manifold be expressed by

*n*variable quantities

*x*

_{1},

*x*

_{2},

*x*

_{3}, and so on to

*x*

_{n}, then …

*ds*= Ö [å (

*dx*)

^{2}] … I will therefore term

*flat*these manifolds in which the square of the line-element can be reduced to the sum of the squares … A decision upon these questions can be found only by starting from the structure of phenomena that has been approved in experience hitherto, for which Newton laid the foundation, and by modifying this structure gradually under the compulsion of facts which it cannot explain.’

[The algebraic Newtonian-equivalent (for weak fields) approximation in general relativity is the Schwarzschild metric, which,

*ds*= (1 – 2

^{2}*GM/r*)

^{-1}(

*dx*

^{2}+

*dy*

^{2}+

*dz*

^{2}) – (1 – 2

*GM/r*)

*d(ct)*

^{2}. This only reduces to the special relativity metric for the impossible, unphysical, imaginary, and therefore totally bogus case of

*M*= 0, i.e., the absence of gravitation. However this does not imply that general relativity proves the postulates of special relativity. For example, in general relativity the velocity of light changes as gravity deflects light, but special relativity denies this. Because the deflection in light, and hence velocity change, is an experimentally validated prediction of general relativity, that postulate in special relativity is inconsistent and in error. For this reason, it is misleading to begin teaching physics using special relativity.]

WARNING: I've made a change to the usual tensor notation below and, apart from the conventional notation in the Christoffel symbol and Riemann tensor, I am indicating covariant tensors by positive subscript and contravariant by negative subscript instead of using indices (superscript) notation for contravariant tensors. The reasons for doing this will be explained and are to make this post easier to read for those unfamiliar with tensors but familiar with ordinary indices (it doesn't matter to those who are familiar with tensors, since they will know about covariant and contravariant tensors already).

Professor Gregorio Ricci-Curbastro (1853-1925) took up Riemann’s suggestion and wrote a 23-pages long article in 1892 on ‘absolute differential calculus’, developed to express differentials in such a way that they remain invariant after a change of co-ordinate system. In 1901, Ricci and Tullio Levi-Civita (1873-1941) wrote a 77-pages long paper on this,

*Methods of the Absolute Differential Calculus and Their Applications,*which showed how to represent equations invariantly of any absolute co-ordinate system. This relied upon summations of matrices of differential vectors. Ricci expanded Riemann’s system of notation to allow the Pythagorean dimensions of space to be defined by a line element or ‘Riemann metric’ (named the ‘metric tensor’ by Einstein in 1916):

**g =**

*ds*The meaning of such a tensor is revealed by subscript notation, which identify the rank of tensor and its type of variance.

^{2}= g_{m n}dx_{-m}dx_{-n}.‘The special theory of relativity … does not extend to non-uniform motion …

*The laws of physics must be of such a nature that they apply to systems of reference in any kind of motion.*Along this road we arrive at an extension of the postulate of relativity…

*The general laws of nature are to be expressed by equations which hold good for all systems of co-ordinates, that is, are co-variant with respect to any substitutions whatever (generally co-variant).*… We call four quantities A

_{v}the components of a covariant four-vector, if for any arbitrary choice of the contravariant four-vector B

^{v}, the sum over v, å A

_{v}B

^{v}= Invariant. The law of transformation of a covariant four-vector follows from this definition.’ – Albert Einstein, ‘The Foundation of the General Theory of Relativity’,

*Annalen der Physik,*v49, 1916.

The rank is denoted simply by the number of letters of subscript notation, so that

*X*is a ‘rank 1’ tensor (a vector sum of first-order differentials, like net velocity or gradient over applicable dimensions), and

_{a}*X*is a ‘rank 2’ tensor (for second order differential vectors, like acceleration). A ‘rank 0’ tensor would be a scalar (a simple quantity without direction, such as the number of particles you are dealing with). A rank 0 tensor is defined by a single number (scalar), a rank 1 tensor is a vector which is described by four numbers representing components in three orthagonal directions and time, a rank 2 tensor is described by 4 x 4 = 16 numbers, which can be tabulated in a matrix. By definition, a covariant tensor (say,

_{ab}*X*) and a contra-variant tensor of the same variable (say,

_{a}*X*) are distinguished by the way they transform when converting from one system of co-ordinates to another; a vector being defined as a rank 1

