PROLOGUE (A brief history of this paper)         Note: this is an old paper. For the new version see: Cavities In A Lambda Vacuum and The Meaning of Matter     Vacuum Density Gradients And The Meaning of Matter    Douglas C. George (Last Revision: March 18, 2007)   Abstract   Inertial motion of a small test object through a hypothetical vacuum energy density gradient of the form    is shown to be equivalent to the motion of a body falling freely in a gravitational field. No forces of attraction are assumed or proposed.   General Relativity requires just such a vacuum energy density gradient in order to balance the flow of momentum through a region and to produce the sort of motion observed for objects falling freely in a gravitational field.   The existence of such density gradients in space implies that matter must consist of holes, or cavities, in the fabric of space.   Article Introduction   The objective of this paper is to show that a simple density gradient of a certain form in the space surrounding a massive object would have exactly the same effect on the motion of other objects as does a gravitational field and would do so without introducing a force of attraction. To show this, we analyze the motion of a small test object moving inertially through such a hypothetical density gradient.   In keeping with general relativity, space (the vacuum) is considered to be a physical medium that can expand, contract or be otherwise distorted. For the sake of argument, we assume space to be an idealized, compressible fluid, with no viscosity and no heat conduction. We assume that space is capable of expanding or contracting locally. Such local deviations from an average universal density are, by definition, density gradients. For our purposes, a density gradient in space means that space can be, in the normal sense of the words, thicker in some places and thinner in other places relative to the average density for the universe.   In a more technical sense, the density of space is taken to be the vacuum energy density of the region under consideration.     Analysis of Motion   The Density Gradient:   Consider a large object of mass M located at the origin of our coordinate system in Figure 1.   (Figure 1 is a composite graph which superimposes a density gradient graph and a momentum graph.)   A small test object of mass m is approaching the origin with an initial, non-relativistic velocity v(0). In the space surrounding the large object, we assume a vacuum energy density gradient of the form                        [1]     (A derivation of this density function for an idealized fluid is given in the Appendix)   where the amplitude of the gradient at some point x is proportional to the mass M and inversely proportional to the absolute value of x.   (Rho average) is the average universal density of space---set to an arbitrary value for the sake of argument---and K is a constant of proportionality to be determined.   To be accurate, the density gradient of the region is equal to the sum of the contributions from both the large and the small objects (the blue-green curve in Figure 1) but the small test object is not moving relative to its own gradient and its mass is considered negligible relative to the large mass M, so the effective gradient being traversed by the small mass is just equation [1], the gradient associated with the large mass M.                       As can be seen in Figure 1, the slope of the density gradient increases as x approaches zero and the amplitude decreases. In other words, the space “thins out” ever more rapidly as x approaches zero.   What happens as the small test object, moving inertially, traverses such a gradient?     The Momentum Curve:   The motion of the test particle depends on its momentum and the momentum is just the negative of the density gradient curve. The reason for this is based on the relationship (explained below) between the energy density of a region of space and the flux of momentum through the region. In our case, where the velocity of the test object is slow compared to the speed of light, the mass remains essentially constant so the momentum curve simply shows the change in velocity of the object as a function of position.   The Einstein Field Equations (EFE) of general relativity equate the geometry of spacetime to the energy density of a region and the flow of momentum through the region:                                   where Gab is the Einstein tensor describing the geometry of spacetime, Tab is the stress-energy tensor describing the flux of energy and momentum and  κ is a constant (from http://en.wikipedia.org/wiki/General_relativity ).   The figure at the right is an illustration showing the components of the stress-energy tensor (from http://en.wikipedia.org/wiki/Stress-energy_tensor).   For an idealized fluid with no viscosity and no heat conduction, the stress tensor takes on the particularly simple form, where all of the components of the tensor are zero except for the diagonal components. As shown in the illustration above, the diagonal components are the energy density (upper left) and the three "pressure" components (highlighted in green) which represent the flux of momentum in x, y, and z directions.   Normally, this idealized fluid is the material that makes up the interior of a massive object such as a star. The vacuum surrounding the object is devoid of matter. In our case, the space exterior to the massive object will be considered to have an energy density as well. The idea that space itself has an energy density is expressed in the well known cosmological constant  (Lamda) which amounts to giving empty space an energy density equal to and pressure equal to . The field equations with the cosmological constant added is:                                      (8 pi = κ)   In the surrounding space, exterior to the surface of the large mass M, there is no contribution from normal matter, so, for the exterior vacuum we get:                           or, writing out the metric,                            For the vacuum to not pick out a preferred notion of `rest', the stress-energy tensor must be proportional to the metric, so, in local inertial coordinates this means that the stress-energy tensor of the vacuum must be:                       The vacuum can therefore be thought of as a perfect fluid with                     vac = - p vac. In general relativity, the total energy and momentum of a region is locally conserved such that the changes in density be countered by a corresponding change in momentum. Because of our proposed density gradient, the energy density of the vacuum decreases as the test object moves toward the origin and this, in turn, reqires the momentum of the test object to increase proportionally.   