In Special Relativity, velocities do not add linearly as they do in Galilean physics. Instead, the combination of velocities must preserve the invariance of the speed of light, as encoded in the Lorentz transformations .
Consider two inertial frames:
Frame S S
Frame S ′ S ′ v v x x S S
An object moves with velocity components (u x ′ u x ′ u y ′ u y ′ u z ′ u z ′ S ′ S ′ u x u x u y u y u z u z S S
Velocity components are defined as:
u x = d x d t , u y = d y d t , u z = d z d t u x = d t d x , u y = d t d y , u z = d t d z
u x ′ = d x ′ d t ′ , u y ′ = d y ′ d t ′ , u z ′ = d z ′ d t ′ u x ′ = d t ′ d x ′ , u y ′ = d t ′ d y ′ , u z ′ = d t ′ d z ′
Our goal is to express ( u x , u y , u z ) ( u x , u y , u z ) ( u x ′ , u y ′ , u z ′ ) ( u x ′ , u y ′ , u z ′ ) v v
The Lorentz transformations for a boost along the x x
t = γ ( t ′ + v x ′ c 2 ) x = γ ( x ′ + v t ′ ) y = y ′ z = z ′ t x y z = γ ( t ′ + c 2 v x ′ ) = γ ( x ′ + v t ′ ) = y ′ = z ′
where:
γ v = 1 1 − v 2 c 2 γ v = 1 − c 2 v 2 1
The Lorentz factor γ v γ v v v not on the particle velocity u u
Differentiate the transformations:
d x = γ v ( d x ′ + v d t ′ ) d x = γ v ( d x ′ + v d t ′ )
d t = γ v ( d t ′ + v c 2 d x ′ ) d t = γ v ( d t ′ + c 2 v d x ′ )
Now compute:
u x = d x d t = d x ′ + v d t ′ d t ′ + v c 2 d x ′ u x = d t d x = d t ′ + c 2 v d x ′ d x ′ + v d t ′
Divide numerator and denominator by d t ′ d t ′
u x = u x ′ + v 1 + v u x ′ c 2 u x = 1 + c 2 v u x ′ u x ′ + v
Since y = y ′ y = y ′ z = z ′ z = z ′
d y = d y ′ , d z = d z ′ d y = d y ′ , d z = d z ′
But time is transformed:
d t = γ v ( d t ′ + v c 2 d x ′ ) d t = γ v ( d t ′ + c 2 v d x ′ )
Thus:
u y = u y ′ γ v ( 1 + v u x ′ c 2 ) u y = γ v ( 1 + c 2 v u x ′ ) u y ′
u z = u z ′ γ v ( 1 + v u x ′ c 2 ) u z = γ v ( 1 + c 2 v u x ′ ) u z ′
Longitudinal component (parallel to v v
u x = u x ′ + v 1 + v u x ′ c 2 u x = 1 + c 2 v u x ′ u x ′ + v
Transverse components (perpendicular to v v
u y = u y ′ γ v ( 1 + v u x ′ c 2 ) u y = γ v ( 1 + c 2 v u x ′ ) u y ′
u z = u z ′ γ v ( 1 + v u x ′ c 2 ) u z = γ v ( 1 + c 2 v u x ′ ) u z ′
We can express this result in matrix form by defining:
u ′ = ( u x ′ u y ′ u z ′ ) , u = ( u x u y u z ) u ′ = ⎝ ⎛ u x ′ u y ′ u z ′ ⎠ ⎞ , u = ⎝ ⎛ u x u y u z ⎠ ⎞
Then, the velocity addition is (not so nicely) represented as:
u = 1 D ( u x ′ + v u y ′ γ v u z ′ γ v ) , D = 1 + v u x ′ c 2 u = D 1 ⎝ ⎜ ⎜ ⎛ u x ′ + v γ v u y ′ γ v u z ′ ⎠ ⎟ ⎟ ⎞ , D = 1 + c 2 v u x ′
There are two distinct Lorentz factors that commonly appear in relativistic kinematics:
γ v γ v v v
Appears in the Lorentz transformations and in the transverse velocity addition formula.
In this page, all γ γ γ v γ v
γ u γ u u u
Appears when computing energy and momentum of a particle.
Not directly involved in the velocity addition formulas derived here.
Example: Plasma Blob in an AGN Accretion Disk Wind
Consider a relativistic plasma blob ejected from an AGN accretion disk wind .
Assume:
The wind frame S ′ S ′ v = 0.6 c v = 0 . 6 c
In S ′ S ′
u x ′ = 0.3 c u x ′ = 0 . 3 c u y ′ = 0.5 c u y ′ = 0 . 5 c u z ′ = 0 u z ′ = 0
γ v = 1 1 − ( 0.6 ) 2 = 1 0.64 = 1 0.8 = 1.25 γ v = 1 − ( 0 . 6 ) 2 1 = 0 . 6 4 1 = 0 . 8 1 = 1 . 2 5
u x = 0.3 c + 0.6 c 1 + ( 0.6 ) ( 0.3 ) 1 = 0.9 c 1 + 0.18 = 0.9 c 1.18 ≈ 0.763 c u x = 1 + 1 ( 0 . 6 ) ( 0 . 3 ) 0 . 3 c + 0 . 6 c = 1 + 0 . 1 8 0 . 9 c = 1 . 1 8 0 . 9 c ≈ 0 . 7 6 3 c
u y = 0.5 c 1.25 × ( 1 + 0.18 ) = 0.5 c 1.25 × 1.18 = 0.5 c 1.475 ≈ 0.339 c u y = 1 . 2 5 × ( 1 + 0 . 1 8 ) 0 . 5 c = 1 . 2 5 × 1 . 1 8 0 . 5 c = 1 . 4 7 5 0 . 5 c ≈ 0 . 3 3 9 c
The longitudinal velocity u x u x c c
The transverse velocity u y u y
Time dilation (factor γ v γ v
Relativity of simultaneity (factor 1 + v u x ′ / c 2 1 + v u x ′ / c 2
Consequences in AGN physics:
Transverse plasma motions in disk winds and jets appear slowed to the observer.
Apparent superluminal motion in jets arises when u x u x
Relativistic velocity addition arises directly from Lorentz transformations.
The longitudinal and transverse components transform differently.
The Lorentz factor γ v γ v frame velocity v v u u
The matrix form reveals the anisotropic nature of the transformation.
These results are essential for understanding the kinematics of relativistic outflows in AGN and black hole environments.
Rindler, W. Introduction to Special Relativity
Jackson, J.D. Classical Electrodynamics , Chapter 11
Urry & Padovani (1995), Unified Schemes for Radio-Loud AGN
Blandford & Königl (1979), Relativistic Jets as Compact Radio Sources