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How gravity works

A.Findlay,♦ P. Haenggi 2013

Abstract

From http://en.wikipedia.org/wiki/Gravitation 15.8.2013

Einstein proposed that spacetime is curved by matter and that free-falling ob-jects are moving along locally straight paths in curved spacetime. These straight paths are called geodesics. Like Newton's first law of motion, Einstein's theory states that if a force is applied on an object, it would deviate from a geo-desic. For instance, we are no longer following geodesics while standing be-cause the mechanical resistance of the Earth exerts an upward force on us, and we are not in an inertial frame when standing on the ground as a result..

When standing on the earth (in a gravitational field) still we are no longer following geodesics. Forces cause acceleration Einstein's theory states that if a force is applied on an object, it would deviate from a geodesic. Therefore when standing still on earth we are accelerating away from the geodesic. Therefore the geodesic is accelerating in the opposite direction to a person standing still on the earth.

Therefore the geodesic is a line of movement , the movement (accelerat-ing) towards the center of the earth….And the closer it gets to the center of mass the faster it accelerates.

The maximum acceleration is proportional to the mass of the body, and the maximum achievable velocity is the speed of light.

By what process could a an object be accelerated along a geodesic? By what process can Matter curve space time? What process limits the ve-locity to the speed of light? We attempt to find a solution

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What is not described and what we will try and show here is how a mass interacts with space to curve it.

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Abstract ....................................................................................................................................... 1

Section 1 Mathematical analysis .............................................................................................. 13

The expansion of the universe – Increased kinetic energy of the masses in the expanded universe, whilst the total energy of the universe remains constant. ..................................... 13

Dark energy ............................................................................................................................... 15

Dark matter ............................................................................................................................... 15

Section 2 - Specific calculations ............................................................................................... 17

To calculate the mass of the smallest possible volume of space that can be absorbed by a particle ....................................................................................................................................... 17

For a cubic meter of Space, in 1 second, ................................................................................. 19

For the universe ........................................................................................................................ 19

Proof of compatibility with General Relativity ...................................................................... 22

Proof of compatibility with Quantum Electrodynamics ............ Error! Bookmark not defined.

How can we experimentally verify this theory? ..................................................................... 25

About constants ......................................................................................................................... 26

REFERENCES ......................................................................................................................... 27

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So let us start. First we will define the universe as being the sum of everything. Therefore there can be no interaction with anything outside the universe. Let’s make a sketch of the universe to help us understand this Diagram 1

At a certain time which we will call T1 it has a certain size.At a later moment in time, lets call it T2 it has expanded. Diagram 2

Time T1 Diameter D1

Time T2 Diameter D2

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What can we say about these the universe at these two different times? They contain the same amount of energy. Of course inside the universe things are going on and types of energy are con-stantly changing form, like chemical energy changing to heat, but still, the sum of all types of energy in the universe remains constant. So let’s have a look at both again after a certain amount of time has passed . At time T1. And let’s have a particular look at two arbitrary objects with mass, a, and b. Diagram 3

Now these objects could be anything, and we can assume that they are not sta-tionary so they will have a kinetic energy, a measure of their energy of move-ment, initially at a distance apart of d1 for our arguments sake we assume they are traveling apart at a velocity of v1.

Time T1 Diameter D1 Total energy E1

Distance apart d1, Relative velocity to

each other v1

a b

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Let’s now look at them again at time T2. As the universe expands, they have now receded away from each other. Diagram 4

As we have observed that objects further away (distance d3) recede faster (ve-locity v3) , we can say that at any particular later moment in time the sum of the kinetic energy of all these objects will be greater than at the time T1. ie d3>d1 and v3>v1. So where does this energy come from?

