Velocity-addition formula

In physics, a velocity-addition formula is an equation that relates the velocities of moving objects in different reference frames.

Galilean addition of velocities

As Galileo observed, if a ship is moving relative to the shore at velocity $v$, and a fly is moving with velocity $u$ as measured on the ship, calculating the velocity of the fly as measured on the shore is what is meant by the addition of the velocities $v$ and $u$. When both the fly and the ship are moving slowly compared to light, it is accurate enough to use the vector sum

$\,{\vec {s}}={\vec {v}}+{\vec {u}}$

where ${\vec {s}}$ is the velocity of the fly relative to the shore.

Special theory of relativity

According to the theory of special relativity, the frame of the ship has a different clock rate and distance measure, and the notion of simultaneity in the direction of motion is altered, so the addition law for velocities is changed. This change isn't noticeable at low velocities but as the velocity increases towards the speed of light it becomes important. The addition law is also called a composition law for velocities. For collinear motions, the velocity of the fly relative to the shore is given by

$s={v+u \over 1+(vu/c^{2})}.$

This is also the law of addition of hyperbolic tangents

$\tanh(\alpha +\beta )={\tanh(\alpha )+\tanh(\beta ) \over 1+\tanh(\alpha )\tanh(\beta )}$

where

${v \over c}=\tanh(\alpha )\,\,\,\,{u \over c}=\tanh(\beta )\,\,\,\,\,{s \over c}=\tanh(\alpha +\beta ),$

which shows that the composition of collinear velocities is associative and commutative. The quantities $\alpha$ and $\beta$ are called rapidities. The reason that the velocities are hyperbolic tangents is because the Lorentz transformation can be thought of as the application of a hyperbolic rotation through a hyperbolic angle which is the rapidity. Suppose the velocity of a line in space-time is the slope of the line, which is the hyperbolic tangent of the rapidity, just as the slope of the $x$-axis after a rotation is given by the tangent of the rotation angle. When a plane is successively rotated by two angles, the final rotation is by the sum of the two angles. So the final slope of the $x$-axis is the tangent of the sum of the two angles. In the same way, the slope of the time axis after two boosts is the hyperbolic tangent of the sum of the two rapidities.

The colinear law of composition of velocities gave the first test of the kinematics of the special theory of relativity. Using a Michelson interferometer, Fizeau measured the speed of light in a fluid moving parallel to the light. The speed of light in the fluid is slower than the speed of light in vacuum, and it changes if the fluid is moving along with the light. The speed of light in a colinear moving fluid is predicted accurately by the colinear case of the relativistic formula.

Derivation

Since a relativistic transformation rotates space and time into each other much as geometric rotations in the plane rotate the $x$ and $y$ axes, it is convenient to use the same units for space and time, otherwise there is a unit conversion factor which converts incessantly between the units. This conversion factor is the speed of light, which turns time intervals measured in seconds to lengths measured in meters. In a system where lengths and times are measured in the same units, the speed of light is dimensionless and equal to 1. All velocities are then fractions of the speed of light. To translate to engineering units, replace $v$ with $v/c$ everywhere.

To find the relativistic transformation law, it is useful to introduce the four-velocities $\scriptstyle (V_{0},V_{1},0)$ and $\scriptstyle (U_{0},U_{1},U_{2})$. The four-velocity is defined to be a four vector with relativistic length equal to 1 tangent to the space-time path of the object. It is convenient to take the $x$-axis to be the direction of motion of the ship, and the $y$-axis so that the $x-y$ plane is the plane spanned by the motion of the ship and the fly. The ordinary velocity is the ratio of the rate at which the space coordinates are increasing to the rate at which the time coordinate is increasing:

$\,{\vec {v}}=(V_{1}/V_{0},0)$

$\,{\vec {u}}=(U_{1}/U_{0},U_{2}/U_{0})$

Since the relativistic length of $V$ is 1, $\scriptstyle V_{0}^{2}-V_{1}^{2}=1$. So $\scriptstyle V_{0}=1/{\sqrt {1-v^{2}}}$ and $\scriptstyle V_{1}=v/{\sqrt {1-v^{2}}}$. The Lorentz transformation matrix which boosts the rest frame to four-velocity $V$ is then:

${\begin{pmatrix}V_{0}&V_{1}&0\\V_{1}&V_{0}&0\\0&0&1\end{pmatrix}}$

This matrix rotates the a pure time-axis vector $\scriptstyle (1,0,0)$ to $\scriptstyle (V_{0},V_{1},0)$, and all its columns are relativistically perpendicular to one another, so it defines a Lorentz transformation.