_{-a}*covariant*tensor. Ricci used lower indices (subscript) to denote the matrix expansion of covariant tensors, and denoted a contra-variant tensor by superscript (for example

**). But even when bold print is used, this is still ambiguous with power notation, which of course means something completely different (the tensor**

*x*^{n}**whereas for powers or indices**

*x*+...^{n}= x^{1 }+ x^{2}+ x^{3 }*x*^{n},*x*). [Another step towards ‘beautiful’ gibberish then occurs whenever a contra-variant tensor is raised to a power, resulting in, say (

^{n}= x_{1}x_{2}x_{3}...x_{n}**)**

*x*^{2}*, which a logical mortal (who’s eyes do not catch the bold superscript) immediately ‘sees’ as*

^{2}**causing confusion.] We avoid the ‘beautiful’ notation by using negative subscript to represent contra-variant notation, thus**

*x*^{4},**is here the contra-variant version of the covariant tensor**

*x*_{-n }**. Einstein wrote in his original paper on the subject, ‘The Foundation of the General Theory of Relativity’,**

*x*_{n}*Annalen der Physik,*v49, 1916: ‘Following Ricci and Levi-Civita, we denote the contravariant character by placing the index above, and the covariant by placing it below.’

This was fine for Einstein who had by that time been working with the theory of Ricci and Levi-Civita for five years, but does not have the clarity it could have. (A student who is used to indices from normal algebra finds the use of index notation for contravariant tensors absurd, and it is sensible to be as unambiguous as possible.) If we expand the metric tensor for m and

*n able to take values representing the four components of space-time (1, 2, 3 and 4 representing the*

*ct, x, y,*and*z*dimensions) we get the awfully long summation of the 16 terms added up like a 4-by-4 matrix (notice that according to Einstein’s summation convention, tensors with indices which appear twice are to be summed over):**g =**

*ds*

^{2}= g_{mn}*dx*

_{-m}dx_{-n}

_{ }**=**

*å*(

*g*

_{m n}dx_{-m}dx_{-n})*= -(*

*g*+_{11}dx_{-1}dx_{-1}*g*+_{21}dx_{-2}dx_{-1}*g*+_{31}dx_{-3}dx_{-1}*g*) + (-_{41}dx_{-4}dx_{-1}*g*+_{12}dx_{-1}dx_{-2}*g*+_{22}dx_{-2}dx_{-2}*g*+_{32}dx_{-3}dx_{-2}*g*) + (-_{42}dx_{-4}dx_{-2}*g*+_{13}dx_{-1}dx_{-3}*g*+_{23}dx_{-2}dx_{-3}*g*+_{33}dx_{-3}dx_{-3}*g*) + (-_{43}dx_{-4}dx_{-3}*g*+_{14}dx_{-1}dx_{-4}*g*+_{24}dx_{-2}dx_{-4}*g*+_{34}dx_{-3}dx_{-4}*g*)_{44}dx_{-4}dx_{-4}The first dimension has to be defined as negative since it represents the time component,

*ct.*We can however simplify this result by collecting similar terms together and introducing the defined dimensions in terms of number notation, since the term

*dx*=

_{-1}dx_{-1}*d(ct)*while

^{2},*dx*=

_{-2}dx_{-2}*dx*=

^{2}, dx_{-3}dx_{-3}*dy*and so on. Therefore:

^{2},**g =**

*ds*=

^{2}*g*(a dozen trivial

_{ct}d(ct)^{2}+ g_{x}dx^{2}+ g_{y}dy^{2}+ g_{z}dz^{2}+*first order*differential terms).

It is often asserted that Albert Einstein (1879-1955) was slow to apply tensors to relativity, resulting in the 10 years long delay between special relativity (1905) and general relativity (1915). In fact, you could more justly blame Ricci and Levi-Civita who wrote the long-winded paper about the invention of tensors (hyped under the name ‘absolute differential calculus’ at that time) and their applications to physical laws to make them invariant of absolute co-ordinate systems. If Ricci and Levi-Civita had been competent geniuses in mathematical physics in 1901, why did they not discover general relativity, instead of merely putting into print some new mathematical tools? Radical innovations on a frontier are difficult enough to impose on the world for psychological reasons, without this being done in a radical manner. So it is rare for a single group of people to have the stamina to both invent a new method, and to apply it successfully to a radically new problem. Sir Isaac Newton used geometry, not his invention of calculus, to describe gravity in his

