EINSTEIN FIELD EQUATIONS OBTAINED ONLY WITH GAUSS CURVATURE AND ZOOM UNIVERSE MODEL CHARACTERISTICS

Transcription

1 EINSTEIN FIELD EQUATIONS OBTAINED ONLY WITH GAUSS CURVATURE AND ZOOM UNIVERSE MODEL CHARACTERISTICS Sergio Garia Chimeno Abstrat Demonstration how to obtain the Einstein Field Equations without using the Stress-Energy Tensor, without using the Bianhi Identities and without using the Energy Conservation to obtain it. Demonstration how to obtain the Einstein Field Equations only using the Gauss Curvature and the zoom universe model harateristis. Gravity in zoom universe model Sphere and Hypersphere example to understand it better Zoom universe model harateristis and zoom speial relativity First we an see: [1] All Speial Relativity Equations Obtained with Galilean Transformation Gauss-Codazzi equations & Riemann Tensor For a Surfae of 2 dimensions embedded in a 3 dimensions spae S : R 2 R 3 expressed by: s = ( x(u 1,u 2 ), y(u 1,u 2 ), z(u 1,u 2 ) ) If we do this: ds = s u1 du 1 + s u2 du 2 = s 1du 1 + s 2du 2 = e 1 du 1 + e 2 du 2 N = e1 e2 e1 e2 We have base vetors and a normal vetor in a point of the surfae And we have: I = ds 2 = ds ds = e 1 e 1 (du 1 ) 2 + e 2 e 2 (du 2 ) e 1 e 2 (du 1 )(du 2 ) = g 11 (du 1 ) 2 + g 22 (du 2 ) g 12 (du 1 )(du 1 ) II = d s d N= e 1 N 1 (du 1 ) 2 + e 2 N 2 (du 2 ) e 1 N 2 (du 1 )(du 2 ) = l 11 (du 1 ) 2 + l 22 (du 2 ) l 12 (du 1 )(du 2 ) The first and seond fundamental form And we have:

2 2 preprint2.nb N 1 = - l 11 e 1 - l 12 e 2 N 2 = - l 21 e 1 - l 22 e 2 The derivation of the normal vetor N expressed in a base vetor oordinates And the Gauss urvature K: K = l11 l22 - l122 g11 g22 - g12 2 = l g The Gauss-Codazzi equations for 2 dimensions an be written as [5]: Γi ab uj - Γ i aj u b + Γ m ab Γi mj - Γ m aj Γi mb = l ab li j - l aj li b, i = 1,2 The Riemann Tensor definition it is: Ri ajb = Γ ab i uj - Γ i aj u b + Γ m ab Γi mj - Γ m aj Γi mb With what for 2 dimensions we an write: Ri ajb = l ab li j - l aj li b, i = 1,2 Rii Tensor & Gauss urvature The Rii Tensor definition it is (for 2 dimensions): R ab = Ri aib = R1 a1b + R2 a2b With what for 2 dimensions we an write: R ab = l ab l i i - l ai l b i, i = 1,2 And we an do: R ab = g ir (l ab l ir - l ai l br ), i = 1,2 The inverse of the metri tensor, we an obtain with the ofator of g, in 2 dimensions we have: g ij = G ij g ij g 11 = g22 g11 g22 -, g 12 = g12 2 -g22 g11 g22 -, g 22 = g12 2 g11 g11 g22 - g12 2 R ab = g ir (l ab l ir - l ai l br ) = g ba (l ab l ir -l ai l br ) g ij If a = b, i = 1,2 = g ba (l ai l br -l ab l ir ) g ij If a b, i = 1,2 We an see if (a = b) (i = r) and if (a b) (i r) with what, knowing the Gauss urvature definition: R ab = g ba [ K ] plane a-b ( in 2 dimensions) We an see that this expression would also be useful for more dimensions as long as we have a diagonal metri and add the urvatures in ommon g = g 11 g 22 g 33 g 44 R 11 = g 11 ( [ K ] plane [ K ] plane [ K ] plane 1-4 ) R 22 = g 22 ( [ K ] plane [ K ] plane [ K ] plane 2-4 ) R 33 = g 33 ( [ K ] plane [ K ] plane [ K ] plane 3-4 ) R 44 = g 44 ( [ K ] plane [ K ] plane [ K ] plane 4-1 )

