A displacement (or metric) is associated with some property of nature, and may be represented as a vector. Five metric vectors are;

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1 A displacement (or metric) is associated with some property of nature, and may be represented as a vector. Five metric vectors are; - spatial vector - temporal vector - matter vector - massive vector - electric vector Each vector has a unique frame of reference. The origins (tails) of each vector may or may not be coincident. The metric vectors may be connected to each other by sharing common components. The spatial and temporal vectors are connected, giving a space-time structure. The electric and massive vectors are also connected to a matter vector. The matter vector is connected to the space-time structure. It also influences the geometry of the space-time structure. All metric vectors are directly or indirectly connected to each other. The Reissner-Nordstrom metric may be derived from the metric vectors. If suitable conditions apply, the RN metric will reduce to the Schwarzschild metric, which may in turn reduce to the Minkowski metric. The General Metric Vector; A metric (R) may be associated with some property of nature (n) and represented as a vector (R n ); R n = R n1 i n + R n j n + R n3 k n Where; (R 1, R, R 3, R 4, R 5 ) are the metric vectors n is a vector identifier (natural property) n = 1 represents a spatial displacement n = represents a temporal displacement n = 3 represents a displacement associated with matter n = 4 represents a displacement associated with mass n = 5 represents a displacement associated with electric charge (i n, j n, k n ) represent a frame of reference associated with a vector n (directional unit vectors) (R n1, R n, R n3 ) represent scalar components of displacement The vector has magnitude; R n = R n4 The magnitude is related to the components; R n1 + R n + R n3 = R n4 (E01)

2 Sub-components (R n0, R n5 ) are; R n0 = R n1 + R n = R n4 - R n3 R n5 = R n + R n3 = R n4 - R n1 (E0) (E03) Angular Geometry; The general metric vector has four angles of interest (X n1, X n, X n3, X n4 ); X n1 + X n3 = ½π X n + X n4 = ½π Connectivity; R n1 = R n0 Cos(X n1 ) and R n = R n0 Sin(X n1 ) (E04) R n0 = R n4 Cos(X n ) and R n3 = R n4 Sin(X n ) (E05) Two vectors may be connected to each other. A vector connection is represented as a connection of their components. Two components are connected if they are co-incident, and have equal magnitudes. A square bracket signifies disconnection (equal magnitudes but no co-incidence). R 44 = R 30 represents connected components R [40] = R [35] represents disconnected components with equal magnitudes. Equal angles may also be disconnected; X [11] = X [] The Spatial Vector; The spatial vector (R 1 ) is; R 1 = R 11 i 1 + R 1 j 1 + R 13 k 1 (E06) The Temporal Vector; The temporal vector (R ) is; R = R 1 i + R j + R 3 k (E07) The temporal vector has any component (R N ) defined as; R N = ct N (E08) Where; T N is a component of time c is the light constant The Space-Time Structure; The spatial vector is connected to the temporal vector; R 14 = R 1 (E09) Angular geometry is also related; X [11] = X [] (E10) This combination gives the space-time structure. May 31, 017 P a g e

3 The Matter Vector; The matter vector (R 3 ) is; R 3 = R 31 i 3 + R 3 j 3 + R 33 k 3 (E11) The matter length scale (R 34 ) may be written as; R 34 = R 31 + R 3 + R 33 = R 30 + R 33 = R 31 + R 35 (E1) The matter vector connects to the massive and electric length scales (R 44, R 54 ); R 30 = R 44 massive connection (E13) R 33 = R 54 electric connection (E14) Giving; R 34 = R 44 + R 54 (E15) (matter length scale) = (massive length scale) + (electric length scale) The matter vector also connects to the space-time structure; X 31 = X R 30 = R 4 R 31 = R 0 R 3 = R 3 (E16) (E17) (E18) (E19) The Massive Vector; The massive vector (R 4 ) is; R 4 = R 41 i 4 + R 4 j 4 + R 43 k 4 (E0) Two conditions apply to the massive vector; R [40] = R [35] X [41] = X [4] (E1) (E) The second condition gives; Cos(X 41 ) = Cos(X 4 ) R 41 /R 40 = R 40 /R 44 R 41 R 44 = R 40 (E3) The massive vector is connected to the matter vector; R 44 = R 30 (E4) The Electric Vector; The electric vector (R 5 ) is; R 5 = R 51 i 5 + R 5 j 5 + R 53 k 5 (E5) The electric vector is connected to the matter vector; R 54 = R 33 (E6) May 31, 017 P a g e 3

