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EXISTENCE OF A 10-DIMENSIONAL SPACETIME

Bernard Marcus We form the 10-dimensional spacetime T × S , where T is the 4-dimensional time continuum of all entities of time in the Special Theory of Relativity (STR) that are coordinatized by Einstein's procedure of synchro-nizing identical clocks in STR, and S is the 6-dimensional space continuum of all entities of space in STR that are coordinatized by purely geometric procedures referred to by Einstein in STR. We extend the Lorentz trans-formations and their relationship with the occurrences of point events to all of T × S .

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I . INTRODUCTION For brevity, inertial reference frames in STR will be called inertial frames. In all that follows, for each inertial frame I in STR, the time continuum of I will be denoted by TI , and the space continuum of I will be denoted by SI . For each inertial frame I in STR, CT I will always denote a coordinate system from STR of TI , CS I will always denote a right-handed Cartesian coordinate system from STR of SI , and CI will always denote a coordinate system from STR of I that is formed from coordinate systems CT I and CS I in the usual fashion in STR. All of these coordinates systems in STR will be constructed by procedures equivalent to those prescribed by Einstein. * Each element of TI will be a point-like entity of time in physical reality which is coordinatized by a coor-dinate system CT I of TI ; each element of SI will be a point-like entity of time in physi-cal reality which is coordinatized by a coordinate system CS I of SI ; and each element of I will be a point-like entity of time and space in physical reality which is coordinatized by a coordinate system CI of I . Let the standard units of length and time in all of these coordinate systems be one centimeter and one second, respectively. Let T be the set of all elements of the time continua of inertial frames, let S be the set of all elements of the space continua of inertial frames and let X be the set of all elements of inertial frames. If STR is correct, then T has four large dimensions, S has six large dimensions and X has seven large dimensions* These two continua T and S are disjoint, and we can coordinatize the elements of either one of these continua with instruments and accompanying procedures that cannot coordinatize the elements of the other continuum. Indeed, T and S differ so much that the coordinates of the elements of T are real numbers, while the coordinates of the elements of S are 3-tuples of real num-bers. Due to this disparity between T and S , they must contribute, in some way, ten large dimensions to the spacetime in physical reality in which we exist. It would be a mis-take to postpone introducing these ten large dimensions, since they do exist if STR is cor-rect, and can be expected to play a major role in the behavior that takes place in physical reality. A brief review of the set theory used here now follows. I I . PERTINENT ASPECTS OF SET THEORY The symbol ∈ denotes "is an element of " and "belongs to" and also "belonging to" when acting as an adjective. Examples are: x ∈ A denotes the phrase "x is an element of A" and also the phrase "x belongs to A" ; x , y ∈ A denotes the phrase "x and y are elements of A" and also the phrase "x and y belong to A" , . . . . In these latter cases, x and y could be equal. " for each x ∈ A " denotes the phrase " for each element x belonging to A" . It is common practice to also use " Let x ∈ A " to denote the phrases "Let x be any element of A" and also " Let x be an arbitrarily chosen element of A " ; and " Let x , y ∈ A" denote "Let x and y be arbitrarily chosen elements of A" and also "Let x and y be any elements of A " , . . . . In these latter cases, x and y could again be equal An example of the use of ∈ is the definition " A ⊆ B iff for each x: if x ∈ A then x ∈ B ." The notation ⊆ is an abbreviation of the phrase " is a subset of " and