Time T2 such that T2>T1 Diameter D3 Total energy E3

d3, v3

a b

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What has changed? Well let us assume that between time T1 and T2 they have not come into contact and therefore not interacted with anything, then the only other change is that the universe has expanded. Or in other words the volume of space has increased. However the volume by which the universe has in-creased is less than the volume by which it would have increased without mass. Diagram 5 Diameter D3 Diameter D2 > D3 >D1

a b

d3 > d1 v3 > v1

Time T2 such that T2>T1 Size D3 Total energy E3 = E2 = E1

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Therefore the universe without mass will have extra volume whilst the universe with mass will have a lower volume but extra kinetic energy such that the total energy of both universes remains the same. Therefore we think the differences in the volume of space must have something to do with the increased kinetic energy of the two objects. So let’s make an assumption. For a given volume of space there is a given amount of space vacuum energy at any time T. An increase in the volume without energy being added or subtracted would cause a decrease in the density of the energy per unit volume. If you would take energy out of a certain volume of vacuum, you would either decrease the energy density, or, decrease the vol-ume or more probably a combination of both. So in a universe with mass in it, such as ours which is expanding with tme, on the one hand there is an increase in kinetic energy of the masses and a reduction in the increase in the volume of the universe as compared to an equivalent uni-verse without mass in it. Missing space (and hence missing vacuum energy) on the one side and addi-tional kinetic energy on the other side? Can we presume that the increase in kinetic energy of an object with mass comes from the object with mass absorbing energy from space? Later, we will show mathematically that under certain conditions, the missing vacuum energy and the additional kinetic energy ae equal. If this was the case, what would be the consequences? The first consequence would be that if all of space has vacuum energy and this vacuum energy is evenly distributed, then as the energy density of space tries to reach zero as time goes on, in order to obey Newton’s second law of thermody-namics, the volume of space would have to increase. So if there is some type of local quantum interaction such as the exchange of a graviton between empty space and a mass at the edge of the two, then the space next to the mass would lose energy and get smaller. Does the mass then gain

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kinetic energy if it gains velocity relative to space, even though it is gains ve-locity relative to space in all six directions, up, down, left, right, forward and backwards? Relative to a stationary point in space such as perhaps the middle point of the universe, yes.

. Would anyone living on an object with mass notice this? Probably at first not, because gaining velocity in all 6 directions effectively leads to a zero change in apparent velocity, making the observer think he is stationary. However as the velocity has increased relative to the surrounding space there would be a rela-tive acceleration. And again the relative acceleration of the object to the sur-rounding space would be in all 6 directions all towards the centre of the object. This acceleration towards the centre of an object with mass is indeed felt by humans when the object is large enough. We have given this acceleration a name, Gravity. Not only that, but as the object withdraws energy from the surrounding space, the energy density of the surrounding space decreases, drawing in energy from the next piece of space and so on. This then is how space is curved by any mass. Spaces then “flows into mass” resulting in

a) moving geodesics

X axis

Z axis

Y axis

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b) accelerating of mass along geodesics c) Maximum acceleration being proportional to the maximum inflow rate

which is propotional to the mass.

X axis

Z axis

Y axis

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How does an object with mass absorb energy from the vacuum? Here we specu-late that this is via a graviton, but that the entire process is over the Planck length and time, making the capture of a graviton experimentally for us humans very difficult. However it does make gravity a local quantum phenomenon, fi-nally linking quantum theory with General relativity. Not only this, it explains the expansion of the universe and explaining dark energy, and by giving a cer-tain volume of space mass (energy) it provides a candidate for certainly a part of the missing dark matter.

X axis

Z axis

Y axis

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Note: Although these thoughts are based on a “universe” the same analysis could be done for an arbitrary sphere of space, providing that the masses in the second “sphere” where very small and the distance between the masses was very large so that their gravitational attraction (and hence gravitational potential energy) would be irrelevant. Therefore even if we live in a “multiverse” or the universe is “infinitely large” the same results occur.

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Section 1 Mathematical analysis

The expansion of the universe – Increased kinetic energy of the masses in the expanded universe, whilst the total energy of the universe remains constant.