If a fly is moving with four-velocity $\scriptstyle (U_{0},U_{1},U_{2})$ in the rest frame, and it is boosted by multiplying by the matrix above, the new four-velocity is $\scriptstyle (S_{0},S_{1},S_{2})$:

$\,S_{0}=V_{0}U_{0}+V_{1}U_{1}$

$\,S_{1}=V_{1}U_{0}+V_{0}U_{1}$

$\,S_{2}=U_{2}$

Dividing by the time component $\scriptstyle S_{0}$ and replacing the four-vectors for $U$ and $V$ by the three vectors $u$ and $v$ gives the relativistic composition law:

$\,s_{1}={v_{1}+u_{1} \over 1+v_{1}u_{1}}$

$\,s_{2}={1 \over V_{0}}{u_{2} \over (1+v_{1}u_{1})}={\sqrt {1-v^{2}}}\,{u_{2} \over 1+v_{1}u_{1}}$

The form of the relativistic composition law can be understood as an effect of the failure of simultaneity at a distance. For the parallel component, the time dilation decreases the speed, the length contraction increase it, and the two effects cancel out. The failure of simultaneity means that the fly is changing simultaneity slices as the projection of $u$ onto $v$. Since this effect is entirely due to the time slicing, the same factor multiplies the perpendicular component, but for the perpendicular component there is no length contraction, so the time dilation adds a factor of $\scriptstyle 1/V_{0}={\sqrt {1-v^{2}}}$.

Vector notation

To translate the formula of the previous section to three-vector notation, replace $u_{1}$ with the component of $U$ parallel to $V$:

General case

$\,{\vec {U}}_{||}={{\vec {V}}\cdot {\vec {U}} \over |{\vec {V}}|^{2}}{\vec {V}}\qquad \qquad {\vec {U}}_{\perp }={\vec {U}}-{\vec {U}}_{||}$

$\,{\vec {S}}={{\vec {V}}+{\vec {U}}_{||}+{\sqrt {1-V^{2}}}\,{\vec {U}}_{\perp } \over 1+{\vec {V}}\cdot {\vec {U}}}$

Special case: parallel velocities

In the case where the velocities are parallel we have

$\,{\vec {U}}_{||}={\vec {U}}\qquad \qquad \qquad {\vec {U}}_{\perp }={\vec {0}}\qquad \qquad \qquad {\vec {V}}\cdot {\vec {U}}=\pm VU$

$\,{\vec {S}}={{\vec {V}}+{\vec {U}} \over 1+{\vec {V}}\cdot {\vec {U}}}$

and, expressed in terms of the speeds:

$\,S=\left|{\frac {V\pm U}{1\pm VU}}\right|$

Special case: orthogonal velocities

In the case where the velocities are orthogonal we have

$\,{\vec {U}}_{||}={\vec {0}}\qquad \qquad \qquad {\vec {U}}_{\perp }={\vec {U}}\qquad \qquad \qquad {\vec {V}}\cdot {\vec {U}}=0$

$\,{\vec {S}}={\vec {V}}+{\sqrt {1-V^{2}}}\,{\vec {U}}$

and, expressed in terms of the speeds:

$\,S={\sqrt {V^{2}+U^{2}-V^{2}U^{2}}}$

General case (engineering units, replaced ${\vec {V}}$ with v / c )

In the general case, the relativistic sum of two velocities $\mathbf {v}$ and $\mathbf {u}$ is given by