*Principia,*because an innovation expressed using new methods makes it too difficult for readers to grasp. It is necessary to use

*familiar*language and terminology to explain

*radical*ideas rapidly and successfully. Professor Morris Kline describes the situation after 1911, when Einstein began to search for more sophisticated mathematics to build gravitation into space-time geometry:

‘Up to this time Einstein had used only the simplest mathematical tools and had even been suspicious of the need for "higher mathematics", which he thought was often introduced to dumbfound the reader. However, to make progress on his problem he discussed it in Prague with a colleague, the mathematician Georg Pick, who called his attention to the mathematical theory of Ricci and Levi-Civita. In Zurich Einstein found a friend, Marcel Grossmann (1878-1936), who helped him learn the theory; and with this as a basis, he succeeded in formulating the general theory of relativity.’ (M. Kline,

*Mathematical Thought from Ancient to Modern Times,*Oxford University Press, 1990, vol. 3, p. 1131.)

General relativity equates the mass-energy in space to the curvature of motion (acceleration) of an small test mass, called the geodesic path. Readers who want a good account of the full standard tensor manipulation should see the page by Dr John Baez or a good book by Sean Carroll, Spacetime and Geometry: An Introduction to General Relativity.

Curvature is best illustrated by plotting a graph of distance versus time and when the line curves (as for an accelerating car) that curve is ‘curvature’. It’s the curved line on a space-time graph that marks acceleration, be that acceleration due to a force acting upon gravitational mass or inertial mass (the equivalence principle of general relativity means that gravitational mass = inertial mass).

This point is made very clearly by Professor Lee Smolin on page 42 of the USA edition of his 1996 book, ‘The Trouble with Physics.’ See Figure 1 in the post here. Next, in order to mathematically understand the Riemann curvature tensor, you need to understand the operator (not a tensor) which is denoted by the Christoffel symbol (superscript here indicates contravariance):

*G*

_{ab}^{c}= (1/2)*g*^{cd}[(d*g*_{da}/d*x*^{b}) + (d*g*_{db}/d*x*^{a}) + (d*g*_{ab}/d*x*^{d})]The Riemann curvature tensor is then represented by:

*R*

^{a}_{cbe}= (*dG*d_{bc}^{a}/*x*^{e}*) – (**dG*d_{be}^{a}/*x*^{c}*) + (G**) – (G*_{te}^{a}G_{bc}^{t}*).*_{tb}^{a}G_{ce}^{t}If there is no curvature, spacetime is flat and things don’t accelerate. Notice that if there is any (fictional) ‘cosmological constant’ (a repulsive force between all masses, opposing gravity an increasing with the distance between the masses), it will only cancel out curvature at a particular distance, where gravity is cancelled out (within this distance there is curvature due to gravitation and at greater distances there will be curvature due to the dark energy that is responsible for the cosmological constant). The only way to have a completely flat spacetime is to have totally empty space, which of course doesn’t exist, in the universe we actually know.

To solve the field equation, use is made of the simple concepts of

*proper lengths*and

*proper times.*The

*proper length*in spacetime is equal to cò (-

*g*ò (

_{mn}dx_{-m}dx_{-n})^{1/2}, while the*proper time*is*g*

_{m n}dx_{-m}dx_{-n})^{1/2}.Notice that the ratio of proper length to proper time is always

*c.*The Ricci tensor is a Riemann tensor contracted in form by summing over a = b, so it is simpler than the Riemann tensor and is composed of 10 second-order differentials. General relativity deals with a change of co-ordinates by using Fitzgerald-Lorentz contraction factor, g = (1 –

*v*

^{2}/

*c*

^{2})

^{1/2}. Karl Schwarzschild produced a simple solution to the Einstein field equation in 1916 which shows the effect of gravity on spacetime, which reduces to the line element of special relativity for the impossible, not-in-our-universe, case of zero mass. Einstein at first built a representation of Isaac Newton’s gravity law

*a = MG/r*(inward acceleration being defined as positive) in the form

^{2}*R*

_{m n}= 4p

*GT*

_{m n}/c

^{2}, where

*T*

_{mn}is the mass-energy tensor,

*T*

_{m n}= r

*u*( This was incorrect since it did not include conservation of energy.) But if we consider just a single dimension for low velocities (g = 1), and remember

_{m}u_{n}.*E = mc*, then

^{2}*T*r

_{m n}=*T*=_{00}*u*= r

^{2}*(g c)*Thus,

^{2}= E/(volume).*T*

_{m n}/c

^{2}is the effective density of matter in space (the mass equivalent of the energy of electromagnetic fields). We ignore pressure, momentum, etc., here:

Above: the components of the stress-energy tensor (image credit: Wikipedia).

