If the universe started as a point (BigBang) and is expanding symmetrically then it would result in an expanding 4 Dimensional sphere. Consider the universe to be a 4 Dimensional sphere with time as the radial Dimension. Then all of space at “present” would be represented by the “3D Surface” of the sphere. Thus each point in space at “present” would be a boundary point. Now the expanding universe would correspond to an increase in the radial dimension or time. If the sphere is to remain a sphere then it would have to be accompanied by an equal increase of the surface or space. Let us take a 2D cross section from this 4D sphere containing its centre. This would be a familiar circle with the radius representing time and the circumference 1D of space. Thus the size of this 1D of space at any time t would be given by 1D space = Constant*2*pi*t Thus d(space)/d(time) = Constant*2*pi = CONSTANT. Could this CONSTANT = C = Speed of light? That is the circumference is increasing at the speed of light? Getting back to the 4D sphere this would mean that each point in space at “present” expands out into a sphere of radius C. Speculating further, if around matter this expansion is changed, that is d(space)/d(time) is not equal to C. It could well be that matter is none other than those areas in which this constant is made less than C. Thus matter would be seen to attract each other or gravity would be proportional to the extent that matter has altered this constant C around it. The increase in entropy or disintegration of matter would be a direct result of the rate of expansion within matter tending towards the rate C which is the rate of empty space Time. This would also explain why the Hubel Constant, the rate of expansion of the visible universe, is far less than C, as it has much matter. In the above model , if space is a 3D spherical surface then a space ship heading in one direction should come back to the starting space coordinates, having travelled the circumference, just like going round the world. However the hitch is he has to travel faster than the speed of light as the circumference is increasing at this rate. Objections. The curvature as measured via cosmic microwave radiation suggest that the universe is flat. However this is also consistent with a rate of expansion = C as if it expanded for 14 Billion years at that rate, the distances measure in back ground radiation measurements , will only be that of a tangential surface, which will always be flat.
As stated what is proposed is in the absence of matter d(space)/d(time) = expansion = C (speed of light, this is for the 2D case)
I thought I will add some calculations to show how hubbles constant is obtained from this model. Hubbels law Consider again the case of the 2 dimensional universe.That is a universe in which the radius of the circle is time and the circumference space. As stated previously d(space)\d(time) = C (speed of light). Now consider 2 objects on the circumference. As this universe expands, these 2 objects will also move apart. However the angle or the arc will always remain constant. S0 be the distance between them at some time T0 . Then (S0 + ds)/(T0 + dt ) = S0/T0 = angle between point at centre of circle = Constant S0 + ds = S0(T0 + dt ) /T0 Dividing by dt and re arranging gives ds/dt = S0(T0 + dt ) /(T0 * dt) - S0 /dt ds/dt = S0 / T0 That is Velocity = ds/dt = (1/T0 ) S0 Which is Hubbles law where (1/T0 ) = H = Hubble’s Constant Thus the inverse of the hubbles constant will give the age of the universe. Here is a Wiki entry confirming this The Hubble constant H0 has units of inverse time, i.e. H0 ~ 2.29×10−18 s−1. “Hubble time” is defined as 1 / H0. The value of Hubble time in the standard cosmological model is 4.35×1017 s or 13.8 billion years. (Liddle 2003, p. 57) The phrase "expansion timescale" means "Hubble time".[2]. If the value of H0 were to stay constant, a naive interpretion of the Hubble time is that it is the time taken for the universe to increase in size by a factor of e (because the solution of dx/dt = xH0 is x = s0exp(H0t), where s0 is the size of some feature at some arbitrary initial condition t = 0). However, over long periods of time the dynamics are complicated by general relativity, dark energy, inflation, etc., as explained above." The huble time is less as the slowing down of the expansion as stated due to matter has not been taken into account in the above As postulated the change in the ratio d(space)/d(time) is what leads to the curvature as given by the Field equations of general relativity. To give an example, take the following solution by Sir Arthur Stanley Eddington to the General Relativity field equations. ds2 = ( 1 + Gm/2C2r)4 (dx2 + dy2 + dz2 ) - C2dt2( 1 - Gm/2C2r)2 / ( 1 + Gm/2C2r)2 Setting ds2 = 0 gives the metric for a beam of light, which is proposed to be also the ratio of space/time created in the expansion around matter. Thus d(space)2 / d(time)2 = C2( 1 - Gm/2C2r)2 / ( 1 + Gm/2C2r)6 What is interesting in the above model is that the arrow of time follows the expansion. That is when we step into the future, we also step into newly created space-time.