First, for simplicity let us assume an idealised universe. Let us say that the uni-verse is a sphere and the edges of the sphere are a distance away from the cen-tre. We observer any two arbitrary masses a and b with combined kinetic ener-gy of Ek and mass energy of Em. We ignore Gravitaional potentially energy by assuming the masses are very small and are very far away. The vacuum energy of cold dark spacetime also has some energy. It could be positive or negative or even zero, but to ensure completeness we need to in-clude it. Let’s call the vacuum energy of cold dark spacetime Est Let us look at the total energy of this theoretical universe. Let us call the total energy of this theoretical universe Et Total Energy Et = Est + Em + Ek At time T1 Et1 = Est1 + Em1 + Ek1

And at time T2 Et2 = Est2 + Em2 + Ek2 Then, as the total energy does not change, Et1 = Et2

= Est1 + Em1 + Ek1

= Est2 + Em2 + Ek2

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In the case that we choose T2 and the velocities such that any gain in mass from the increased velocity is negligible we can then set Em1 = Em2

And conclude that the change in Vacuum Energy = Change in Kinetic Energy

Est1 - Est2 = Ek2 – Ek1

Conclusion based on the analysis of the whole universe: The point we are making here is that any increase in kinetic energy must come from the vacuum itself. Where does space time contract (or expand) because of these changes? How fast does spacetime contract or decrease its density? If the rate at which the vacuum energy of space time between the particles was transferred to the particles in form of kinetic energy was at a constant rate, then there would appear to be AN ACCELERATION between the particles. But this is exactly what we observe. This is what we call gravity. This energy is supplied by the vacuum energy. This energy is absorbed at a constant rate at all times by a mass dependent only on the mass of the object – It gives the object the property of mass! The rate of acceleration is however proportional to the local energy density of the spacetime at the particle. Nearest the particle the energy density is lowest. Further away from the particle are areas with higher density. The second law of thermodynamics dictates that different energy densities will tend to cause a flow of vacuum energy from re-gions of higher density towards regions of lower density. (From which the “flatness” of the universe arises) This will cause the local regions of space time to change size and shape depend-ing on their local vacuum energy densities, which would have the effect of mass

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causing local bending of space time. The fact that space time remains bent around a mass infers that there is a constant transfer of energy of space time to the mass. The path of any particles in spacetime due to such an effect and the rate of change of velocity of the particles would be identical to what we call gravity. But there is no actual attraction between the masses, they absorb the vacuum energy at a constant rate and hence spacetime next to them and thus the effect would be that they accelerate towards each other. So what would happen if as we suggest, the vacuum energy of spacetime was transferred to the particles? Then the particles would start to move towards each other and the space time between them decreases. If this was a constant process it would look like a force of gravity attracting the two particles, causing them to accelerate towards each other. So we would have the same effect as a force of gravity. However there is no “gravitational” force acting at a distance between the two particles, and the two particles would be just absorbing energy locally from its spacetime surroundings. An objection might be “But particles would heat up if they constantly absorbed energy?” No, they would speed up and gain kinetic energy. Why have we not noticed this change in velocity? It is not obvious but this sim-ple analysis shows that all masses constantly accelerate and change velocity. We know that our universe is expanding. And we know that mass in our uni-verse should cause a slowing down of this expansion.

Dark energy If the vacuum energy density decreases over time, but the entire energy of the universe remains constant, then the volume of the universe must increase. This would then also explain the dark energy causing the measured expansion of the universe. This would follow from the second law of thermodynamics.

Dark matter

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In areas of high mass (such as around galaxies) the stress caused by the bend-ing of space time due to the mass absorbing energy and increasing their kinetic energy would result in a net gain of energy around the mass, such that it would appear that the (energy and hence) mass of the region had increased. So this would also explain at least a part of the dark matter and the dark matters halo like distribution around mass. If this is true and as it seems to logically follow then we should be able to cal-culate some real values.

