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I am currently reading the book "Classical charge particle" by Fritz Rohrlich, and I struggle a lot with the appendix "space-like planes and Gauss's integral theorem".

He says "the world line of a particle determines uniquely its now-plane at every instant and as seen by any inertial observer". My question is basically : what is a "now-plane", and qualitatively, what am I suppose to understand from this sentence? (basically : why is it worth stating that?).

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    $\begingroup$ Presumably the "now-plane" is the plane spanned by the instantaneous velocity and acceleration vectors (as seen by any inertial observer). $\endgroup$ – Qmechanic Jul 28 '18 at 14:10
  • $\begingroup$ Hum, I'm not quite sure to understand. Basically my point is that : time depends on the inertial observer, so how talking about a "now-plane" (which sounds like a global, observer-independent object) can make sense ? $\endgroup$ – Ryuzaki Jul 28 '18 at 15:45
  • $\begingroup$ Some people call it "the hyper-slice of simultaneity". $\endgroup$ – JEB Jul 28 '18 at 20:23
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Elaborating on the @Qmechanic comment...
at each event P on the particle worldline one can do the following construction.

  • Draw a future-timelike hyperboloid [which is asymptotic to the light-cone of P], with radius chosen small enough so that the worldline is approximately inertial in this neighborhood.
  • At the future event where the worldline intersects this hyperboloid, construct the tangent hyperplane to this hyperboloid. As Minkowski defined, this hyperplane is Minkowski-orthogonal to the worldline. (Intuitively, the "tangent" is perpendicular to the "radius".)
  • Through P, construct the parallel hyperplane.
    This is the "now plane" for the particle worldline at event P.
    [And all frames of reference who diagram this situation would agree on this hyperplane.]

    Physically...
    For the particle (and the instantaneous inertial frame moving with the particle at P), all events on this hyperplane would be simultaneous with P. In other words, it is the instantaneous "notion of space" at this instant for this particle. So, any "integrals over all space" would use this hyperplane.
  • If the particle is not inertial, future "now planes" might not be parallel to this hyperplane at P.

Here's a visualization I did that displays this.
Try changing the E parameter to see the Euclidean and Galilean analogues.
https://www.desmos.com/calculator/wm9jmrqnw2

robphy-Desmos-hyperplane

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The worldline of an object is the sequence of spacetime events, corresponding to the history of the object.

The worldline is a time-like curve in spacetime. Each point of the worldline is an event that can be labeled by the time and spacial position of that object.

To answer the question, that is what he means by now-plane, at every instant, it is an event, that is labeled by the time and spatial position.

Why plane? For easy visualization, of four dimensions, two space coordinates are often suppressed, the event is then represented by a point on a Minkowski-diagram, which is a plane usually plotted with a time coordinate.

enter image description here

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  • $\begingroup$ I think "plane" is really "hyperplane", which has one less dimension than the dimensionality of space[time]. On a typical (1+1)-dimensional spacetime diagram, as you have drawn, for each event on the worldline, the "plane" appears as line through that event which is Minkowski-perpendicular to the worldline. $\endgroup$ – robphy Jul 28 '18 at 21:27

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