Classification of events and curves in Minkowskian spacetime

[Wox]

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 $\eta$ with signature (-+++), two events $p$ and $q$ can be timelike ($\eta(\vec{pq},\vec{pq})<0$), spacelike ($\eta(\vec{pq},\vec{pq})>0$) or lightlike related ($\eta(\vec{pq},\vec{pq})=0$). A curve $w \colon I\subset\mathbb{R}\to C\subset M_{p}^{4}$ is timelike/spacelike/lightlike when its tangent vectors $w'(t)$ 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?
$$w \colon I\subset\mathbb{R}\to C\subset M_{p}^{4}\colon t\mapsto w_{v}(t)+o$$
$$w_{v} \colon I\subset\mathbb{R}\to C\subset M_{v}^{4}\colon t\mapsto w_{v}(t)$$
$$w'(t) =\lim_{h\to 0}\frac{(w_{v}(t+h)+o)-(w_{v}(t)+o)}{h}=w_{v}'(t)$$
where $M_{p}^{4}$ point space, $o\in M_{p}^{4}$ and $M_{v}^{4}$ vector space with Minkowskian inner product $\eta$ (i.e. inner product but weaken positive-definite to non-degenerate).

 

[Matterwave]

Space-like separated events can only be connected by space-like 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").

 

[dx]

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

 

[Matterwave]

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

What? The geodesic of massive particles are timelike curves, how can events lying on the worldline of a particle be spacelike separated?

 

[robphy]

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.

 

[Matterwave]

A general curve which changes from timelike to spacelike or vice-versa 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.

 

[PAllen]

A general curve which changes from timelike to spacelike or vice-versa 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.

Correct, a geodesic is timelike, null, or spacelike, no mixture. An arbitrary curve can be a timelike in one portion and spacelike in another. Obviously, the world line of an arbitrary body (geodesic or not) must be timelike everywhere. And a massless particle must follow a null path in vacuum. Mixed paths are can't really represent anything physical, so far as I know.

 

[robphy]

While geodesics are nicely-behaved curves, there are physically important non-geodesic curves...like worldlines of piecewise-inertial 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.

 

[PAllen]

While geodesics are nicely-behaved curves, there are physically important non-geodesic curves...like worldlines of piecewise-inertial 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.

A causal curve might play a role in a proof. So might any curve. But no physical entity can follow curve that is part timelike and part null.

 

[Matterwave]

While geodesics are nicely-behaved curves, there are physically important non-geodesic curves...like worldlines of piecewise-inertial 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.

Since the OP was talking about Minkowski space-time, the causal structure is pretty trivial, so I didn't think such complications are necessary.

 

[Wox]

Since the OP was talking about Minkowski space-time, the causal structure is pretty trivial, so I didn't think such complications are necessary.

Thanks for you answer. As for which curves to consider: any curve that has physical meaning. I'm really just trying to understand the Minkowski space-time structure (not trivial to me :-)) and one of the questions I asked myself was: "How do I get from one event to another?" One example to get from event p to event q: a timelike curve (physical meaning: path the a point mass follows in space-time with arc-length the elapsed time according to this mass). Another example (not sure whether it's correct) would be a spacelike curve which arc-length would be a Euclidean distance (physical meaning: a projection of the path of a point mass?) One can also think about piecewise curves and mixed curves and ...

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

 

[Matterwave]

To preserve causality (and for various other reasons), physical objects can only move along time-like curves (or null in the case of photos). The time-like curves in Minkowski space-tie 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.

 

[Wox]

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 $\vec{u}$ are called simultaneous and this is an equivalence relation in Minkowskian point space. The difference vectors form a three-dimensional spacelike subspace $\vec{u}^{\perp}$. When restricted to this space, the inner product is positive definite, so the induces norm in $\vec{u}^{\perp}\subset M_{v}^{4}$ and metric in $Sim_{p}\subset M_{p}^{4}$ are like the Euclidean ones ($Sim_{p}$ is the equivalence class that contains event p). Curves in equivalence class $Sim_{p}$ must be spacelike and their arc-lengths 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 double-cone brings up something else I was wondering. Since light rays in vacuum are timelike lines, their speed (norm of the velocity) would be $\left\|w'(t)\right\|=1$. 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 $M_{p}^{4}$ induces by the Minkowski inner product (which is actually not an inner product since it is not possitive definite).

 

[bobc2]

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...

Very intriguing question, Wox. If we could classify the admissable curves, would we not essentially have the laws of physics rendered as statements of topology and geometry?

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 3-D 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 2-D sheet of paper, appears to approach and bounce off of the black curve--simulating 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 4-dimensional 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 3-dimensional structure driven by "laws of physics" and causality. Rather we would have a static 4-dimensional universe structure populated by admissible curves and geometric patterns.