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" is contained in ". An example of the use of ⊆ is the property that for all sets A and B , A = B iff A ⊆ B and B ⊆ A where " iff " is an abbreviation of " if and only if " . We will introduce the standard notation used when defining a set by its properties. (1) " { " denotes the phrase “the set of all” , (2) " | " denotes the phrase "such that" (some authors use " : " ) and (3) " } " denotes the end of the definition of this set. Each definition of a set by its properties can then be put into the form { A | P } , where A is a variable or a set described by variables and P is a property satisfied by the variable A or by the variables describing A . The use of the notation { A | P } is ubiquitous in set theo-ry because { A | P } stands out clearly when used; and in set theory, virtually all sets are defined by properties. Examples of the use of this notation now follow. First, let A and B be any given sets. (1) { a1 , a2 , . . . , an } = {x | x = ak for some k = 1, 2 , …, n } (2) A × B = { ( a , b ) | a ∈ A and b ∈ B } is the set of all ordered pairs ( a , b) such that a ∈ A and b ∈ B ." (3) { ( x , y) | x , y ∈ A } is the set of all ordered pairs ( x , y) such that x , y ∈ A . Note x and y are called "dummy variables" since {( x , y) | x , y ∈ A } = {( a , b) | a , b ∈ A } = A × A = A2 . (4) {( x , y ) | x ∈ A } is " the set of all ordered pairs ( x , y) such that x ∈ A ." The set defined in (4) depends upon what y turns out to be. The variable y is called a free variable, it is not a dummy variable. (5) ∪A = { x | x ∈ y for some y ∈ A } . ∪A is called the union of the set A . (6) A ∪ B = { x | x ∈ A or x ∈ B } = ∪{A , B } For convenience, instead of { x | x ∈ A and P } , we can also use the notation { x ∈ A | P } where P is a property satisfied by all x ∈ A . A relation is a set of ordered pairs. If θ is a relation, then the domain of θ is the set { a | ( a , b ) ∈ θ for some b } and denoted by dom( θ ) ; and the range of θ is the set { b | ( a , b ) ∈ θ for some a } and denoted by ran( θ ) . A function is a relation f such that if ( a , b ) and ( a , b' ) belong to f then b = b' . Since every function is a relation, then every function must have a domain and range, as defined above, and which are denoted by dom( f ) and ran( f ) , respectively. The expression f (a) denotes the unique element b in ran( f ) such that ( a , b ) ∈ f . If f and g are functions such that ran ( f ) ⊆ dom( g ) , then g o f (sometimes denoted by ( g o f ) for clarity) is the function defined by g o f ( a ) = ( g o f )( a ) = g ( f ( a )) , for each a ∈ dom( f ) . We call g o f the composition of g with f . Note, g o f need not equal f o g , nor need they both be defined. If f and g are arbitrarily chosen functions, then f /\ g is the unique function such that if (a , b) ∈ dom( f ) × dom(g) , then f /\ g ( a , b ) = ( f ( a) , g ( b ) ) . Then dom ( f /\ g ) = dom ( f ) × dom (g) and ran (f /\ g ) = ran ( f ) × ran (g) .

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I I I . THE SPACETIME T × S In all that follows let Λ denote the set of all inertial frames. Then X = ∪Λ , T = ∪{ TI | I ∈ Λ } and S = ∪{ SI | I ∈ Λ } . The Cartesian product T × S = { ( t , s ) | t ∈ T and s ∈ S } has the simplest defi-nition of any set containing the elements of X that also preserves the disparity between T and S . So, T × S will be our initial choice to mathematically model, in set theory, that part of physical reality which is formed from the 4-dimensional time continuum T and the 6-dimensional space continuum S . To introduce into T × S what we have achieved so far, our mathematical model of each inertial frame I in STR will now be the Cartesian product TI × SI . Then for each I ∈ Λ : I = TI × SI ⊆ T × S . We will call TI × SI an Einstein frame or an E-frame. If I and J are arbitrarily chosen inertial frames that need not be equal, then TI × SJ will be called a generalized Einstein frame or G-frame. Then every E-frame is a G-frame, but not all G-frames are E-frames. Let Γ = { TI × SJ | I , J ∈ Λ } . Then Γ is the family of all G-frames. Then Λ ⊆ Γ , but Λ ≠ Γ . Since the family { TI | I ∈ Λ } ∪ { SI | I ∈ Λ } is pairwise disjoint, then Λ and Γ must each be pairwise disjoint. In Theorem 2 below, we show T × S = ∪Γ . To coordinatize T × S , let I , J ∈ Λ . If CT I is an coordinate system from STR for TI and CS J is an coordinate system from STR for SJ then CT I /\ CS J will be a coordinate system for TI × SJ , since by its definition, if t ∈ TI and s ∈ SJ , and if t is the coordinate of t in CT I and ( x , y , z) are the coordinates of s in CS J , then CT nnI /\ CS J ( t , s ) = ( CT I ( t ) , CS J ( s ) ) = ( t , ( x , y , z) ) . From now on, when working with T × S we will use coordinate systems of the form CT nnI /\ CS J for both G-frames and E-frames. We will also denote CT nnI /\ CS J by CIJ and CT nnI /\ CS I by CI . Then every element of T × S will be an ordered pair ( t , s ) such that t is a point-like location in the time continuum TI of some inertial frame I and s is a point-like lo-cation in the space continuum SJ of some inertial frame J , and I need not equal J . Then ( t , s ) will be a point-like location in T × S . When convenient, we will denote ( t , s ) by r as we do with inertial frames. We assume that all of the elements of T × S are these point-like locations r in T × S . Let R3