$\mathbf {w} =\mathbf {v} \oplus \mathbf {u} ={\frac {\mathbf {v} +\mathbf {u} _{\parallel }+\alpha _{\mathbf {v} }\mathbf {u} _{\perp }}{1+{\frac {\mathbf {v} \cdot \mathbf {u} }{c^{2}}}}},$

where $\mathbf {u} _{\parallel }$ and $\mathbf {u} _{\perp }$ are the components of $\mathbf {u}$ parallel and perpendicular, respectively, to $\mathbf {v}$, and $\alpha _{\mathbf {v} }$ is ${\sqrt {1-{\frac {|\mathbf {v} |^{2}}{c^{2}}}}}$ the reciprocal of the gamma factor $\gamma _{\mathbf {v} }$.

If there is now put as before,

$\,{\mathbf {u} }_{||}={\mathbf {v} \cdot \mathbf {u} \over |\mathbf {v} |^{2}}\mathbf {v} \qquad \qquad {\mathbf {u} }_{\perp }=\mathbf {u} -{\mathbf {u} }_{||}$

the equation may easily be transformed to the form used by Ungar

$\mathbf {w} =\mathbf {v} \oplus \mathbf {u} ={\frac {1}{1+{\frac {\mathbf {v} \cdot \mathbf {u} }{c^{2}}}}}\left\{\mathbf {v} +{\frac {1}{\gamma _{\mathbf {v} }}}\mathbf {u} +{\frac {1}{c^{2}}}{\frac {\gamma _{\mathbf {v} }}{1+\gamma _{\mathbf {v} }}}(\mathbf {v} \cdot \mathbf {u} )\mathbf {v} \right\}$

Using coordinates this becomes:

${\begin{pmatrix}w_{1}\\w_{2}\\w_{3}\\\end{pmatrix}}={\frac {1}{1+{\frac {v_{1}u_{1}+v_{2}u_{2}+v_{3}u_{3}}{c^{2}}}}}\left\{\left[1+{\frac {1}{c^{2}}}{\frac {\gamma _{\mathbf {v} }}{1+\gamma _{\mathbf {v} }}}(v_{1}u_{1}+v_{2}u_{2}+v_{3}u_{3})\right]{\begin{pmatrix}v_{1}\\v_{2}\\v_{3}\\\end{pmatrix}}+{\frac {1}{\gamma _{\mathbf {v} }}}{\begin{pmatrix}u_{1}\\u_{2}\\u_{3}\\\end{pmatrix}}\right\}$

where $\gamma _{\mathbf {v} }={\frac {1}{\sqrt {1-{\frac {v_{1}^{2}+v_{2}^{2}+v_{3}^{2}}{c^{2}}}}}}$.

Einstein velocity addition is commutative only when $\mathbf {u}$ and $\mathbf {v}$ are parallel. In fact

$\mathbf {u} \oplus \mathbf {v} =gyr[\mathbf {u} ,\mathbf {v} ](\mathbf {v} \oplus \mathbf {u} )$

Also it is not associative and

$\mathbf {u} \oplus (\mathbf {v} \oplus \mathbf {w} )=(\mathbf {u} \oplus \mathbf {v} )\oplus gyr[\mathbf {u} ,\mathbf {v} ]\mathbf {w}$

where gyr is the mathematical abstraction of Thomas precession into an operator called Thomas gyration and given by

$gyr[\mathbf {u} ,\mathbf {v} ]\mathbf {w} =\ominus (\mathbf {u} \oplus \mathbf {v} )\oplus (\mathbf {u} \oplus (\mathbf {v} \oplus \mathbf {w} ))$

for all w.

The gyr operator forms the foundation of gyrovector spaces.

