Accession of Velocities

 13 July 02:30   

    In classical physics, velocities artlessly add. If an item moves with acceleration u in one advertence frame, which is itself affective at v with account to a additional frame, the item moves at acceleration u+v in that additional frame.

    This is inconsistant with relativity because it predicts that if the acceleration of ablaze is c in the first anatomy it will be v+c in the second.

    We charge to acquisition an another blueprint for accumulation velocities. We can do this with the Lorentz transform.

    Because the agency v/c will accumulate alternating we shall alarm that arrangement β.

    We are because three frames; anatomy O, anatomy O which moves at acceleration u with account to anatomy O, and anatomy O which moves at acceleration v with account to anatomy O.

    We wish to understand the acceleration of O with account to anatomy O,U which would classically be u+v.

    The transforms from O to O and O to O can be accounting as cast equations,

    :$$

    egin x \ ct end =

    gamma egin 1 & - eta \ -eta & 1 end

    egin x \ ct end quad

    egin x \ t end = gamma

    egin 1 & - eta \ -eta & 1 end

    egin x \ ct end

    where we are defining the βs and γs as

    :

    eta = frac & gamma = frac} \beta^prime = frac &

    gamma^prime = frac^2 }}

    end

    We can amalgamate these to get the accord amid the O and O coordinates artlessly by adding the matrices, giving

    :$$

    egin x \ ct end = gamma gamma^prime

    egin 1+eta eta & - (eta + eta) - (eta + eta) & 1+eta eta end

    egin x \ ct end quad (1)

    This should be the aforementioned as the Lorentz transform amid the two frames,

    :$$

    egin x \ ct end =

    gamma egin 1 & - eta \ -eta & 1 end

    egin x \ ct end quad (2) mbox

    egin eta & = & frac \gamma & = & frac^2 }} end

    

    These two sets of equations do attending similar. We can create them attending added agnate still by demography a agency of 1+ββ out of the cast in (1) giving#

    :$$

    egin x \ ct end = gamma gamma (1+eta eta)

    egin 1 & - frac - frac & 1+ end

    egin x \ ct end

    This will be identical with blueprint 2 if

    :

    gamma = gamma gamma (1+eta eta) mbox

    Since the two equations haveto accord identical results, we understand these altitude haveto be true.

    Writing the βs in agreement of the velocities blueprint 3a becomes

    :$frac=frac\}$

    which tells us U in agreement of u and v.

    A little algebra shows that this implies blueprint 3b is aswell true

    Multiplying by c we can assuredly write.

    $U\; =\; frac\}$

    Notice that if u or v is abundant abate than c the denominator is about 1, and the velocities about add but if either u or v is c then so is U, just as we expected.