< c be the set of all 3-tuples belonging to R3 whose Euclidian norms are less

than the speed of light c . We can place T × S into one-to-one correspondence with (R × R3

< c)×(R3× R3< c) as follows. We first choose a single coordinate system CT nnI /\ CS I

for each inertial frame I in STR. We let M be an arbitrarily chosen inertial frame in STR which we hold will fixed throughout the construction of this correspondence. For all ( t , s ) ∈ T× S , let F( t , s ) = ( ( t , (v1 , v2 , v3 ) ) , ( (x , y , z ) , (w1 , w2 , w3 ) ) ) ,

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where (v1 , v2 , v3 ) is the velocity, relative to CT nnM /\ CS M , of the unique inertial frame I such that t ∈ TI and t is the coordinate of t in CT nnI ; and (w1 , w2 , w3 ) is the velocity, relative to CT nnM /\ CS M , of the unique inertial frame J such that s ∈ SJ and (x , y , z ) are the coordinates of s in CS J . We can show that F is a one-to-one correspondence between T × S and (R × R3

< c)×(R3× R3< c) . In this sense, T× S then has

ten large dimensions. Since (R × R3< c)×(R3× R3

< c) can be placed into one-to-one corre-spondence with (R × R3)×(R3

< c × R3< c) the set of all G-frames can be placed into one-to-

one correspondence with the 6-dimensional set R3< c × R3

< c . In order to preserve and use the properties of the 7-dimensional spacetime X and its 4-dimensional event space X / ~ that we have previously derived, we will decompose T × S in the following fashion. For each G-frame TI × SJ , the weight of TI × SJ will be the speed of J relative to I . We will denote this weight by δ ( TI × SJ ) . Then δ ( TI × SJ ) = δ ( TJ × SI ) . As always, let c denote the speed of light. For each real number w such that 0 ≤ w < c , let Xw = ∪{ TI × SJ | δ ( TI × SJ ) = w }. Then T × S = ∪{ Xw | 0 ≤ w < c } and X0 = X . The signed direction from Xw to Xw' equals w' – w , and is the negative of

the signed direction from Xw' to Xw . IV. SIMPLE PROPERTIES OF T × S The following theorems from elementary set theory are very simple derivations of properties satisfied by the structure of T × S and by occurrences of point events in T × S .

Theorem 1. For each (t , s) ∈ T × S , there is one-and-only-one G-frame in Γ to which (t , s ) belongs.

Proof . Let (t , s) ∈ T × S . Then t ∈ T and s ∈ S . Now T = ∪{ TK | K ∈ Λ } and S = ∪{ SK | K ∈ Λ } , where { TK | K ∈ Λ } and { SK | K ∈ Λ } are each pairwise disjoint. Since { TK | K ∈ Λ } is pairwise disjoint and t ∈ ∪{ TK | K ∈ Λ }, then there is one-and-only-one I ∈ Λ such that t ∈ TI . Since { SK | K ∈ Λ } is pairwise disjoint and s ∈ ∪{ SK | K ∈ Λ } , then there is one-and-only-one J ∈ Λ such that s ∈ SJ . Since t ∈ TI and s ∈ SJ are true if and only if (t , s ) ∈ TI × SJ is true, then there is one-and-only-one I ∈ Λ and one-and-only-one J ∈ Λ such that (t , s ) ∈ TI × SJ . So, there is one-and-only-one G-frame in Γ to which (t , s ) belongs, namely TI × SJ . QED. Theorem 2. T × S = ∪{ TI × SJ | I , J ∈ Λ } = ∪Γ .