3 preprint2.nb 3 Rii Salar & Gauss urvature The Rii Salar definition it is (for 2 dimensions): R = g ab R ab = g 11 R 11 + g 12 R 12 + g 21 R 21 + g 22 R 22 With what for 2 dimensions we an write ( δ = Kroneker delta ): R = g ab R ab = g ab g ba [ K ] plane a-b = δ a a [ K ] plane a-b = 2 [ K ] plane a-b We an see that this expression would also be useful for more dimensions as long as we have a diagonal metri and add all urvatures g = g 11 g 22 g 33 g 44 R = 2 [ K ] plane [ K ] plane [ K ] plane [ K ] plane [ K ] plane [ K ] plane 3-4 Einstein Tensor & Gauss urvature The Einstein tensor definition it is: G ab = R ab g ab R With what, always as long as we have a diagonal metri g = g 11 g 22 g 33 g 44 G ab = R ab g ab R = g ba [ K ] plane a-b - g ab [ K ] all planes = - g ab [ K ] planes exept plane a-b G 11 = R g 11 R = - g 11 ( [ K ] plane [ K ] plane [ K ] plane 4-3 ) G 22 = R g 22 R = - g 22 ( [ K ] plane [ K ] plane [ K ] plane 4-3 ) G 33 = R g 33 R = - g 33 ( [ K ] plane [ K ] plane [ K ] plane 4-2 ) G 44 = R g 44 R = - g 44 ( [ K ] plane [ K ] plane [ K ] plane 1-3 ) Zoom universe model & Gauss urvature Now first let s alulate how would be the urvature of eah plane if we do the zoom level fixed ( R ) we would have a hypersphere with 3 dimensions on its surfae ( α, θ, ϕ ) and a radius R. But for eah 2 dimensions plane ( plane α-θ ) ( plane α-ϕ ) ( plane ϕ-θ ) we have a 2 dimensions sphere with same radius R and same spherial Gauss urvature 1 R 2 [ K ] plane α-θ or [ K ] plane α-ϕ or [ K ] plane ϕ-θ = 1 R 2 We know for [1] that: U = - F R (potential energy, work done on an objet is found by multiplying fore and distane) [ K ] plane α-θ or [ K ] plane α-ϕ or [ K ] plane ϕ-θ = 1 R 2 = FF UU Multiplying both sides for Area and radius ( A R ): [ K ] plane α-θ or [ K ] plane α-ϕ or [ K ] plane ϕ-θ = 1 R 2 = FF UU = FFAR UUAR Now we know [1] that U = - F R = m 2, we know that the sphere area it s A = 4 R 2, we know that the Pressure it s P = F A = F R A R = -U Vol, (