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Section 2 - Specific calculations

To calculate the mass of the smallest possible volume of space that can be absorbed by a particle

We know that Newtonian mechanics is a good approximate at low speeds and over short periods, so, assuming two identical masses (A and P) initially at rest are attracting each other mutually by “gravitation”, and there are no other forces acting on them. (Idealised situation) Mp = Mass of P Kg Ma = Mass of A Kg Fp = Force acting on P N G = Gravitational constant m3/Kgs2 R = Distance between A and P m Sp = Distance travelled by P m Vp = Velocity of P m/s Ap = Acceleration of P m/s2 T = an amount of time s Ms = Mass of smallest amount of space Kg Vps =Velocity of P by itself due to absorption of vacuum energy m/s Then the force acting on the mass P is given approximately by Fp = GMp Ma/R

2 1 And is also given by Fp = Mp Ap 2 Therefore the acceleration is Ap = G Ma/ R

2 3 Therefore the velocity is Vp = G MaT/ R2 4 And the distance travelled is Sp = G MaT

2/2 R2 5 Assuming the smallest amount of space would be a cube with sides the size of a Planck length then,

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at the Planck distance where R = Sp Vp= G MaT/Sp 12 And Sp = G MaT

2/2Sp 2

13 The above equations are based on the acceleration of two identical particles with mass It follows then that the acceleration of a single particle will then have to be ad-justed When there is only one particle and the acceleration continues anyway, it fol-lows that the total mass in the space is given by Ms = (Ma)/2 which can be written Ma = 2Ms 21 The mass of space can then be calculated as follows Ms = Sp

3/ GT2 22 Generalising equation 22 we have M= R3/GT2 23 This is the generalised gravitation equation for any single mass. Putting in Planck units to work out the maximum rate at which the mass of this smallest possible volume of space that can be absorbed, we have the mass of space as: M = (1.6616x10-35)3/ (6.674x10-11 x (5.391x10-44) 2) M = 2.18x10-08 Kg This is the Planck Mass.!!! This is the mass of the smallest quantum of space time that can be absorbed by a particle in a Planck second. Therefore the (quantum of) energy in the smallest volume of space that can be absorbed is 1.93x109 J. This is of course the Planck energy

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Therefore gravity is a local quantum phenomenon, of the constant transfer of a quantum’s of vacuum energy of spacetime proportional to the mass of a particle to the particle which increases its kinetic energy. From Wikipedia… “Significance of the Planck mass The Planck mass is an idealized mass that is thought to have special signifi-cance for quantum gravity when General Relativity and the fundamentals of quantum physics become mutually important to describe mechanics.” http://en.wikipedia.org/wiki/Planck_mass “The Planck energy is not only the energy needed (in principle) to probe the Planck length, but is probably also the maximum possible energy that can fit into a region of that scale.

For a cubic meter of Space, in 1 second,

M1 = 13 / (6.674 x10-11 x 12) =1/G = 1.498 x1010 Kg Thus the universal gravitational constant G is not just an arbitrary constant, but, (Actually 1/G) is the maximum amount of mass available (in the form of vacu-um energy) in one cubic meter of space that can be made available to a body with mass within one second at the present time.

For the universe

Mu = Ru

3/G Tu2

(Estimates from Wikipedia) Ru= 46.5 billion light years = 4.4x1026 m Tu= 13.8 Billion years = 4.35x1017 s

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Therefore the vacuum energy mass equivalent of the universe is Mu = 6.73x1054 Kg Eu = 6.05 x 10 71 Vu = 1.14 x 10 80 Energy density = 5.33 x10-9 J/ m3 Mass density = 5.93 x 10 -26 kg / m3 As the estimates are within three orders of magnitude, (volume is three orders of magnitude due to the radius being approximate to one order of magnitude) the mass density is then Between 5.593 x 10 -23 kg / m3 and 5.93 x 10 -29 kg / m3 From Wikipedia

Here is how we got the numbers. Using the Λ-CDM model, the Wilkinson Mi-crowave Anisotropy Probe estimates that Ω&Lambda = 0.726 ± 0.015. This means that the energy density of the vacuum is about 0.726 times the critical density. The critical density, in turn, is defined to be