Proof Let (t , s) ∈ T × S . By Theorem 1, there are unique E-frames K and L such that (t , s) ∈ TK × SL ∈ { TI × SJ | I , J ∈ Λ } . Then by definition of the union of a set, (t , s) ∈ ∪{ TI × SJ | I , J ∈ Λ } . Then T × S ⊆ ∪{ TI × SJ | I , J ∈ Λ } .

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Conversely, let (t , s) ∈ ∪{ TI × SJ | I , J ∈ Λ } . Then there are E-frames K , L ∈ Λ such that (t , s) ∈ TK × SL . Then t ∈ TK and s ∈ SL , implying t ∈ T and s ∈ S , implying (t , s) ∈ T × S . Then ∪{ TI × SJ | I , J ∈ Λ } ⊆ T × S . Then T × S = ∪{ TI × SJ | I , J ∈ Λ } = ∪Γ . QED Theorem 3. Each point event occurs at one-and-only one element in each G-frame. Proof. Let e be an arbitrarily chosen point event. Let TI × SJ be an arbitrarily chosen G-frame, where I and J are E-frames. In STR, e occurs at exactly one element in I and at exactly one element in J . Then e occurs at exactly one element in TI and at exactly one element in SI ; and e occurs at exactly one element in TJ and at exactly one element in SJ . Then e occurs at exactly one element in TI and at exactly one element in SJ . So, e occurs at exactly one element in TI × SJ . Since e and TI × SJ were arbitrarily chosen and could be any point event and any G-frame, respectively, each point event then occurs at one-and-only one element in each G-frame. QED V. AN EXTENSION OF LORENTZ TRANSFORMATIONS TO T × S In all that follows, for each inertial frame I in STR, let CT I be a coordinate sys-tem of the time continuum TI of I in STR and let CS I be a coordinate system of the space continuum SI of I in STR. Then the coordinate system CI of I in STR can be modeled by CI = CT I /\ CS I . When dealing with Lorentz transformations, we will define our ordered 4-tuples so that (t, x, y, z ) = (t, (x, y, z )) . Let I and J be arbitrarily chosen E-frames, where I ≠ J. Let w be the speed of I relative to J . Then c > w > 0 . Let βw = w/c and γw = ( 1 – βw

2 ) – 1 / 2 . We know that in STR we can always construct coordinate systems CT I /\ CS I and CT J /\ CS J of I and J, respectively, so that the ordered pair (CT I /\ CS I , CT J /\ CS J ) is positively oriented, by which we mean a point event can occur at both ( t , x , y , z ) in ran(CI) and at ( t' , x' , y' , z' ) in ran(CI') if and only if ( t' , x' , y' , z' ) = (γw( t – xβw /c ) , γw(x – wt ) , y , z ) .

Case 1. Let t0' be an element of TJ and s0 be an element of SI . Let e be a point event which occurs at both t0' and s0 . Theorem 3 implies one and only one point event e can occur at both t0' and s0 . Let t0' be the time coordinate of t0' in ran(CJ) . Let ( x0 , y0 , z0) be the space-coordinates of s0 in ran(CI) . Let t be the time-coordinate in ran(CI) of the occurrence of e in TI , and let ( x', y', z' ) be the space-coordinates in ran(CI') of the occurrence of e in SJ . Then the point event e can occur at both (t , x0 , y0 , z0 ) in ran(CI) and ( t0' , x' , y' , z' ) in ran(CJ) if and only if (t0', x', y', z' ) = (γw( t – x0βw /c ) , γw(x0 – wt ) , y0 , z0) . ( 1 ) By equation ( 1 ) , t0' = γw( t – x0βw /c ) . Solving for t , we have:

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t = γw–1t0' + x0βw /c . ( 2 )

By equations ( 1 ) and ( 2 ), x' = γw(x0 – wt) = γw[x0 – w(γw

–1t0' + x0βw /c )] = γwx0 – w(t0' + γwx0βw /c) = γwx0(1 – wβw /c ) – wt0' = γw

–1x0 – wt0' . ( 3 ) Then t = γw

–1t0' + x0βw /c , x' = γw–1x0 – wt0' , y' = y0 , z' = z0 .