4 4 preprint2.nb Vol = volume ) and in one volume we an have n partiles P = F A = F R A R = -U Vol = - n m2 Vol,with what: [ K ] plane α-θ or [ K ] plane α-ϕ or [ K ] plane ϕ-θ = 1 = FF R 2 UU = FFAR UUAR = - m2 4 R 2 n m 2 U 2 R Vol We know that the mass density ρ m = n m Vol, we know the light esape veloity 2 = 2 G m, ( G it is the gravitational onstant ) and we know [1] R that: v = γ [ K ] plane α-θ or [ K ] plane α-ϕ or [ K ] plane ϕ-θ = 1 R 2 = FF UU = FFAR UUAR = - m2 4 R 2 n m 2 U 2 R Vol = - m 2 G m 4 R2 ρm 2 m 2 4 R R = - 8 G 4 ρ m γ 2 v 2 We an see what all mass and mass density in this equation it s refer to all mass and mass density into all universe plane sphere The rest of planes in Zoom universe model & Gauss urvature For alulating the rest of planes we an do one of this dimensions fixed ( α, θ, ϕ ) and r would be part of the surfae: with α fixed ( Α ) [ K ] plane r-θ, [ K ] plane r-ϕ with θ fixed ( Θ ) [ K ] plane r-α, [ K ] plane r-ϕ with ϕ fixed ( Φ ) [ K ] plane r-θ, [ K ] plane r-α All Gauss urvature will be 1 (radius) 2 = 1 Α 2 or 1 Θ 2 or 1 Φ 2 and all plane sphere ontains all universe plane sphere mass and mass density regardless of the dimension used as a radio [ K ] rest of planes = 1 = 1 or 1 or 1 = FF (radius) 2 Α 2 Θ 2 Φ 2 UU = FFA(radius) UUA(radius) Now we an see the energy used to be in a position on the surfae of the sphere it is the Kineti energy ( K ), U = - F (radius) = K = mv 2, and the Pressure P = F A = F (radius) A (radius) = -K Vol = - n mv 2 Vol [ K ] rest of planes = 1 = 1 or 1 or 1 = FF (radius) 2 Α 2 Θ 2 Φ 2 UU = FFA(radius) KKA(radius) = - mv2 (radius) 2 n mv 2 K 2 (radius) Vol We know that the mass density ρ m = n m Vol, we know the esape veloity v2 = 2 G m, ( G it is the gravitational onstant ) and we know for [1] v R = γ [ K ] rest of planes = 1 = 1 or 1 or 1 = FF (radius) 2 Α 2 Θ 2 Φ 2 UU = FFA(radius) KKA(radius) = - mv2 (radius) 2 n mv 2 = - 4 m 2 G m (radius)2 ρm v 2 = - 8 G ρm v2 γ 4 K 2 (radius) Vol m 2 v 4 (radius) (radius) m 2 = - 8 G 4 ρ 4 m γ 2 2 We an see that the veloity of r dimension either as a radius or as a surfae should be, that agrees with the exposed in [1] [ light transmission medium and waves veloity traveling in it ( ) ] Join Einstein Tensor & Zoom universe model Now if we join all equations ( the sphere have a diagonal metri tensor ) ( index 1 = r, index 2 = α, index 3 = θ, index 4 = ϕ): G ab = R ab g ab R = - g ab [ K ] planes exept plane a-b = g ab 8 G 4 ρ m γ 2 2 veloity planes We an see that the sum of 3 bi-dimensional veloity omponents squared gives us a veloity squared in 3 dimensions G 11 = R g 11 R = 8 G ρ 4 m γ 2 g 11 v2 plane v2 plane v2 plane 3-4 = 8 G ρ 4 m γ 2 g 11 v2 ube planes G 22 = R g 22 R = 8 G ρ 4 m γ 2 g 22 2 plane plane v2 plane 3-4 = 8 G ρ 4 m γ 2 g 22 v2 ube planes G 33 = R g 33 R = 8 G ρ 4 m γ 2 g 33 2 plane plane v2 plane 2-4 = 8 G ρ 4 m γ 2 g 33 v2 ube planes G 44 = R g 44 R = 8 G ρ 4 m γ 2 g 44 v2 plane plane plane 3-1 = 8 G ρ 4 m γ 2 g 44 v2 ube planes 1-2-3

