Four Vectors

 13 July 02:34   

    We accept apparent that a beachcomber is declared by four numbers, the apparatus of the spatial agent k, and the abundance ω

    In appropriate relativity these four numbers anatomy a four-vector

    It is alleged a four-vector because it has 3 spacelike components, basic a vector, and one timelike basic if there are 3 amplitude dimensions. It is alleged a four-vector because of the way it behaves if we change advertence frames.

    The spacelike basic of the beachcomber four-vector is just $mathbf$ if there are 3 amplitude dimensions, while the timelike basic is $omega/c$ area the c is in the denominator to accord the timelike basic the aforementioned ambit as the spacelike component.

    Let us ascertain some terminology. We announce a four-vector by underlining and address the apparatus in the afterward way: $underline\; =\; (k\; ,\; omega\; /c\; )$, area $underline$ is the beachcomber four-vector, $k$ is its spacelike component, and $omega\; /c$ is its timelike component. For three amplitude dimensions, area we accept a beachcomber agent rather than just a wavenumber, we address $underline\; =\; (k\; ,\; omega\; /c\; )$.

    Another archetype of a four-vector is artlessly the position agent in spacetime, $underline\; =\; (\; x,\; ct\; )$, or $underline\; =\; (\; mathbf,\; ct\; )$ in three amplitude dimensions. The $c$ multiplies the timelike basic in this case, because that is what is bare to accord it the aforementioned ambit as the spacelike component.

    In three ambit we ascertain a agent as a abundance with consequence and direction. Extending this to spacetime, a four-vector is a abundance with consequence and administration in spacetime. Absolute in this analogue is the angle that the vectors consequence is a abundance absolute of alike arrangement or advertence frame. We accept apparent that the invariant breach in spacetime is

    :$I\; =\; sqrt$,

    so it makes faculty to analyze this as the consequence of the position vector. This leads to a way of defining a dot artefact of four-vectors.

    Given two four-vectors

    :$underline\; =\; (\; mathbf\; ,A\_t\; )\; quad$

    underline = ( mathbf , B_t )

    then the dot artefact is

    :$underline\; cdot\; underline\; =\; mathbf\; cdot\; mathbf\; -$

     A_t B_t quad mbox

    This is constant with the analogue of invariant breach if we set

    :$underline\; =\; underline\; =\; underline$,

    since then

    :$underline\; cdot\; underline\; =\; x^2\; -\; c^2\; t^2\; =\; I^2$.

    Now, the key point about dot articles for 3-vectors is that they are scalars, absolute of the observer. They do not change if the axes are rotated, as was accurate earlier.

    For our analogue of the dot artefact of four-vectors to be useful, it should aswell be absolute of the observer. In particular, it should not depend on the assemblage velocity, abroad it would breach the assumption of relativity.

    We can calmly analysis that our analogue does amuse this criterion.

    Its bright that its absolute of rotation, back it is the aberration amid a dot artefact and the artefact of two scalars, both of which agreement arent afflicted by alternating the coordinates.

    Is it aswell anatomy acceleration independent?

    To check, first we charge to be able to address down our four-vectors in the new advertence frame. We understand how to do this for the position agent — use the Lorentz transform. It can be apparent that the aforementioned transform haveto authority for all vectors , so the apparatus of a four agent in the new advertence frame, affective at acceleration v forth the x-axis with account to the antecedent one, are

    :

    A_x & = & gamma left( A_x - frac A_t
ight)

    \ A_y & = & A_y \ A_z & = & A_z A_t & = & gamma left( -frac A_x + A_t
ight)

    end

    The dot artefact in this anatomy is

    :$underline\; cdot\; underline\; =$

     A_x B_x + A_y B_y + A_z B_z - A_t B_t

    Simplifying, we get

    :

    underline cdot underline & = &

    gamma^2 left( A_x B_x -frac (A_x B_t + A_t B_x) + frac A_t B_t
ight) & & + A_y B_y + A_z B_z & & & - gamma^2 left( fracA_x B_x -frac (A_x B_t + A_t B_x) + A_t B_t
ight) & = & left(1- frac
ight)^ left( A_x B_x + frac A_t B_t
ight) & & + A_y B_y + A_z B_z \

    & & - left(1- frac
ight)^ left( frac A_x B_x - A_t B_t
ight) & = & A_x B_x + A_y B_y + A_z B_z - A_t B_t

    end

    which is just the dot artefact in the aboriginal frame, absolutely as we wanted.

    We now understand that the dot artefact of two four-vectors is a scalar result, i. e., its amount is absolute of alike system. This can be acclimated to advantage on occasion.

    In the odd geometry of spacetime it is not accessible what erect means. We accordingly define two four-vectors $underline$ and $underline$ to be erect if their dot artefact is zero, in the aforementioned way as with three-vectors.

    :$underline\; cdot\; underline\; =\; 0$

    Because the dot artefact is a scalar, if vectors are erect in one frame, they will be erect in all frames.

    We can aswell accede the dot artefact of a four-vector $underline$ which resolves into $(A\_x\; ,\; A\_t\; )$ in the unprimed frame. Let us added accept that the spacelike basic is aught in some abreast frame, so that the apparatus in this anatomy are (0, A_t ) The actuality that the dot artefact is absolute of alike arrangement agency that

    :$underline\; cdot\; underline\; =\; A\_x^2\; -\; A\_t^2\; =\; -\; A\_t^2$

    This constitutes an addendum of the spacetime Pythagorean assumption to four-vectors additional then the position four-vector. Thus, for instance, the wavenumber for some beachcomber may be aught in the abreast frame, which agency that the wavenumber and abundance in the unprimed anatomy are accompanying to the abundance in the abreast anatomy by $k^2\; -\; omega^2\; /\; c^2\; =\; -\; omega^2\; /\; c^2$.

    Classically, the banausic derivative, d/dt acts like a scalar so we can accumulate a agent by it, and get addition vector.

    In relativity t is allotment of a four-vector, which agency d/dt aswell is, so we deceit artlessly differentiate vectors with account to t and apprehend to apprehend to get vectors.

    For example, the position of a anchored atom is (0, ct).

    Viewed from a anatomy affective at v to the right, its position becomes (-vτ, cτ), area τ=γt is the time as abstinent in the affective frame.

    If we differentiate with account to τ the acceleration would be (-v, c)

    If we differentiate with account to t, we get (0, c) in the anchored frame, which would be (using the Lorentz transform) (-γv, -γc) in the affective frame, if this were a four vector.

    These two expressions alter by a agency of γ, if abstinent in the aforementioned frame, so this can not be a four vector.

    However, if the affective eyewitness divides by γ, which is the time dilation, they will get the aforementioned agent as the anchored observer.

    Doing this is agnate to appropriate by the time in the particles own blow frame. Back this works for the position vector, we can apprehend it to plan for all vectors.

    The time abstinent in a particles blow anatomy is alleged its able time.

    Differentiating a agent with account to able time gives addition vector, which is the relativistic agnate of the banausic derivative.