Hence, if e occurs at (t0' , x0 , y0 , z0) in ran(CT J /\ CS I) then (i) e has to occur at ( γw –1t0' + x0βw /c , x0 , y0 , z0 ) in ran(CT I /\ CS I) , (ii) e has to occur at ( t0' , γw–1x0 – wt0' , y0 , z0) in ran(CT J /\ CS J) , (iii) e has to occur at (γw –1t0' + x0βw /c , γw

–1x0 – wt0' , y0 , z0) in ran(CT I /\ CS J) . So the Lorentz transformations for each (t0' , x0 , y0 , z0) ∈ ran(CT J /\ CS I ) are: (i) L(t0' , x0 , y0 , z0) = ( γw

–1t0' + x0βw /c , x0 , y0 , z0 ) ∈ ran(CT I /\ CS I ) (ii) L(t0' , x0 , y0 , z0) = ( t0' , γw–1x0 – wt0' , y0 , z0) ∈ ran(CT J /\ CS J ) (iii) L(t0' , x0 , y0 , z0) = ( γw

–1t0' + x0βw /c , γw–1x0 – wt0' , y0 , z0) ∈ ran(CT I /\ CS J ) .

Case 2. Let t0 be a primitive element of time in TI and s0' be a primitive ele-ment of space in SJ . Then Theorem 3 implies one and only one point event e can occur at both t0 and s0' . Let e be a point event which occurs at both t0 in TI and s0' in SJ . Let t0 be the time coordinate of t0 in ran(CI) . Let ( x0' , y0' , z0' ) be the space-coordinates of s0' in ran(CJ) . Let t' be the time-coordinate in ran(CJ) of the occurrence of e in TJ , and let ( x , y , z ) be the space-coordinates in ran(CI) of the occurrence of e in SI . Then the point event e can occur at both ( t0 , x , y , z ) in ran(CI) and (t' , x0' , y0' , z0' ) in ran(CJ) if and only if (t', x0', y0', z0' ) = (γw( t0 – xβw /c ) , γw(x – wt0 ) , y , z) . ( 4 ) By equation ( 4 ) , x must satisfy γw(x – wt0 ) = x0' . Solving for x , we have: x = γw

–1x0' + wt0 . ( 5 ) Then x must equal γw

–1x0' + wt0 . By equations ( 4 ) and ( 5 ), t' must satisfy t' = γw( t0 – xβω /c ) = γw[t0 – (γw

–1x0' + wt0)βw /c ] = γwt0 – (x0' + γwwt0 )βw /c = γwt0(1 – wβw /c ) – x0'βw /c = γw

–1t0 – x0'βw /c . ( 6 )

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Then t' = γw–1t0 – x0'βw /c , x = γw

–1x0' + wt0 , y = y0' and z = z0' . Hence, if e occurs at (t0 , x0' , y0' , z0' ) in ran(CT I /\ CS J) , then (1) e has to occur at (γw –1t0 – x0'βw /c , x0' , y0' , z0' ) in ran(CT J /\ CS J) , (2) e has to occur at (t0 , γw–1x0' + wt0', y0' , z0' ) in ran(CT I /\ CS I) , (3) e has to occur at (γw –1t0 – x0'βw /c , γw–1x0' + wt0 , y0' , z0' ) in ran(CT J /\ CS I) . So the Lorentz transformations for each (t0 , x0' , y0' , z0' ) ∈ ran(CT I /\ CS J ) are: (i) L(t0 , x0' , y0' , z0' ) = ( γw