This is the relationship shown by the mirror-image density and momentum curves in Figure 1.   The small object m, as it approaches the origin, moves through regions containing progressively less vacuum energy. In order, then, to satisfy the conservation of energy/momentum, the test object m must constantly increase its momentum. For the non-relativistic speeds assumed here, where the mass remains constant, this increase in momentum is due solely to the increase in velocity of the small object.   The incremental change, d[mv(0)], in the initial momentum of the small object is a function of its position in the gradient multiplied by its inital velocity. Holding the mass constant, we have:                            dv/dx = f(x)   Specifically, the decrease in the energy density of space associated with the large mass M must be mirrored by m times the increase in velocity of the small mass.  If the vacuum density falls off as a function of -MK/|x|  , the velocity of the moving object must increase as a function of +MK/|x|.                           dv = f(x)dx   where                           f(x) = MK/|x|       [change in the initial velocity v as a function of position x (in meters per second per meter)]           [2]       Acceleration:   The acceleration a, then, of our test particle of mass m at postion x is just the slope of the momentum curve at position x times the initial velocity:                           a = v(0)d(f(x))/dx     [meters per second times meters per second per meter = meters per second squared]   Substituting from equation [2] for f(x), we get:                           a = v(0)d(MK/\x\)/dx        Taking the derivative, we obtain the acceleration a of an object moving inertially through a density gradient:                                              [3]   By way of comparison, let us examine the acceleration according to Newtonian theory of an object falling freely in a gravitational field. The assumed gravitational force between two masses (M and m) is given by:                                  [G = gravitational constant and the minus sign indicates an attractive force]   But, from Newton’s law, , so,                               Substituting for F, we get                              Which reduces to:                                        [4]   We see that equation [4], the acceleration of a body falling freely in a gravitational field,  is---except for the initial velocity v(0)---identical in form to equation [3] that describes inertial motion through our space density gradient.   The v(0) is necessary in equation [3] because there is no assumed force of attraction in the analysis. With no initial velocity relative to the density gradient, the small mass m would, in our accounting, not accelerate. It would remain motionless. Conveniently for us, this situation could never arise in any real situation simply because, if nothing else, there would always be quarks in motion relative to the gradient. This is discussed further under conclusions, below.     Conclusions   Equivalence to a Gravitational Field   A space density gradient of the form   would result in the acceleration of an object moving inertially through the gradient that is identical to motion through a gravitational field.   No forces of attraction are assumed in this analysis. In agreement with general relativity, acceleration is seen not to be the result of a force per se but is due simply to the conservation of momentum constraints on inertial motion for an object moving through a space of varying density.   At Rest in the Gradient   What happens if the test object is initially at rest with respect to the gradient?   Nothing is ever completely at rest. Random vibrations of the atoms in the test object and/or the motion of their quarks relative to the gradient would start the object moving along the gradient. Any motion “downhill” through the gradient would be exaggerated by positive acceleration and any motion “uphill” through the gradient would be retarded. The net effect would be to start the center of gravity of the test object moving in the downhill direction.   The existence of vacuum density gradients combined with the impossibility that any real, physical object could be absolutely at rest relative to the gradient would, then, give us an alternate accounting for why, if no forces are assumed, physical objects begin to gravitate toward each other.   All masses accelerate at the same rate   Since an object’s own mass does not influence its acceleration (it’s never moving relative to its own gradient), an object of any mass located at position x, would accelerate at the same rate---a rate that is determined only by how much the density gradient due to other objects in the region is changing per unit distance. This result is in agreement with general relativity, Newtonian theory and observation.   Zero Density   We have assumed an arbitrary value for the average density of the universe (shown in Figure 1). From this, the density gradient value associated with the large mass M is subtracted to obtain the density at some distance x. The result is the black density curve shown Figure 1.   Because the gradient function goes to minus infinity as it approaches the origin, the resultant density must necessarily go to zero---i.e., the gradient curve must intersect the x axis somewhere---no matter what value we assume for the average density.   Furthermore, since the density gradient curve approaches the origin from both the positive and negative sides---intersecting the x axis in two places---the entire region between the points of intersection (the width of the circle M in Figure 1) represents a region of zero density.   The curve shown in Figure 1 is a one-dimensional representation of the situation but we should keep in mind that, in three dimensions, this region of zero density would be spherical.   And, finally, because we consider the energy density to be a measure of the actual, physical density of a fluid-like space, this region of zero density must, logically, represent a spherical hole in the fabric of space.   