5 preprint2.nb 5 Join Metri Tensor & veloity ube planes squared The veloity squared an be expressed in terms of the first fundamental form: v2 ube planes = ds 2 = ds ds = e 1 e 1 (du 1 ) 2 + e 2 e 2 (du 2 ) 2 + e 3 e 3 (du 3 ) e 1 e 2 (du 1 )(du 2 ) + 2 e 1 e 3 (du 1 )(du 3 ) + 2 e 2 e 3 (du 2 )(du 3 ) = g 11 (du 1 ) 2 + g 22 (du 2 ) 2 + g 33 (du 3 ) g 12 (du 1 )(du 2 ) + 2 g 13 (du 1 )(du 3 ) + 2 g 23 (du 2 )(du 3 ) = g d (du )(du d ) v2 ube planes = g d (du ) du d, with index = 1, 2, 3, with index d = 1, 2, 3 v2 ube planes = g d (du ) du d, with index = 2, 3, 4, with index d = 2, 3, 4 v2 ube planes = g d (du ) du d, with index = 1, 3, 4, with index d = 1, 3, 4 v2 ube planes = g d (du ) du d, with index = 1, 2, 4, with index d = 1, 2, 4 Now let s multiply those metri tensors of 3 dimensions ( g d ) with the metri tensor of 4 dimensions ( g ab ): ( We must bear in mind that the missing dimension in g d will always be in g d so that in the end we have 2 metri tensors of 4 dimensions, I develop a ase to make it look better ): g ab g d (du )(du d ) = ( e a e b ) ( e e d ) (du )(du d ) = ( e a e d ) ( e e b ) (du )(du d ) = g ad g b (du )(du d ) g 11 v2 ube planes = g 11 g d (du )(du d ), with index = 2, 3, 4, with index d = 2, 3, 4 = g 11 g 22 (du 2 )(du 2 ) + g 11 g 23 (du 2 )(du 3 ) + g 11 g 24 (du 2 )(du 4 ) + g 11 g 32 (du 3 )(du 2 ) + g 11 g 33 (du 3 )(du 3 ) + g 11 g 34 (du 3 )(du 4 ) + g 11 g 42 (du 4 )(du 2 ) + g 11 g 43 (du 4 )(du 3 ) + g 11 g 44 (du 4 )(du 4 ) = g 12 g 12 (du 2 )(du 2 ) + g 12 g 13 (du 2 )(du 3 ) + g 12 g 14 (du 2 )(du 4 ) + g 13 g 12 (du 3 )(du 2 ) + g 13 g 13 (du 3 )(du 3 ) + g 13 g 14 (du 3 )(du 4 ) + g 14 g 12 (du 4 )(du 2 ) + g 14 g 13 (du 4 )(du 3 ) + g 14 g 14 (du 4 )(du 4 ) With what: g ab 2 veloity planes = g ad g b (du )(du d ), with (index and d) (index a and b) And we have: R ab g ab R = 8 G 4 ρ m γ 2 g ad g b (du )(du d ), with (index and d) (index a and b) Finally some tensor algebra Multiplying both sides by g bi : R a i δ a i R = 8 G 4 ρ m γ 2 g ad δ i (du )(du d ) Multiplying both sides by g aj : R ij gij R = 8 G 4 ρ m γ 2 δ d j δ i (du )(du d ) R ij gij R = 8 G 4 ρ m γ 2 (du i )(du j ) The derivative of a oordinate is the veloity omponent in that oordinate ( Remember that the speed in oordinate 1 ( r ) it is ): R ij gij R = 8 G 4 ρ m γ 2 v i v j This expression ρ m γ 2 v i v j it is the stress energy tensor T ij with what, we have the Einstein field equations:

6 6 preprint2.nb R ij gij R = 8 G 4 T ij Gravity We an see that the previous equation was obtained with spheres in one point that inludes the whole universe, with all its radius and all its mass-energy density ( beause mass and energy are related by E = m 2 ). This mass-energy density It ause the all universe spherial urvature form. Obviously if we have a smaller mass-energy, we will have a smaller sphere with a smaller radius. And the urvature of this 2 dimension plane we an alulate it with the radius of that s smaller sphere, similar to how we alulate the urvature of a 1 dimension line using the radius of irles. It was already known that gravity was aused by the urvature of the 4 dimensions of spae-time, but with this zoom universe model and its 4 dimensions of spae-zoom it may look better. A objet with onstant veloity auses a entrifugal aeleration as shown in [1] your veloity will be perpendiular and entrifugal aeleration it travels for the radius dimension ( r ) there is no urvature. But if that objet have a aeleration, this aeleration will be added to the entrifugal aeleration with what will bend that radius and how the veloity is perpendiular, we see how the aeleration of an objet urves the spae-zoom. And if an aeleration auses a urvature, then a urvature then auses an aeleration. That's the gravity

7 preprint2.nb 7 Sphere example (drawn with one dimension less) (without 3d perspetive) First we draw a irle with yours polar oordinates ( x, y ) ( r, ϕ ): x = r os ϕ y = r sin ϕ Now we take the radium dimension and we replae it with another irle perpendiular to the rest of the dimensions r z x = r os θ z y = r sin θ We an see that the z y is a new dimension and the z x mathes the radius, with that: z = r sin θ x = r os θ os ϕ y = r os θ sin ϕ But if we want the last angle to begin with the last dimension: z = r os θ x = r sin θ os ϕ y = r sin θ sin ϕ We have the spherial oordinates ( x, y, z ) ( r, θ, ϕ ) ds 2 = dr 2 + r 2 dθ 2 + r 2 sin 2 θ dϕ 2 N = (x,y,z) r = ( sin θ os ϕ, sin θ sin ϕ, os θ ) If we make onstant r dr = g = g 22 g 23 g 32 g 33 = l = l 22 l 23 l 32 l 33 = l θθ g θθ g θϕ g ϕθ g ϕϕ = l θϕ r 2 r 2 sin 2 θ = -r l ϕθ l ϕϕ -r sin 2 θ K plane θ-ϕ = 1 r 2, K plane r-ϕ =, K plane θ-r = Rii Tensor = R 22 R 23 R 32 R 33 = R θθ R θϕ = 1 R ϕθ R ϕϕ sin 2 θ, Rii Salar = 2 r 2 Einstein Tensor = G 22 G 23 G 32 G 33 = G θθ G θϕ = G ϕθ G ϕϕ We observed that the flat oordinate and the urved oordinates ould be hosen differently