–1t0' + x0βw /c , x0 , y0 , z0 ) ∈ ran(CT I /\ CS I ) , (ii) L(t0 , x0' , y0' , z0' ) = ( t0' , γw–1x0 – wt0' , y0 , z0) ∈ ran(CT J /\ CS J ) , (iii) L(t0 , x0' , y0' , z0' ) = (γw –1t0 – x0'βw /c , γw–1x0' + wt0 , y0' , z0' ) , ∈ ran(CT I /\ CS J ) . Case 3. Let t0 be a primitive element of time in TI and s0 be a primitive ele-ment of space in SI . Then Theorem 3 implies one and only one point event e can occur at both t0 and s0 . Let ( t0 , x0 , y0 , z0) be the coordinates of the occurrence of e in ran(CI ) . From STR we know ( t' , x' , y' , z' ) must be the coordinates of the occurren-ce of e in ran(CJ where ( t' , x' , y' , z' ) = (γw( t0 – x0βw /c ) , γw(x0 – wt0 ) , y0 , z0 ) . (7) Then t' = (γw( t0 – x0βw /c ) , x' = γw(x0 – wt0 ) , y' = y0 , z' = z0 . Hence, if e occurs at ( t0 , x0 , y0 , z0) in ran(CT I /\ CS I , then (1) e has to occur at (γw( t0 – x0βw /c ) , γw(x0 – wt0 ) , y0 , z0 ) in ran(CT J /\ CS J , (2) e has to occur at (γw(t0 – x0βw /c ) , x0 , y0 , z0 ) in ran(CT J /\ CS I , (3) e has to occur at (t0 , γw(x0 – wt0 ) , y0 , z0 ) in ran(CT I /\ CS J . So the Lorentz transformations for each (t0 , x0 , y0 , z0) ∈ ran(CT I /\ CS I ) are: (i) L(t0 , x0 , y0 , z0) = (γw(t0 – x0βw /c ) , γw(x0 – wt0 ) , y0 , z0 ) ∈ ran(CT J /\ CS J ) , (ii) L(t0 , x0 , y0 , z0) = (γw(t0 – x0βw /c ) , x0 , y0 , z0 ) ∈ ran(CT J /\ CS I ) , (iii) L(t0 , x0 , y0 , z0) = (t0 , γw(x0 – wt0 ) , y0 , z0 ) ∈ ran(CT I /\ CS J ) . Case 4. Let t0' be a primitive element of time in TJ and s0' be a primitive ele-ment of space in SJ. Then STR implies one and only one point event e can occur at both t0' and s0' . Let ( t0' , x0' , y0' , z0' ) be the coordinates of the occurrence of e in the coordinate system CJ in J in STR. From STR, we know ( t , x , y , z ) must be the coordinates of the occurrence of e in CI where ( t , x , y , z) = (γw( t0' + x0'βw /c ) , γw(x0' + wt0' ) , y0' , z0' ) . (8) Then t = γw( t0' + x0'βw /c ) , x = γw(x0' + vt0' ) , y = y0' , z = z0' .

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Hence, if e occurs at ( t0' , x0' , y0' , z0') in CT J /\ CS J , then (1) e has to occur at (γw( t0' + x0'βw /c ) , γw(x0' + wt0' ) , y0' , z0' ) in CT I /\ CS I , (2) e has to occur at (γw(t0' + x0'βw /c ) , x0' , y0' , z0' ) in CT I /\ CS J , (3) e has to occur at (t0' , γw(x0' + wt0' ) , y0' , z0' ) in CT J /\ CS I . So the Lorentz transformations for each (t0' , x0' , y0' , z0') ∈ ran(CT J /\ CS J ) are: (i) L(t0' , x0' , y0' , z0') = (γw( t0' + x0'βw /c ) , γw(x0' + wt0' ) , y0' , z0' ) ∈ ran(CT I /\ CS I ) , (ii) L(t0' , x0' , y0' , z0') = (γw(t0' + x0'βw /c ) , x0' , y0' , z0' ) ∈ ran(CT I /\ CS J ) , (iii) L(t0' , x0' , y0' , z0') = (t0' , γw(x0' + wt0' ) , y0' , z0' ) ∈ ran(CT J /\ CS I ) . Case 5. Clearly, we can add to the twelve Lorentz transformations above four more uniquely defined Lorentz transformations: the identity mapping on ran(CT J /\ CS I) , the identity mapping on ran(CT I /\ CS J) , the identity mapping on ran(CT I /\ CS I) and the identity mapping on ran(CT J /\ CS J) . We don’t need to use primes or subscripts in our results above as long as we state what coordinate systems our variables belong to. For example, we can rewrite the results of Case 1 as follows. Hence, if e occurs at (t , x , y , z) in ran(CT J /\ CS I) then (1) e has to occur at ( γw –1t + xβw /c , x , y , z ) in ran(CT I /\ CS I) , (2) e has to occur at ( t , γw–1x – wt , y , z) in ran(CT J /\ CS J) , (3) e has to occur at ( γw –1t + xβw /c , γw