4 The General Metric; The temporal sub-component (R 0 ) is; R 0 = (R 1 + R ) ½ R 0 = R 1 + R R 4 Cos (X ) = R 1 + R c T 4 Cos (X ) = R 1 + R c T 4 Cos (X ) = R 14 + R c T 4 Cos (X ) = (R 10 + R 13 ) + R c T 4 Cos (X ) = R 11 /Cos (X [11] ) + R 13 + R c T 4 Cos (X ) = R 11 /Cos (X ) + R 13 + R c T 4 Cos (X 31 ) = R 11 /Cos (X 31 ) + R 13 + R c T 4 (R 31 /R 30 ) = R 11 (R 30 /R 31 ) + R 13 + R c T 4 (R 30 + R 33 - R [35] )/R 30 = R 11 (R 30 /R 31 ) + R 13 + R c T 4 (R 30 + R 33 - R [40] )/R 30 = R 11 (R 30 /R 31 ) + R 13 + R c T 4 (R 30 + R 54 - R 41 R 44 )/R 30 = R 11 (R 30 /R 31 ) + R 13 + R c T 4 (R 44 + R 54 - R 41 R 44 )/R 44 = R 11 (R 30 /R 31 ) + R 13 + R (see E0) (see E05) (see E08) (see E09) (see E0) (see E04) (see E10) (see E16) (see E04) (see E1) (see E1) (see E3, E6) (see E13, E14) c T 4 (1 + R 54 /R 44 - R 41 /R 44 ) = R 11 (R 30 /R 31 ) + R 13 + R The general metric equation may be written as; c T 4 (1 + R 54 /R 44 - R 41 /R 44 ) = R 11 (1 + R 54 /R 44 - R 41 /R 44 ) -1 + R 13 + R or; c T 4 (1 + R 54 /R 44 - R 41 /R 44 ) = R 11 (1 + R 54 /R 44 - R 41 /R 44 ) -1 + R 13 + c T (see E08) A general velocity equation is; c (1 + R 54 /R 44 - R 41 /R 44 ) = (R 11 /T 4 )(1 + R 54 /R 44 - R 41 /R 44 ) -1 + R 13 /T 4 + R /T 4 c (1 + R 54 /R 44 - R 41 /R 44 ) = v 11 (1 + R 54 /R 44 - R 41 /R 44 ) -1 + v 13 + v Velocity Definitions; The velocities have differential definitions; v 11 = R 11 /T 4 = ꝺr/ꝺt v 13 = R 13 /T 4 = ꝺu Ω /ꝺt v = R /T 4 = ꝺs/ꝺt Where; Ω = U Ω /r and rꝺω = ꝺu Ω May 31, 017 P a g e 4

5 Matter Components; The components of matter may be written as length scales; R 41 = r S R 44 = r R 54 = r Q (the Schwarzschild radius) (massive length scale) (electric length scale) Substitution of velocities and length scales gives a differential velocity equation; c (1 + r Q /r - r S /r) = (ꝺr /ꝺt )( 1 + r Q /r - r S /r) -1 + ꝺu Ω /ꝺt + ꝺs /ꝺt Giving the Reissner-Nordstrom metric; c ꝺt (1 + r Q /r - r S /r) = ꝺr ( 1 + r Q /r - r S /r) -1 + ꝺu Ω + ꝺs c ꝺt (1 + r Q /r - r S /r) = ꝺr ( 1 + r Q /r - r S /r) -1 + r ꝺω + ꝺs If; r Q = 0 Then the Schwarzschild metric is obtained; c ꝺt (1 - r S /r) = ꝺr ( 1 - r S /r) -1 + r ꝺω + ꝺs c ꝺt (1 - r S /r) = ꝺr ( 1 - r S /r) -1 + r ꝺω + c ꝺt If; r Q = 0 and r S = 0 Then the Minkowski metric is obtained; c ꝺt = ꝺr + r ꝺω + ꝺs c ꝺt = ꝺr + r ꝺω + c ꝺt Conclusion; Metric vectors may be connected, giving the Reissner-Nordstrom metric equation. May 31, 017 P a g e 5