Velocity composition paradox

Since in general u$\oplus$v ≠ v$\oplus$u this raises the question as to which velocity is the real velocity. The paradox is resolved as follows. There are two types of Lorentz transformation: boosts which correspond to a change in velocity, and rotations. The outcome of a boost followed by another boost is not a pure boost but a boost followed by or preceded by a rotation (Thomas precession). So unlike Galilean composite transformations, in special relativity, boost composition is parameterized not by velocities alone, but by velocities and orientations, so u$\oplus$v and v$\oplus$u both describe correctly but partially the boost composition B(u)B(v). If B(u)B(v) is parameterized by u$\oplus$v, the rotation gyr[u,v] associated with the composite boost B(u)B(v) is applied before the boost B(u$\oplus$v), whereas if B(u)B(v) is parameterized by v$\oplus$u, the boost B(v$\oplus$u) of v$\oplus$u is followed by the rotation gyr[u,v]. Since this rotation must be taken into account, neither u$\oplus$v or v$\oplus$u on their own give the direction of the composite velocity, only the magnitude.

In the above, a boost can be represented as a 4x4 matrix. The boost matrix B(v) means the boost B that uses the components of v, i.e. v₁, v₂, v₃ in the entries of the matrix, or rather the components of v/c in the representation that is used in the section Lorentz transformation#Matrix form. The matrix entries depend on the components of the 3-velocity v, and that's what the notation B(v) means. It could be argued that the entries depend on the components of the 4-velocity because 3 of the entries of the 4-velocity are the same as the entries of the 3-velocity, but the usefulness of parameterizing the boost by 3-velocity is that the resultant boost you get from the composition of two boosts uses the components of the 3-velocity composition u$\oplus$v in the 4x4 matrix B(u$\oplus$v). But the resultant boost also needs to be multiplied by a rotation matrix because boost compostion (i.e. the multiplication of two 4x4 matrices) results not in a pure boost but a boost and a rotation, i.e. a 4x4 matrix that corresponds to the rotation gyr[u,v] to get B(u$\oplus$v) = B(u$\oplus$v)gyr[u,v] = gyr[u,v]B(v$\oplus$u). The notation gyr[u,v] was previously used as the rotation applied to a 3-velocity, whereas here the same notation is used to mean the 4x4 matrix rotation, but it is essentially referring to the same thing: the rotation that arises from the composition of two boosts.

If the 3x3 matrix form of the rotation applied to 3-velocities in u$\oplus$v = gyr[u,v]B(v$\oplus$u) is given by gyr[u,v], then the 4x4 matrix rotation applied to 4-coordinates is given by:

${\begin{pmatrix}1&0\\0&gyr[\mathbf {u} ,\mathbf {v} ]\end{pmatrix}}$.

Doppler shift

A notion of velocity addition can also be formulated in the theory of the nonrelativistic, one dimensional Doppler shift. When the source of a wave is moving with nonrelativistic velocity $s$ toward the receiver, the frequency of the waves is increased by a factor of ${1/(1-s/c)}$. If the receiver is moving with velocity $v$, the frequency of the waves detected is decreased by a factor of $(1-v/c)$. When both the source and the receiver are moving, the frequency measured is given by:

$f'=f{1-v/c \over 1-s/c}\,.$

If a receiver measures velocities using Doppler shifts, and it determines that an object coming towards it is moving with velocity $u$, it is actually determining the shift in frequency, from which it calculates the velocity. Suppose that the receiver itself is moving with velocity $v$, but it does not take this into account in the calculation. It calculates the value $u$ falsely assuming that it is at rest. the velocity $u$ can then be thought of as the inferred velocity relative to the ship from Doppler shifts alone. What, then, is the actual velocity of the object relative to the medium?

Since the ship determined $u$ from the frequency, the frequency shift factor relative to the ship is

${1 \over 1-u/c}\,.$

But this factor is not the frequency shift relative to a stationary receiver. For a stationary observer, it must be corrected by dividing by the frequency shift of the ship:

$f'=f{1 \over (1-u/c)(1-v/c)}\,.$

The velocity of the object relative to the medium is then given by

$s=v+u-{vu \over c}\,.$

This is the true velocity of the object. Unlike the relativistic addition formula, the velocity $u$ is not the physical velocity of the object.

