Relativistic velocity tangential to movement¶
Consider two inertial reference frames: one fixed (lab frame) and one tied to the moving object (proper frame). The proper frame is moving with some velocity \(\vec v\) relative to the lab frame. According to the theory of special relativity, the velocity of the object relative to lab frame is not equal to the sum of its velocity in the proper frame and the velocity of the proper frame relative to the lab frame.
Notes:
One can get the same expression for \({\vec u'}_\perp\) in terms of \(\vec u\) by replacing \(\vec v\) with \(-{\vec v}\). This is essentially the inverse Lorentz transformation from lab frame to proper frame that uses the fact that the lab frame can be viewed as moving with velocity vector \(-{\vec v}\) relative to the proper frame.
Conditions:
Works in special relativity.
Links:
- tangential_velocity_in_lab_frame¶
Component of the velocity vector relative to the lab frame tangential to \(\vec v\). See
speed.
- Symbol:
u_t- Latex:
\({\vec u}_\text{t}\)
- Dimension:
velocity
- Symbol:
u'- Latex:
\({\vec u'}\)
- Dimension:
velocity
- Symbol:
v- Latex:
\({\vec v}\)
- Dimension:
velocity
- law¶
u_t = v * (dot(u', v) * dot(v, v)^(-1) + 1) * (1 + dot(u', v) / c^2)^(-1)- Latex:
- \[{\vec u}_\text{t} = {\vec v} \left(\left( {\vec u'}, {\vec v} \right) \left( {\vec v}, {\vec v} \right)^{-1} + 1\right) \left(1 + \frac{\left( {\vec u'}, {\vec v} \right)}{c^{2}}\right)^{-1}\]