ρc = 3H2/8πG

where H is the Hubble constant and G is the gravitational constant. The WMAP data estimate the Hubble constant at 70.5 ± 1.3 kilometers per second per me-gaparsec, and the gravitational constant is known much more accurately, at 6.67384 ± .00008 × 10-11 meters3 per kilogram second2. This puts the critical density between 9.0 × 10-27 and 9.7 × 10-27 kilograms per cubic meter, and the energy density of the vacuum between 6.4 × 10-27 about 7.2 × 10-27 kilograms per cubic meter. Please check our math, and our data!

http://math.ucr.edu/home/baez/vacuum.html Plausibility check:

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The mass of visible matter is estimated to be about Mv = 3.0x1052 Kg estimate presumed to be within 1 order of magnitude (Estimate - from Wikipedia) Thus 3x1051 Kg <Mv<3x1053 Making the visible mass between 0.05 % to 5% of the total mass of the universe And the mass density calculate done on the basis of WMAP data is in the middle of the calculation limits set 5.5 x 10 -23 kg / m3 < 6.4 × 10-27 about 7.2 × 10-27 27 kilograms per cubic meter < 5.93 x 10 -29 kg / m3 Again, from Wikipedia… “However, as noted in the "matter content" section, the WMAP results in com-bination with the Lambda-CDM model predict that less than 5% of the total mass of the observable universe is made up of visible matter such as stars, the rest being made up of dark matter and dark energy. ”http://en.wikipedia.org/wiki/Observable_universe

An analysis showing that the new theory is consistent with all existing ex-periments. As far as we are aware General Relativity is consistent with all existing experi-ments. General Relativity is a theory of fields. By assuming the energy distri-bution properties of the vacuum are compliant with the energy distribution of the fields of general relativity such that the results would be identical to that of general relativity as a must criteria for any detailed calculations, this theory would then be consistent with general relativity and therefore consistent with all existing experiments relating to space time.

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Proof of compatibility with General Relativity In General relativity there is a POSTULATE that the speed of light is the max-imum velocity attainable and hence remains the same maximum in every refer-ence frame no matter how the reference frame is moving. We use our theory to show that the speed of absorption of the vacuum energy is limited to a maxi-mum, and we calculate this theoretical maximum. Under the assumption that this calculated theoretical maximum turns out to be the speed of light we see that it is proven that our theory is compatible with GR. Further, we then do not need just a POSTULATE for GR we, then have the reason why this postulate is true. We start by looking at the formula for escape velocity V= Sqrt of 2GM/R Where V is the escape velocity in m/s G is Newton’s Gravitational constant (or at least its current value) M is the mass of the escaping object and R is the distance between the escaping object and the centre of gravity. Of the mass the object is escaping from As mentioned above (equation 21) as we are not dealing with 2 masses, but one mass and the energy of space, then just as above we need to modify this equa-tion for the velocity at which space is absorbed (which equals the escape veloci-ty with opposite sign) by dividing the term 2GM/R by 2. We have the new formula that the velocity at which the vacuum is absorbed Vv = -sqrt of GM/R Where Vv is the velocity of the vacuum being absorbed.

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Now as the maximum amount of energy in vacuum space is limited to the Planck energy in Planck space time, we can then say the maximum mass (ener-gy) of vacuum space that can be absorbed is the Planck mass in the Planck time. Now, if this mass is absorbed in the shortest possible time, the velocity of ab-sorption will be a maximum. The minimum time is the Planck time, which equates to a distance of the Planck length. Therefore we have all the variables to find Vvmax. Inserting the amounts in our equation of the velocity of absorption, we get, us-ing

G = 6.673 84 x 10-11 m3 kg-1 s-2

M = 2.176 51 x 10-8 kg

R = 1.616 199 x 10-35 m

Vvmax = 299 792 533 ms-1 Comparing this to the Codata measured value of C C= 299 792 458 ms-1 We find that the results agree. The difference is a function of the accuracy of which we know the values of the physical constants, especially G. Therefore the maximum velocity attainable in our universe is a constant independent of any frame of reference, and this velocity is c, proving that our theory is compatible with the conclusions and experiments of General Relativity We know that the value of G is not known experimentally to a very good de-gree of precision, so working from the Codata measured values we can predict a more accurate value of G G= RV2/M We therefore predict a more accurate value of G would be G = 6.6738366 x 10-

11 m3 kg-1 s-2 vs the Codata value of 6.67384 x 10-11 m3 kg-1 s-2 which is within the measured discrepancy to which G has been found by experiments to date.

