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Classification of events and curves in Minkowskian spacetime 
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#1
Feb2312, 04:55 AM

P: 71

I'm trying to understand the causal structure of Minkowskian spacetime and I was wondering whether something can be said about the relation between the classification of events and curves.
To clarify: for Minkowskian inner product [itex]\eta[/itex] with signature (+++), two events [itex]p[/itex] and [itex]q[/itex] can be timelike ([itex]\eta(\vec{pq},\vec{pq})<0[/itex]), spacelike ([itex]\eta(\vec{pq},\vec{pq})>0[/itex]) or lightlike related ([itex]\eta(\vec{pq},\vec{pq})=0[/itex]). A curve [itex]w \colon I\subset\mathbb{R}\to C\subset M_{p}^{4}[/itex] is timelike/spacelike/lightlike when its tangent vectors [itex]w'(t)[/itex] are. The question is now: can any type of events be connected by any type of curve? A secondary question: can the velocity (tangent vector) of a curve in Minkowskian spacetime be defined as below? [tex] w \colon I\subset\mathbb{R}\to C\subset M_{p}^{4}\colon t\mapsto w_{v}(t)+o [/tex] [tex] w_{v} \colon I\subset\mathbb{R}\to C\subset M_{v}^{4}\colon t\mapsto w_{v}(t) [/tex] [tex] w'(t) =\lim_{h\to 0}\frac{(w_{v}(t+h)+o)(w_{v}(t)+o)}{h}=w_{v}'(t) [/tex] where [itex]M_{p}^{4}[/itex] point space, [itex]o\in M_{p}^{4}[/itex] and [itex]M_{v}^{4}[/itex] vector space with Minkowskian inner product [itex]\eta[/itex] (i.e. inner product but weaken positivedefinite to nondegenerate). 


#2
Feb2312, 11:21 AM

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P: 2,698

Spacelike separated events can only be connected by spacelike curves. Time like events with time like curves, and null events with null. This is because in every reference frame, these designations do not change.
You can define the the tangent vector the way you did, but one should note that the tangent vectors should be defined for an equivalence class of curves which cross that point and have "the same tangent" so to speak (i.e. if two tangent vectors have the same components in some coordinate system, then they should be "the same tangent vector"). 


#3
Feb2312, 06:05 PM

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There's no such thing as a timelike curve or a spacelike curve. Events on curves can be spacelike or timelike seperated depending on where on the curve they are.



#4
Feb2312, 07:26 PM

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Classification of events and curves in Minkowskian spacetime



#5
Feb2312, 08:40 PM

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PF Gold
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A timelike curve in spacetime is a curve whose tangent vector at each event is timelike [in the tangent vector space associated with that event], as the stated in the original post.
It seems that there is an implicit assumption that the curves under discussion here have tangent vectors that don't change their type or temporal orientation. 


#6
Feb2312, 08:58 PM

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A general curve which changes from timelike to spacelike or viceversa cannot be a geodesic of the spacetime because geodesics parallel transport their own tangent vectors, so it seems like these more general curves are more limited in their application.



#7
Feb2312, 10:20 PM

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#8
Feb2312, 10:33 PM

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While geodesics are nicelybehaved curves, there are physically important nongeodesic curves....like worldlines of piecewiseinertial particles and worldlines of accelerated particles.
In the proofs of some theorems involving causal structure, there are "causal curves" whose tangents can be timelike or null. 


#9
Feb2412, 01:08 PM

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#10
Feb2412, 05:08 PM

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#11
Feb2712, 04:10 AM

P: 71

So what I'm looking for are all possible means of relating events and the physical implicating of this relations (elapsed time, Euclidean distance,...) 


#12
Feb2712, 05:18 AM

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To preserve causality (and for various other reasons), physical objects can only move along timelike curves (or null in the case of photos). The timelike curves in Minkowski spacetie are exactly those curves which are always moving at no more than 45 degrees away from the vertical axis (assuming t is your vertical axis).
From any event P in Minkowski space, one an draw 2 cones, one going forward and one going backward, the edges of which make a 45 degree angle with the vertical axis. Everything in the future directed cone are events which an object at event P can affect, everything in the past directed cone are events which could have affected event P. These are the tirival chronological future and chronological past of the event P. If you include the edges of the cones, then these define the causal future and causal past. 


#13
Feb2712, 07:47 AM

P: 71

Thanks, that has been very helpful. But this is only one relation between events (causal relation). Aren't there others? For example, wouldn't there be a distance between simultaneous events which is equivalent to the Euclidean distance? As I understand, all events with difference vectors perpendicular to some timelike vector [itex]\vec{u}[/itex] are called simultaneous and this is an equivalence relation in Minkowskian point space. The difference vectors form a threedimensional spacelike subspace [itex]\vec{u}^{\perp}[/itex]. When restricted to this space, the inner product is positive definite, so the induces norm in [itex]\vec{u}^{\perp}\subset M_{v}^{4}[/itex] and metric in [itex]Sim_{p}\subset M_{p}^{4}[/itex] are like the Euclidean ones ([itex]Sim_{p}[/itex] is the equivalence class that contains event p). Curves in equivalence class [itex]Sim_{p}[/itex] must be spacelike and their arclengths are the normal Euclidean ones. Furthermore, can't we also project timelike curves on their spacelike component and consider a "spacial (Euclidean) distance" that an object traveled?
Your mentioning of the 45 degree doublecone brings up something else I was wondering. Since light rays in vacuum are timelike lines, their speed (norm of the velocity) would be [itex]\left\w'(t)\right\=1[/itex]. How can we connect this to physical units (e.g. m/s)? Same problem for the elapsed proper time along a timelike curve: how do we "connect" it to seconds? And more in general: what unit is connected with the Minkowski metric (which is actually not a metric since it is e.g. not subadditive) in [itex]M_{p}^{4}[/itex] induces by the Minkowski inner product (which is actually not an inner product since it is not possitive definite). 


#14
Feb2712, 10:28 AM

P: 848

I suspect that many of these allowable curves would somewhat resemble Feynman diagrams. A vast number of curves may be simply drawn by hand on a sheet of paper using a pen (any curve you can draw on a piece of paper would be manifestly admissable). An example curve drawn with black ink in this instance would perhaps turn out to be (upon microscopic examination) a collection of black ink deposits making a curved pattern in 3D and extending along their 4th dimension world line (along with the paper on which it is drawn). I can add another red curve that, on the 2D sheet of paper, appears to approach and bounce off of the black curvesimulating the interaction of two objects. We can create a host of admissable curves playing a game of pool. And the interactions are easily understood. These curves could perhaps be organized in a way that expresses many of the laws of physics. Perhaps a fundamental question would be, can the laws of physics be replaced by listing the allowable 4dimensional geometric patterns? Perhaps Einstein's unified field theory would have been a set of differential equations whose solutions are the admissable curves you seek to classify. This would mean that it is not necessary to understand the universe as a dynamically evolving 3dimensional structure driven by "laws of physics" and causality. Rather we would have a static 4dimensional unverse structure populated by admissible curves and geometric patterns. 


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