The scalar term sum or "trace" of the stress-energy tensor is of course the sum of the diagonal terms from the top left to the top right, hence the trace is just the sum of the terms with subscripts of 00, 11, 22, and 33 (i.e., energy-density and pressure terms).

The velocity needed to escape from the gravitational field of a mass (ignoring atmospheric drag), beginning at distance

*x*from the centre of mass, by Newton’s law will be

*v*= (2

*GM/x*)

^{1/2}, so

*v*= 2

^{2}*GM/x*. The situation is symmetrical; ignoring atmospheric drag, the speed that a ball falls back and hits you is equal to the speed with which you threw it upwards (the conservation of energy). Therefore, the energy of mass in a gravitational field at radius

*x*from the centre of mass is equivalent to the energy of an object falling there from an infinite distance, which by symmetry is equal to the energy of a mass travelling with escape velocity

*v*. By

*Einstein’s principle of equivalence*between inertial and gravitational mass, this gravitational acceleration field produces an identical effect to ordinary motion. Therefore, we can place the square of escape velocity (

*v*= 2

^{2}*GM/x*) into the Fitzgerald-Lorentz contraction, giving g = (1 –

*v*

^{2}/

*c*

^{2})

^{1/2}= [1 – 2

*GM*/(

*xc*

^{2})]

^{1/2}.

However, there is an important difference between this gravitational transformation and the usual Fitzgerald-Lorentz transformation, since length is only contracted in one dimension with velocity, whereas length is contracted equally in 3 dimensions (in other words,

*radially outward in 3 dimensions*, not sideways between radial lines!), with spherically symmetric gravity. Using the binomial expansion to the first two terms of each: Fitzgerald-Lorentz contraction effect: g =

*x/x*=

_{0}*t/t*=

_{0}*m*= (1 –

_{0}/m*v*

^{2}/

*c*

^{2})

^{1/2}= 1 – ½

*v*

^{2}/

*c*

^{2}+ ... . Gravitational contraction effect:

*g =*where for spherical symmetry (

*x/x*=_{0}*t/t*=_{0}*m*[1 – 2_{0}/m =*GM*/(*xc*^{2})]^{1/2}= 1 –*GM*/(*xc*^{2}) + ...,*x = y = z = r*), we have the contraction spread over three perpendicular dimensions not just one as is the case for the FitzGerald-Lorentz contraction:

*x/x*+

_{0}*y/y*+

_{0}*z/z*= 3

_{0}*r/r*. Hence the radial contraction of space around a mass is

_{0}*r/r*= 1 –

_{0}*GM*/(

*xc*

^{2}) = 1 –

*GM*/[(

*3rc*]. Therefore, clocks slow down not only when moving at high velocity, but also in gravitational fields, and distance contracts in all directions toward the centre of a static mass. The variation in mass with location within a gravitational field shown in the equation above is due to variations in gravitational potential energy. The contraction of space is by (1/3)

^{2}*GM*/

*c*

^{2}. This physically relates the Schwarzschild solution of general relativity to the special relativity line element of spacetime.

This is the 1.5-mm contraction of earth’s radius Feynman obtains,

*as if there is pressure in space*. An equivalent pressure effect causes the Lorentz-FitzGerald contraction of objects in the direction of their motion in space, similar to the wind pressure when moving in air, but without molecular viscosity (this is due to the Schwinger threshold for pair-production by an electric field: the vacuum only contains fermion-antifermion pairs out to a small distance from charges, and beyond that distance the weaker fields can't cause pair-production - i.e., the energy is below the IR cutoff - so the vacuum contains just bosonic radiation without pair-production loops that can cause viscosity; for this reason the vacuum compresses macroscopic matter without slowing it down by drag). Feynman was unable to proceed with the LeSage gravity and gave up on it in 1965.

More information can be found in the earlier posts here, here, here, here, here and here.