Since the hole in space is where the object of mass M supposedly resides, what does this say about matter?   The Meaning of Matter   We know from general relativity that space and matter do not exist as independent entities. The Einstein Field Equations (EFE), through the conservation of momentum, equate the geometry of spacetime to the effects of the flow of mass/energy through a region.   The normal interpretation of this relationship is that matter "distorts" the space in which it is embedded, a view that, in our opinion, fails to recognize the true extent of the interconnectedness. This paper proposes that space and matter are bound together in a much more intimate way: matter, simply put, may just be distorted space and nothing more. Here is the reasoning:   Matter (according to Einstein) does not simply occupy space but is, instead, spatially extended:    “… space-time is not necessarily something to which one can ascribe a separate existence, independently of the actual objects of physical reality. Physical objects are not in space, but these objects are spatially extended. In this way the concept of “empty space” loses its meaning. “ A. Einstein, Relativity, June 9th, 1952.   Einstein is saying two things above:        a) that there is no such thing as space that is separate from matter and      b) matter is spatially extended.   Taken literally, these two conditions would appear to make space and extended matter essentially indistinguishable. Furthermore, since atoms are said to be limited in size to tiny regions of otherwise empty space, this "spatially extended" matter must include the gravitational fields in the space surrounding the atoms---fields which can extend across the visible universe.   We see, then, that space, matter and the gravitational field blend seamlessly into a continuously interactive mixture wherein matter is said to somehow distort its surrounding space and the space so distorted becomes the gravitational field which (because it too has energy) becomes yet another source of gravitation just like the original matter. We may be forgiven if all of this seems somewhat circular.   Is it possible to untangle these circularly interconnected concepts and arrive at a simple and logically consistent discription of space, matter and the gravitatioanl field? We maintain that it is possible if we take seriously the necessary logical step of abandoning the idea that matter is something separate from the space in which it is imbedded---just as Einstein says above. But how exactly would it work?   The simple solution is that matter does not distort its surrounding space. Matter is the distortion of space.   We must accept that a physical, fluid-like space is all there is and that mass/energy consists simply of the distortions thereof. The question remains, of course, as to what, if not matter, distorts the space. This question is addressed further on below but, for now, we will adopt the "space-is-all-there-is" paradigm at face value and see where it leads.   So, in keeping with general relativity and the conclusions reached above about the nature of matter, this paper will now consider our assumed vacuum energy density gradient and the object of mass M to be part and parcel of the same thing, i.e., the density gradient used in this paper includes the massive object M as well as its surrounding space. Both matter and its surrounding space, taken together, are represented by the density gradient and the density gradient, in turn, depicts a hole in the fabric of space.   We should note here that a hole in the fabric of spacetime is not a new concept. One of the more drastic consequences of general relativity is the possibility that space and time may exhibit "holes" or "edges" called spacetime singularities. (for a discussion go to  http://www.einstein-online.info/en/spotlights/singularities/index.html  )   To answer the question, then, of what such a hole in space says about matter, we may conclude that, if vacuum density gradients of the sort presumed in this paper actually exist, matter, itself, would have to be interpreted to mean the following:   Matter consists of holes in space that distort the space surrounding the holes.   This interpretation of the meaning of matter is the logical and necessary consequence of the existance of vacuum energy density gradients like (or similar to) the one used in this paper. The fact is that, as argued above, general relativity requires a vacuum energy density gradient in order to balance the flux of momentum through a region which, in turn, results in the observed motion of freeling falling objects. Of course, it is also true that we could have constructed this paper by starting with the observed motion of an inertially moving object and deriving the form of the density gradient from the stress tensor relationships.   To address the question asked above of what, exactly, could produce such holes in the fabric of spacetime, we offer up the following possibility: the holes in space are cavitation bubbles resulting from the negative pressure produced by an expanding universe---a process analagous to the production of cavitation bubbles under water caused by a ship's propeller. The rapid expansion of the very young universe would have produced cavitation bubbles in space. Space that started out smooth and dense would have been quickly boiled by the Big Bang into the foaming, Swiss Cheese universe we see today filled with countless cavitation bubbles that we ultimately experience as matter.   In support of the above senario, it turns out that---as can be seen in the Appendix---the derived density gradient surrounding a hypothetical cavitation bubble (a hole) in an ideal fluid is of the exact same form as the function we originally assumed.   There are numerous ramifications if the view of matter and space proposed in this paper is correct. One is that individual cavitation bubbles (which can, of course, be considered as tiny black holes) could, under the gravitational pressure of large numbers of them, coalesce to form ever larger holes in space, eventually growing in size until they become the familiar black holes of cosmological fame. Since black holes could grow by the simple accretion of tiny bubbles merging with the surface of a larger cavity, the mathematical problem encountered when atoms are supposedly crushed down to a singularity becomes a moot point.   