How do Vectors transform from one inertial reference frame to another inertial reference frame in special relativity. A bound vector in an inertial reference frame (x,ct) has its line of action as one of the space axis in that frame and is described by x*i*,then what would it be in form of new base vectors (a) and (b) in a different inertial x‘,ct‘) moving with respect to the former inertial system with v*i* velocity.Let (i) and (j) be the two bounded unit vectors with the line of action as co-ordinate axis(x) and (ct) respectively and senses in the positive side of co-ordinates and similarly (a) and (b) are defined for co-ordinates (x‘) and (ct‘) respectively.

easternerjx

easternerjx

Answered question

2022-09-17

How do Vectors transform from one inertial reference frame to another inertial reference frame in special relativity.
A bound vector in an inertial reference frame ( x, c t) has its line of action as one of the space axis in that frame and is described by x* i*,then what would it be in form of new base vectors ( a) and ( b) in a different inertial system ( x , c t ) moving with respect to the former inertial system with v*i* velocity.Let (i) and (j) be the two bounded unit vectors with the line of action as co-ordinate axis( x) and ( c t) respectively and senses in the positive side of co-ordinates and similarly ( a) and ( b) are defined for co-ordinates ( x ) and ( c t ) respectively.

Answer & Explanation

Ruben Horn

Ruben Horn

Beginner2022-09-18Added 7 answers

Well, vectors (3D vectors) don't really transform linearly. Unlike Galilean transformations, you need now know anything "extra" when transforming a vector. Here, due to the "mixing" of space and time, you do. To transform displacement, you need to know time, and vice versa. Same with energy and momentum.
Four-vectors, on the other hand, transform linearly. These are four-dimensional vectors which transform linearly via the Lorentz matrix ( β = v c , γ = 1 1 β 2 ):
L = [ γ β γ 0 0 β γ γ 0 0 0 0 1 0 0 0 0 1 ]
For example, if you want to transform position and/or time, you use the four-position
X = [ c t x y z ]
this can be compactly written as ( c t , x )-this just means that you can expand the second "vector" term to get the next three four-vector components.
Anyway, the four vector transforms as:
X = L × X
(matrix product)
This, expanded, is:
[ c t x y z ] = [ γ β γ 0 0 β γ γ 0 0 0 0 1 0 0 0 0 1 ] [ c t x y z ] ,
which are your normal lorentz transformations. A property of four vectors is that if we are talking about the same four vector ( a , b ) in two frames, the value of a 2 | b | 2 is the same in both. For four-position, you get c 2 t 2 x 2 y 2 z 2 = c 2 t 2 x 2 y 2 z 2
Other four vectors are:
Four-velocity: ( γ c , γ u )
Four-momentum: ( E / c , p )
Four-current density: ( γ ρ , J )
Four-potential: ( ϕ c , A )

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