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In standard books on relativity, a material point is modelled in space-time as a world-line parameterized by a time-like vector $\tau$ . Given that time and space are given an equal footing in relativity: Is there also a space-like vector associated with this world line?

Physiks lover
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  • World-lines are generally parametrized by the (scalar) proper time (commonly denoted by $\tau$). What timelike vector are you talking about here? – Sandejo Jan 25 '23 at 01:53
  • "Time and space are on equal footing" doesn't mean "every timelike vector has an associated spacelike vector". – Javier Jan 25 '23 at 02:18

1 Answers1

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In 2D Euclidean geometry,

  • Construct a circle centered at the tail of the given vector, with radius determined by the tip of the given vector.
  • At the tip, the tangent line to the circle is orthogonal to the vector. This defines a vector orthogonal to the given vector.
  • The "orthogonal axis" of that coordinate system is the line through the tail of the given vector that is parallel to that tangent line.

Follow the same procedure in special relativity (1+1)-Minkowski spacetime, where the "circle" is now the future branch of a hyperbola. (The tips of all 4-velocities [unit future-pointing timelike vectors] lie on this hyperbola.)

  • At the tip of the given 4-velocity vector, the tangent line to the hyperbola (the "circle" in this geometry) is Minkowski-orthogonal to the given 4-velocity. (This is a set of simultaneous events for the observer with that 4-velocity... simultaneous with that observer's t=1 on her wristwatch.)
    This defines a spacelike vector orthogonal to the given 4-velocity.
  • The observer's "spatial axis" is the line through the tail of the 4-velocity that is parallel to that tangent line. (This is a set of simultaneous events with that observer's t=0 on her wristwatch.)

UPDATE: The above is based on the construction suggested by Minkowski.

(From my answer https://physics.stackexchange.com/a/638018/148184 to Is simultaneity in SR merely an artifact of coordinate systems? ... )

From Minkowski's "Space and Time"...

We decompose any vector, such as that from O to x, y, z, t into four components x, y, z, t. If the directions of two vectors are, respectively, that of a radius vector OR from O to one of the surfaces ∓F = 1, and that of a tangent RS at the point R on the same surface, the vectors are called normal to each other. Accordingly, $$c^2tt_1 − xx_1 − yy_1 − zz_1 = 0$$ is the condition for the vectors with components x, y, z, t and $x_1$, $y_1$, $z_1$, $t_1$ to be normal to each other.

As mentioned in the linked article, you can play around with this idea using my spacetime diagrammer for relativity https://www.desmos.com/calculator/kv8szi3ic8 .
Tune the E-slider: 1 is Minkowski, 0 is Galilean, and -1 is Euclidean.

robphy
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