–1x – wt , y , z) in ran(CT I /\ CS J) . V. AN EQUIVALENT EXTENSION We can also tabulate the results in our five cases above by using compositions of whatever Lorentz transformations we wish as long as they are well defined and possess the appropriate domains and ranges. The important property to rely on in this development is the fact that Lorentz transformations are one-to-one functions. We start our with the three results from Case 1. The following is an example. 1. Case 1(2). The Lorentz transformation from CT J /\ CS I to CT J /\ CS J is that one-to-one mapping λ from the range of CT J /\ CS I to the range of CT J /\ CS J defined by the formula λ (t, x , y , z) = (t , γw–1

x – wt, y , z ). (9) 2. Case 1(1). The Lorentz transformation from CT J /\ CS I to CT I /\ CS I is that one-to-one mapping ξ from the range of CT J /\ CS I to the range of CT I /\ CS I defined by the formula ξ (t , x , y , z) = ( γw

–1t + xβw /c, x , y , z ). (10)

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3. Case 1(3). The Lorentz transformation from CT J /\ CS I to CT I /\ CS J is that one-to-one mapping µ from the range of CT J /\ CS I to the range of CT I /\ CS J defined by the formula µ (t , x , y , z) = ( γw

–1t + xβw /c , γw–1x – wt, y , z ). (11)

4. Case 4(3). The Lorentz transformation from CT J /\ CS J to CT J /\ CS I is the one-to-one mapping λ –1 from the range of CT J /\ CS J to the range of CT J /\ CS I and hence defined by the formula λ –1(t , x , y , z ) = (t , γw (x + wt ) , y , z ). (12) To prove this, show (λ o λ –1)(t , x , y , z ) = λ (t , γw (x + wt ) , y , z ) = (t , x , y , z ) . 5. Case 3(2). The Lorentz transformation from CT I /\ CS I to CT J /\ CS I is the one-to-one mapping ξ

–1 from the range of CT I /\ CS I to the range of CT J /\ CS I and hence defined by the formula ξ

–1(t , x , y , z ) = ( γw(t – xβw /c) , x , y , z ). (13) To prove this, show (ξ o ξ –1)(t , x , y , z ) = ξ (γw(t – xβw /c) , x, y , z ) = (t , x , y , z ) . 6. Case 2(3). The Lorentz transformation from CT I /\ CS J to CT J /\ CS I is the one-to-one mapping µ –1 from the range of CT I /\ CS J to the range of CT J /\ CS I and hence defined by the formula µ

–1(t , x , y , z ) = ( γw–1t – xβw /c, γw

–1x + wt , y , z ). (14) To prove this, show (µ o µ –1)(t , x , y , z ) = µ (γw–1t – xβw /c, γw

–1x + wt , y , z ) = (t , x , y , z ) . 7. Case 3(1). The Lorentz transformation from CT I /\ CS I to CT J /\ CS J is the com-position λ o ξ –1 from the range of CT I /\ CS I to the range of CT J /\ CS J and hence defined by the formula (λ o ξ

–1 )(t , x , y , z ) = λ ( ξ –1 (t , x , y , z))

= (γw(t – xβw /c) , γw(x – wt ) , y , z ). (15) 8. Case 4(1). The Lorentz transformation from CT J /\ CS J to CT I /\ CS I is the inverse mapping (λ o ξ –1 )–1 = ξ o λ –1 from the range of CT J /\ CS J to the range of CT I /\ CS I and hence defined by the formula (ξ o λ

–1 )(t , x , y , z ) = ξ ( λ –1 (t , x , y , z))

= (γw(t + xβw /c) , γw(x + wt ) , y , z ). (16) 9. Case 2(2). The Lorentz transformation from CT I /\ CS J to CT I /\ CS I is the compo-sition ξ o µ –1 from the range of CT I /\ CS J to the range of CT I /\ CS I and hence defined by the formula (ξ o µ