8 8 preprint2.nb If we make onstant θ dθ = and: z = θ os r x = θ sin r os ϕ y = θ sin r sin ϕ K plane θ-ϕ =, K plane r-ϕ = 1 θ 2,K plane θ-r = If we make onstant ϕ dϕ = and: z = ϕ os θ x = ϕ sin θ os r y = ϕ sin θ sin r K plane θ-ϕ =, K plane r-ϕ =, K plane θ-r = 1 ϕ 2 And we an use that s spheres to determine the urvature of any pseudo-spherial objet Hypersphere example (drawn with two dimension less) (without 3d perspetive) (without 4d perspetive) First we draw a sphere with ours spherial oordinates ( x, y, z ) ( r, θ, ϕ ): z = r os θ x = r sin θ os ϕ y = r sin θ sin ϕ Now we take the radium dimension and we replae it with another irle perpendiular to the rest of the dimensions: r z y = r os α z z = r sin α We an see that the z z is a new dimension and the z y mathes the radius, with that:

9 preprint2.nb 9 z z = r sin α z = r os α os θ x = r os α sin θ os ϕ y = r os α sin θ sin ϕ But if we want the last angle to begin with the last dimension: z z = r os α z = r sin α os θ x = r sin α sin θ os ϕ y = r sin α sin θ sin ϕ We have the hyperspherial oordinates ( x, y, z, z z ) ( r, α, θ, ϕ ) ds 2 = dr 2 + r 2 dα 2 + r 2 sin 2 α dθ 2 + r 2 sin 2 α sin 2 θ dϕ 2 N = (x,y,z,zz) r = ( sin α sin θ os ϕ, sin α sin θ sin ϕ, sin α os θ, os α ) If we make onstant r dr = g = l = g 22 g 23 g 24 g 32 g 33 g 34 = g 42 g 43 g 44 l 22 l 23 l 24 l 32 l 33 l 34 = l 42 l 43 l 44 g αα g αθ g αϕ g θα g θθ g θϕ = g ϕα g ϕθ g ϕϕ l αα l αθ l αϕ l θα l θθ l θϕ = l ϕα l ϕθ l ϕϕ r 2 r 2 sin 2 α -r -r sin 2 α r 2 sin 2 α sin 2 θ -r sin 2 α sin 2 θ K plane α-θ = 1 r 2, K plane α-ϕ = 1 r 2, K plane θ-ϕ = 1 r 2 K plane r-α =, K plane r-θ =, K plane r-ϕ = Rii Tensor = R 22 R 23 R 24 R αα R αθ R αϕ R 32 R 33 R 34 = R θα R θθ R θϕ = R 42 R 43 R 44 R ϕα R ϕθ R ϕϕ G 22 G 23 G 24 G αα G αθ G αϕ Einstein Tensor = G 32 G 33 G 34 = G θα G θθ G θϕ = G 42 G 43 G 44 G ϕα G ϕθ G ϕϕ 2 2 sin 2 α 2 sin 2 α sin 2 θ -1 -sin 2 α -sin 2 α sin 2 θ, Rii Salar = 6 r 2 We observed that the flat oordinate and the urved oordinates ould be hosen differently, as in the example of the sphere. And as in the example of the sphere we an use spheres to determine the urvature of any pseudo-hyperspherial objet, as the zoomuniverse model.

10 1 preprint2.nb Referenes [1] All Speial Relativity Equations Obtained with Galilean Transformation [2] Superfiies-1 [3] Superfiies-2 [4] Superfiies-3 [5] El Teorema Egregium de Gauss [6] Curvatura en variedades [7] La Teoría de la Relatividad [8] Teoria da Relatividade Geral