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This therefore implies that if G increases every year then C will increase every year and we calculate the increase per year to be 0.011m/s or 11mm/s per year. This will then be easier to find experimentally than the change in G. Now you may be sceptical about this because it is commonly thought that rela-tivity demands that C is constant but; From http://en.wikipedia.org/wiki/Einstein’s_constant

About constants

The Einstein field equation has zero divergence. The zero divergence of the stress-energy tensor is the geometrical expression of the conservation law. So it appears constants in the Einstein equation cannot vary, otherwise this postulate would be violated.

However since Einstein's constant had been evaluated by a calculation based on a time-independent metric, this by no mean requires that G and c must be unvarying constants themselves, but that the only absolute constant is their ra-tio:

Of course we as humans can always decide we like the idea of defining the speed of light as a constant because that makes many things a lot easier . Unfortunately we would then have to live with the consequence that we then have to accept that the total energy of the universe is increasing, and we are creating the increase in energy artificially by deciding to define the speed of light as being constant… As much as we would like to keep the definitions of the fundamental measure-ment quantities in physics as they currently are based on a constant speed of light, the consequence mentioned above unfortunately leads us to to the conclu-sion that the “cost” of this would be too high a price to pay.

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How can we experimentally verify this theory?

1) By measuring G more accurately to see if it is our predicted value of G = 6.6738366 x 10-11 m3 kg-1 s-2 vs the current the Codata value of G = 6.67384 x 10-11 m3 kg-1 s-2 2) G may change over time. By calculating the change in G with time and comparing it observed change. Assuming Mu = Ru

3/G Tu2

Therefore = Ru3 = Mu G Tu

2

As Eu = Mu c

2

Then Ru

3 = G Eu Tu

2 /c2

Thus G = Ru3 c2 / Eu

Tu2

This makes sense, as our reasoning suggested 1/G should be the “current energy density of the vacuum” As the universe expands, the density must decrease so 1/G must get smaller with time. Therefore G must get bigger with time and G must be proportional to T squared As Eu and G/c2 are constants then R3 must be proportional to T2. Therefore the universes expansion must be accelerating.

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Therefore we make the prediction that if we found an accelerated expansion of the universe, this would be an experimental proof for our theory. Further, as we can defne gravitational waves as moving changes in energy den-sity of the vacuum, we should be able to detect such waves by noting the changes in G caused by such waves. From http://en.wikipedia.org/wiki/Einstein’s_constant

About constants

The Einstein field equation has zero divergence. The zero divergence of the stress-energy tensor is the geometrical expression of the conservation law. So it appears constants in the Einstein equation cannot vary, otherwise this postulate would be violated.

However since Einstein's constant had been evaluated by a calculation based on a time-independent metric, this by no mean requires that G and c must be unvarying constants themselves, but that the only absolute constant is their ra-tio:

Of course we as humans can always decide we like the idea of defining the speed of light as a constant because that makes many things a lot easier . Unfortunately we would then have to live with the consequence that we then have to accept that the total energy of the universe is increasing, and we are creating the increase in energy artificially by deciding to define the speed of light as being constant… As much as we would like to keep the definitions of the fundamental measure-ment quantities in physics as they currently are based on a constant speed of

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light, the consequence mentioned above unfortunately leads us to to the conclu-sion that the “cost” of this would be too high a price to pay.

REFERENCES

Please note. Although we are aware that papers submitted for publication in research journals should be placed in the context of current research, with ap-propriate citations to the literature, we have not used any such information di-rectly. All the information we have used from external sources is referenced below

[1] Wikipedia as mentioned in the text

[2] http://physics.nist.gov/cuu/Constants/index.html