How can we visualize timelike curves and surfaces in 3-dimensional Minkowski space? I mean, we have to live in 3-dimensional Euclidean space now, so the images must be different. For example, the arc length of a spacelike curve converges as a complex integral. Since the metric is different, I guess we should have a Lorentz-Minkowski eyewear for us to see it. :D
The timelike curves in Minkowski 3-spacetime have their tangent vectors of timelike character for any value of the parameter. That means that its image is always on the interior of the 2-dimensional lightcone with direction parallel to a straitght line that lives in that interior too. To build an image of a timelike curve in our local space-time we must consider the 4-dimensional Minkowski space-time in wich the lightcone is 3-dimensional and is defined by all non cero vectors of lightlike character. Then a timelike curve in our local 4-dimensional (Minkowski) space-time is always in the interior of that 3-dimensional lightcone with direction parallel to a straitgh line that lives in that interior too. That interior always contains the time axis. As you can see, is the same property but in one more dimension (3 dimensions for space and one for the time). And is the same in any finite dimensional Minkowski space-time. The timelike surfaces in 3-dimensional Minkowski space-time have its tangent plane with normal direction of spacelike character for any point of the surface. For timelike surfaces in any finite dimensional Minkowski space-time we have the same definition.
Thank you for your answer. What about the arc length?
Is it possible to see this curve with the naked eye? Or is it just in theory? In some articles, these curves are plotted in Euclidean space, is this the right approach?
What do you mean for a naked eye? By definition, a timelike curve only exists in a semi-Euclidean space, which is topologically a Euclidean space but the metric is non-euclidean. The pictures can be done in Euclidean space but definitely the metric can not be Euclidean if we must consider timelike curves. Minkowski space-time is semi-Euclidean with signature one. If the metric is Euclidean (signature cero), only spacelike curves can exist.
I think this paper is good answer. I found it yet.
http://www.rpi.edu/dept/phys/Courses/Astronomy/CurvedSpacetimeAJP.pdf
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