There is a group of transformations in one space and one time dimension for which this operation forms the addition law. The group is defined by all matrices:

${\begin{pmatrix}1&0\\{-v \over 1-v/c}&{1 \over 1-v/c}\end{pmatrix}}\,$

When they act on $\scriptstyle (t,x)$, they produce the transformations

$\,t'=t\qquad x'={x-vt \over 1-v/c}\,$

which is a Galilean boost accompanied by a rescaling of the $x$ coordinate. When two of these matrices are multiplied, the quantity $v$ (the velocity of the frame), combines according to the Doppler addition law.

The physical meaning can be extracted from the transformation. Time is the same for both frames, but the rescaling of the $x$ axis keeps the right-moving speed of sound fixed in the moving frame. This means that if the ship uses this transformation to define its frame, the ruler that it uses is the distance that the waves move to the right in one unit of time. The velocity $u$ can now be given a physical interpretation, although an unusual one. It is the velocity of the object as measured from the ship using a Doppler contracted ruler.

Relativistic Doppler shift

In the theory of the relativistic Doppler shift, the case where the speed of the wave is equal to the speed of light is special, because then there is no preferred rest-frame. In this case the frequency of the received waves can only depend on the relativistic sum of the velocities of the emitter and the receiver. But when the speed of the wave $\scriptstyle c\neq 1$, meaning that the phase velocity of the wave is different from that of light, the relativistic Doppler shift formula does not depend only on the relative velocities of the emitter and receiver, but on their velocities with respect to the medium.

In the rest frame of the medium, the frequency emitted by a relativistic source moving with velocity $v$ is decreased by the time dilation of the source:

$f'={f{\sqrt {1-v^{2}/c^{2}}} \over 1-v/c}$

If the receiver is moving with a velocity $u$ through the fluid perpendicular to the wave fronts, the frequency received is determined by the proper time between the events where the receiver crosses crests. The fluid frame time between crest-crossings does not require changing frames and is the same as in the nonrelativistic case:

$\Delta t'={\Delta t \over 1-u/c}$

In this time, the receiver has moved (in the fluid frame) an amount

$\Delta x'=u\Delta t'\,$

And the proper time between the two crest crossing is

$\Delta \tau '=\Delta t'^{2}-\Delta x'^{2}=\Delta t{{\sqrt {1-u^{2}/c^{2}}} \over 1-u/c}\,$

And this is the time between crest-crossings as measured by the receiver. From this, the received frequency can be read off:

$f'=f{\frac {1-u/c}{\sqrt {1-u^{2}/c^{2}}}}\,$

Multiplying the two factors for the emitter and receiver gives the relativistic Doppler shift:

$f'=f{{\sqrt {1-v^{2}/c^{2}}} \over (1-v/c)}{(1-u/c) \over {\sqrt {1-u^{2}/c^{2}}}}\,$

When $c=1$, it simplifies:

$f'=f{\sqrt {1+v \over 1-v}}{\sqrt {1-u \over 1+u}}\,$

and then

${\sqrt {(1+v)(1-u) \over (1-v)(1+u)}}={\sqrt {1+(v-u)/(1-vu) \over 1-(v-u)/(1-vu)}}\,,$

so that the relativistic Doppler shift of light is determined by the relativistic difference of the two velocities.

It is also possible to determine, in the relativistic case, the actual velocity of a source, when a moving ship falsely determines it from a Doppler shift without taking its own motion into account. Just as in the non-relativistic case, this is the velocity at which a source would have to be moving in order to make the Doppler shift factor for a moving receiver equal to the Doppler shift factor for the velocity $u$. It is the solution of the equation:

${{\sqrt {1-s^{2}/c^{2}}} \over (1-s/c)}={{\sqrt {1-v^{2}/c^{2}}}{\sqrt {1-u^{2}/c^{2}}} \over (1-v/c)(1-u/c)}$

which is quadratic in $s$. This is the relativistic analog of the Doppler velocity addition formula. When $c$ is not the speed of light, the velocity $u$ is not the velocity of anything, just a false inferred velocity from the point of view of the moving ship. In the relativistic case, there is no group of transformations for which this is the velocity addition law, since it is impossible to independently rescale time and distance measurements.