Angle between space-like and time-like vectors

In summary: A while back I thought about defining the angle you seek in terms of the angle between a timelike vector and a timelike-vector-orthogonal-to-the-spacelike-vector...of course, all three vectors in a common plane. However, it seems that one needs to play around with signs to get things to be consistent... however, this scheme looked rather unnatural to me... and didn't seem to have an immediate physical or geometrical interpretation.The angle between two future-timelike vectors can be determined by the intercepted arc-length of a unit hyperbola. The rapidity is the hyperbolic angle that corresponds to this.
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I would like to learn about the angle between space-like vectors and time-like vectors. Is there anyone who can help me repeatly, please?
 
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  • #2
There's no such thing.Angles and directions can be thoroughly defined only for spaces with positively defined metric (because they're defined using a scalar product).Minkowski space doesn't have this property,be it curved or flat.

Daniel.
 
  • #3
It seems the best you can do is determine that "a nonzero spacelike vector is orthogonal to a nonzero timelike one" if their Minkowski-dot product is zero.

Angle measure is usually defined as the ratio of arc-length on a "circle" to the radius. With two future-timelike vectors, one can use the intercepted arc-length of a unit hyperbola to define the angle between two future-timelike vectors. (This is called the rapidity.) However, there is no such hyperbola for a timelike and a spacelike vector. (One could also think of an angle as a parameter in the Lorentz Transformation to "rotate" one vector into another... However, Lorentz Transformations preserve the timelike (or, respectively, spacelike or null) nature of a vector.)

A while back I thought about defining the angle you seek in terms of the angle between a timelike vector and a timelike-vector-orthogonal-to-the-spacelike-vector...of course, all three vectors in a common plane. However, it seems that one needs to play around with signs to get things to be consistent... however, this scheme looked rather unnatural to me... and didn't seem to have an immediate physical or geometrical interpretation.
 
  • #4
I can say at "Lorentz space" if v and w at the same timecone than,

g(v,w)= -ııvıı.ııwıı.chQ, Q>=0

if v and w are not at the same timecone than,

ı g(v,w) ı = ııvıı.ııwıı.chQ

if v and w spacelike vectors,

g(v,w)= ııvıı.ııwıı.cosQ, 0<=Q<=pi

Note that if v and w are together timelike or spacelike vectors than v are not

orthogonal to w ( if v orthogonal to w than a once is timelike the other is must be spacelike)
 
  • #5
robphy said:
Angle measure is usually defined as the ratio of arc-length on a "circle" to the radius. With two future-timelike vectors, one can use the intercepted arc-length of a unit hyperbola to define the angle between two future-timelike vectors. (This is called the rapidity.)

If I've understood this right, the rapidity is a hyperbolic angle, which is defined a little differently to a circular angle. The absolute value of a hyperbolic angle corresponds to the area under the unit hyperbola (on both sides of the axis), rather than arc length. Rapidity is labelled u on the diagram at the top of the first link:

http://hubpages.com/hub/Hyperbolic-Functions
http://en.wikipedia.org/wiki/Rapidity
http://www.chartwellyorke.com/sketchpad/downloads/Minkowski_Overview.pdf [Broken]
 
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What is the angle between a space-like and a time-like vector?

The angle between a space-like and a time-like vector is not well-defined in the context of special relativity. This is because the concept of "angle" relies on the notion of a Euclidean space, which does not apply to the non-Euclidean spacetime of relativity.

Can the angle between two spacetime vectors be measured?

No, the angle between two spacetime vectors cannot be measured in the same way as angles in Euclidean geometry. However, there are other mathematical techniques that can be used to compare the orientations of different vectors in spacetime.

Why is the angle between space-like and time-like vectors important?

The distinction between space-like and time-like vectors is crucial in understanding the behavior of objects in relativity. Space-like vectors represent distances and lengths, while time-like vectors represent intervals of time. The angle between them can indicate the relative motion of objects and how events are perceived by different observers.

Is the angle between space-like and time-like vectors always zero?

No, the angle between space-like and time-like vectors can vary. In fact, in some cases, it can be imaginary or undefined. This is because the angle is determined by the inner product of the two vectors, which can be complex or non-existent depending on the nature of the vectors.

How does the angle between space-like and time-like vectors change in different reference frames?

The angle between space-like and time-like vectors is relative and can change depending on the observer's reference frame. This is known as the relativity of simultaneity, where two events that are simultaneous in one frame of reference may not be simultaneous in another frame. Therefore, the angle between the space-like and time-like vectors may also change.

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