And finally, for what it's worth, when the constant of integration is set to 2MG/c^2, the Schwarzchild solution to the EFE for a spherically symmetric black hole with no charge and zero angular momentum (the simplest possible spacetime surrounding any massive object), is just the same Newtonian acceleration for an object falling freely in a gravitational field that was derived in this paper using our assumed density gradient function. http://en.wikipedia.org/wiki/General_relativity (scroll down to the section on Gravitation)   This paper does not address the added complexities of viscous forces, heat transfer and electric charge. We will simply say, for now, that all of these effects, including electric charge, could be considered to be the physical properties of space, itself. In this way, all of reality could be attributed to the properties of space and space alone.       Appendix   The density gradient surrounding a stable cavitation bubble in an ideal, compressible fluid.     The assumed density gradient    in the main article was chosen because its derivative is of the same mathematical form as is acceleration due to gravity---in other words, it was chosen because it works. The surprising implication, however, that matter has to be a hole in the fabric of space, leads to the question of what, on theoretical grounds, the density gradient around a hypothetical hole in space should be like.   To address this question, we assume here (for the sake of simplicity) that space is an ideal, compressible fluid in a static condition with no viscous forces and no heat conduction. The only forces assumed to be acting in this fluid are cohesive forces---like in a stretched membrane.   Assume the fluid to be of infinite extent and, in its undisturbed state, to have an average density  (Rho average). A stable cavity (see Figure A1) of radius  is created by the application of sufficient negative pressure to the fluid---think of the rapid universal expansion of space during the Big Bang. In two dimensions, the cavity is represented by the circle of radius .   Pressure at the cavity boundary   At the boundary of the cavity, the forces created by the negative fluid pressure  must equal the force created by the surface tension (Sigma).   The force trying to collapse the cavity is the surface tension times the circumference of the circle:   Inward force =           [A1]   The force trying to expand the cavity is coming from the surrounding fluid which is under negative pressure. To calculate this force at the boundary of the cavity we consider the void inside the cavity to have a virtual positive pressure  (an imaginary pressure used as a computational device to find the pressure at the boundary of the cavity).  The force, then, to expand the cavity is the virtual pressure times the area of the circle:     Outward force =         [A2]   For the bubble to be stable, the two forces must be equal:                                            [A3]   Therefore, simplifying and rearranging terms, the virtual pressure at the boundary of the cavity is:                               [virtual pressure at the cavity boundary]    [A4]   The pressure in the fluid at the boundary is the negative of the virtual pressure, so, the fluid pressure  at the cavity boundary is:                                [fluid pressure at the cavity boundary]    [A5]   Pressure at an arbitrary point   To find the fluid pressure at some arbitrary radius R, consider an imaginary shell (see Figure A1) of radius  imbedded in the fluid outside of and concentric to the cavity. As with a stretched membrane, the surface tension  at the boundary of the cavity is transferred by cohesive forces to the shell. To find the tension  at the shell, we note that the ratio of the tension at the shell to the tension at the cavity boundary is, for geometric reasons, inversely proportional to the ratio of their respective circumferences:                                           [A6]   Simplifying and rearranging terms, we find the tension at the shell to be:                                     [A7]   To balance the tension at the shell, the negative pressure at the boundary of the cavity must, in a likewise manner, transfer its force to the shell and, as with the tension, the pressure  at the shell is inversely proportional to the ratio of the respective circumferences:                                    [A8]   Which reduces to:                                     [A9]   Substituting for  from equation [A5] we get:                                 [A10]   Which reduces to:                                     [A11]   We see that he pressure in the fluid surrounding the cavity increases as an inverse function of the radial distance R.   In terms of the linear distance x from the origin (see Figure A2), we have the pressure at distance x:                                [pressure at distance x]   [A12]     Density at an arbitrary point   The density of our idealized, compressible fluid, with no viscosity and no heat conduction, is equal to the pressure times a proportionality constant K.                                           [A13]   Therefore, substituting the value for P from equation [A12],  the density function  at some distance x from the origin is:                            [density function at distance x]    [A14]   The resultant value  for the density is equal to the average density at infinity  minus the value of the density function at x. We see, then, that the mathematical form for the density gradient surrounding a cavitation bubble is:                          [resultant density at distance x]  [A15]   Summation: Three Surprises   a) The first surprising finding in the main body of this paper was that a density gradient of the form shown in equation [1]---and here in equation [A15]---would result in motion identical to that produced by gravity.   b) The second surprise was that such a density gradient in space would strongly imply that matter consists of a cavity, or a hole, in the fabric of space.   c) And, finally, in this appendix we have the third surprise in that the theoretical density gradient surrounding a cavitation bubble (a hole) in an ideal fluid turns out to be of the exact same form as the space density gradient needed to replicate motion due to gravity.