–1 )(t , x , y , z ) = ξ ( µ –1 (t , x , y , z))

= ( t , γw–1x + wt , y , z ). (17) 10. Case 2(1). The Lorentz transformation from CT I /\ CS J to CT J /\ CS J is the com-position λ o µ –1 from the range of CT I /\ CS J to the range of CT J /\ CS J and hence defined by the formula (λ o µ

–1 )(t , x , y , z ) = λ ( µ –1 (t , x , y , z))

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= ( γw–1t – xβ w /c , x , y , z ). (18)

11. Case 3(3). The Lorentz transformation from CT I /\ CS I to CT I /\ CS J is the com-position µ o ξ –1 from the range of CT I /\ CS I to the range of CT I /\ CS J and hence defined by the formula (µ o ξ

–1 )(t , x , y , z ) = µ ( ξ

–1 (t , x , y , z)) = (t , γw(x – wt ) , y , z ). (19) In this derivation and in the one that follows, the following identity is useful. γw

–1a + γw b = γw [ ( 1 – βw2 ) a + b ] .

12. Case 4(2). The Lorentz transformation from CT J /\ CS J to CT I /\ CS J is the com-position µ o λ –1 from the range of CT J /\ CS J to the range of CT I /\ CS J and hence defined by the formula (µ o λ

–1 )(t , x , y , z ) = µ ( λ –1 (t , x , y , z))

= (γw(t + xβw /c ) , x , y , z ). (20) 13. The Lorentz transformation from CT J /\ CS I to CT J /\ CS I is then the identity mapping λ –1 o λ . 14. The Lorentz transformation from CT I /\ CS I to CT I /\ CS I is then the identity mapping ξ o ξ –1 . 15. The Lorentz transformation from CT J /\ CS J to CT J /\ CS J is then the identity mapping λ o λ –1 . 16. The Lorentz transformation from CT I /\ CS J to CT I /\ CS J is then the identity mapping µ o µ –1 . Theorem 4. The occurrence of a point event at any arbitrarily chosen G-frame uniquely determines its occurrence at every other G-frame. Proof. In STR, given any coordinate systems CI and C'I of any inertial frame I , if these systems have the same standard units of time and space, then the system CI can be transformed into the system C'I by an initial translation of the time coordinates and space coordinates of CI followed by a rotation of the space coordinates of the coordinate sys-tem resulting from this initial translation. At the end of this process, the coordinates in CI of a point-like location r in I will be transformed into the coordinates of r in CI' . Consequently, if we know the coordinates in CI of any point-like location r in I then we can determine coordinates of r in C'I . Let K , L , M and N be arbitrarily chosen inertial frames. Let e be a point event occurring at an arbitrarily chosen point-like location r of TK × SL . We can always construct coordinate systems CT K /\ CS K and CT L /\ CS L of K and L , re-spectively, so that the ordered pair (CT K /\ CS K , CT L /\ CS L ) is positively oriented. Then we can first determine the coordinates of the occurrence of e in both CT K /\ CS K and CT L /\ CS L and then determine the coordinates of r in CT K /\ CS L by using equati-on (19) and setting K = I and L = J . We can then construct coordinate systems

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C'T L /\ C'S L and CT M /\ CS M of L and M , respectively, so that the ordered pair (C'T L /\ C'S L , CT M /\ CS M ) is positively oriented. Then we can use the coordinates of the occurrence of e in C'T L /\ C'S L to determine the coordinates in CT M /\ CS M of the unique point-like location r'' of TM × SM at which e occurs by using equation (15), setting C'T L = CT I , C'S L = CS I and M = J . We then construct coordinate sys-tems CT 'M /\ C'S M and CT N /\ CS N of M and N , respectively, so that the ordered pair (C'T M /\ C'S M , CT N /\ CS N ) is positively oriented. Then we use the coordinates of r'' in C'T M /\ C'S M to determine the coordinates in CT M /\ CS N of the unique point-like location r''' of TM × SN at which e occurs by using equation (19) , setting C'T M = CT I, C'S M = CS I and N = J . QED VII. BIBLIOGRAPHY

Marcus, B., Elementary Set Theory and the Special Theory of Relativity.