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Geometric object with magnitude (length) and direction.

0
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two vectors in Euclidean space, the angle could be described as the arc-length of an arc of the unit-circle intercepted by those vectors. Then the inner product could be expressed in terms of the … future-directed 4-vectors in Minkowski, the angle [a.k.a. the rapidity] could be described as the Minkowski arc-length of the intercepted arc of the unit-Minkowski circle [a hyperbola] in the plane …
answered Mar 9 '18 by robphy
2
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the rapidity $\theta$ (the Minkowski angle between two timelike vectors), that zeroth component is essentially $\cosh\theta$. In practice, [in geometric units] the 4-velocity is a unit-timelike …
answered Jan 12 '18 by robphy
3
votes
The Lorentz factor $\gamma=\frac{1}{\sqrt{1-v^2}}$ can expressed in terms of the rapidity as $\gamma=\cosh\theta$, where $\theta$ is the rapidity between 4-velocities (which are timelike unit-vectors
answered Aug 3 by robphy
0
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If you intend your $\theta$ to be the [real-valued] rapidity (arc-length of a unit hyperbola in Minkowski spacetime), then your 4-vectors should be both future-timelike [or both past]. I think it can … work with spacelike vectors as long as they are coplanar with the future-timelike vectors, correspondingly orthogonal to the timelike-vectors, and probably must both be in the forward [or backward …
answered Mar 16 '17 by robphy
1
vote
, a sign might be included for convenience in each of the above. On the other hand, lightlike vectors can't be normalized. So, there is no lorentz-invariant sense of a unit vector for a lightlike direction. …
answered Jun 7 '18 by robphy
10
votes
The magnetic field is not a [polar] vector, but a pseudovector. In fact, the cross-product of a vector (e.g. Velocity) and a pseudovector (Magnetic Field) is a [polar] vector (e.g. Force). In a more …
answered Oct 16 '17 by robphy
0
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This answer amplifies @PiKindOfGuy's answer and @Dale's comment. I think there is a confusion in the use and/or interpretation of $d\vec x$ as an infinitesimal element of a directed path and as …
answered Nov 11 '18 by robphy