Special Relativity Special Relativity Consider again the case of the 2 dimensional universe. Consider 2 reference frames O and O’ such that at some time To O’ is So distance from O. Thus from 2 above its velocity as seen by O is given by ds/dt = So / To = Vo which is a constant. It is worth noting that this corresponds to newtons first law as Vo will remain constant with expansion. Let the units of the Time dimension be such that the constant C is 2pi. (It has the same unit’s as S that is space as such it correspond to a simple circle) Thus we have 2piTo = Do the Circumference of the circle or total of all space at the time To . D/T = 2pi will be maintained as this 2d universe expands. Consider a beam of light leaving O at To and Travelling towards O’ at a velocity 2*pi or the speed of light in this universe. In O’ this light beam would have to travel a distance So . Lets say it takes a time dT’. The we get 2pi(dT’) = So = To*Vo ----- 3 Consider the passage of the light as seen by O. As in the time the light travels, O’ is moving away from O at a velocity Vo,it will see as having to travel a greater distance. Thus we get 2pi*(dT) = So + dS = To* Vo + (dT)*Vo Substituting from 3 2pi*(dT) = 2pi*(dT’) + (dT)*Vo Rearranging 2pi*(dT’) = 2pi*(dT) - (dT)*Vo Dividing both sides by 2pi*(dT) we get (dT’) / (dT) = 1 - Vo/2pi = gamma ------- 4 4 is the time dilation equation of special relativity. To see the correspondence of this coordinate and units with that normally used, we can equate the constants gamma as it is a ratio and should be the same. Thus if O’ is moving away from O at a velocity Vx we get. gamma = 1 - Vo/2pi = SQRT (1 - Vx^ 2/ C^2) ------5 Where C is the velocity of light in the common coordinate system .Squaring both sides and solving 5 for Vx Vx = SQRT( ( Vo/pi)( 1- Vo/4pi)) ---------6 It can be seen by substitution when Vo = 2pi the velocity of light in this coordinate system we get Vx = C the velocity of light, as is to be expected. By substituting (dT’) = dS’/Vo and (dT) = dS/Vo We get (dS’)/(dS) = = 1 - Vo/2pi = gamma ------- 7 Which is length contraction of Special relativity. The intent of the demonstration above is not to propose that the shape of the universe is spherical but rather to demonstrate that if the universe had a shape, and maintained that shape with expansion, then the expansion alone can account for the laws of motion.
Red Shift Red Shift and Distance Consider a segment of space or a length λe at some time Te. This segment of length will always subtend a fixed angle θ at the origin with the progress of time. Thus consider it length λo some time later at To. The angles θ is constant as given in equation 1 Thus we have λe / Te. = λo/ To. -----(1) Rearranging λo / λe = To./ Te = 1 + Z ( where Z = redshift) Te = To/(1+Z) Time Taken D = To - Te = To - To/(1+Z) D = (To + ZTo - To)/ (1+Z) D = Z*To/(1+Z) ------(2) . I used Data from the NASA/IPAC Extragalactic database master list of galaxy distances. Compiled by Barry F.Madore and Ian P. Steer for the following galaxies Galaxy Name NGC7131 2dFGRS S839Z607 MCG -01-02-001 CGCG 409-007 NGC 0105 [P96] J003618.17+112334.7 NGC 0252 UGC 00607 UGC 00646 CGCG 307-023 These are some galaxies for which distance has been measured independent of redshift. The red sgift is also available as such I was able to do a linear regresion for the original hubbel's relation and the one proposed above The correlation coefficients (R) for the 2 relations: D <=> Z ---------- (3) Hubbell’s original version D <=> θ = Z/(Z+1) --(4) The proposed improvement was calculated. It can be seen that both 3 and 4 give high values close to 1 for the correlation coefficients (R). Thus supporting both linear relationships. However, it is observed that for this data set, the proposed improvement (4) has a R value 0.993699854 > 0.992997374 the R value for the original equation (3). The linear regression equation for the 2 cases can be seen to be D = 4698.365(Z) + 8.654325 ---- for (1.1) D = 5038(Z/(Z+1)) + 13.30627 ---for (1.2) As can be seen in both cases the intercept for Z = 0 will be ignored. Thus then the 2 equations become D = 4698.365(Z) ----------------------(5) Hubbell’s original with Hubbell’s Constant H = C/(4698.365) D = 5038(Z/(Z+1)) = 5038(θ) -------(6) Where T = 5038(Mpc) the age of the universe .The units of this constant can be in Distance unit or in Seconds. The conversion from distance to time is the speed of light C , just as the original equation. It is important to recognize that equation 2 will give a distance measure in light years that is always less than the age of the universe as z/(z+1) < 1
