Congruence vs Family of Worldlines: Taking Notes

[kent davidge]

I'm taking some notes about what I'm studying, and I would like to know if I can substitute the word "congruence" by "family" of worldlines. Is there any difference? In the literature, it seems that the former is favoured over the latter.

 

[PeterDonis]

I would like to know if I can substitute the word "congruence" by "family" of worldlines. Is there any difference?

The term "congruence" has a specific standard meaning. The term "family" does not; it is sometimes used as a synonym for "congruence", but it is not a standard term.

 

[kent davidge]

If one goes by the Wikipedia definition https://en.wikipedia.org/wiki/Congruence_(general_relativity)
the conclusion would be that a congruence is always a family, but the converse is not true.

Correct?

 

[PeterDonis]

If one goes by the Wikipedia definition

Wikipedia is not using "family" as a technical term, just as an ordinary language term. "Family" has no technical definition that I am aware of. "Congruence" does.

 

[robphy]

If the details are important, maybe it's a good idea to DEFINE your terms,
even if it is to say " by X , I really mean Y ".

From your wikipedia link, (bolding mine)

In general relativity, a congruence (more properly, a congruence of curves) is the set of integral curves of a (nowhere vanishing) vector field
.
.
.

In general relativity, a timelike congruence in a four-dimensional Lorentzian manifold can be interpreted as a family of world lines of certain ideal observers in our spacetime.

In particular, a timelike geodesic congruence can be interpreted as a family of free-falling test particles.
Null congruences are also important, particularly null geodesic congruences, which can be interpreted as a family of freely propagating light rays.

So, "family of worldlines" without something akin to the bolded phrases allows other kinds of worldlines that would not comprise a congruence.

 

[pervect]

What's mathematically important is that the tangent vectors of the congruence create a non-vanishing vector field at every event in space and time. Or, as Wiki says, that the congruence is a set of integral curves of a nowhere vanishing vector field.

Does a "family" have this property? If so, it's fine. I would tend to think it does, but as others point out it's a bit vague.

I've not seen it explicitly spelled out, but I, at least, think of the vector-field associated with the congruence as representing the velocity of an "observer". In the standard formalism it's the 4-velocity, though, not the 3-velocity. The wordlines themselves I regard as being the worldlines of "observers". And we require that one unique worldline (observer) pass through every event in space-time.

[afterthoughts, added later]
If "integral curves" are not familiar, it's worth looking them up and reading about them. Basically, the related math says that if you define a non-vanishing vector field at every point, you also define a set of curves whose tangent vectors are the vector field. This is akin to the process of integration, where you specify the derivative of a function at every point, and compute the function via integration, which is uniqute up to a constant factor. But in this case we sepcify the tangent vector of a curve for instance $\partial t / \partial \tau, \partial x / \partial \tau, \partial y / \partial \tau, \partial z / \partial \tau$, to find the curve itself, $t(\tau), x(\tau), y(\tau), z(\tau)$.

 

[PeterDonis]

I, at least, think of the vector-field associated with the congruence as representing the velocity of an "observer".

More precisely, this is what a timelike congruence describes--the worldlines of observers. A null congruence describes the worldlines of light rays. Mathematically, one could also construct a spacelike congruence, but I have never seen such a construction used.

 

[pervect]

More precisely, this is what a timelike congruence describes--the worldlines of observers. A null congruence describes the worldlines of light rays. Mathematically, one could also construct a spacelike congruence, but I have never seen such a construction used.

Ah yes, I was indeed describing a time-like congruence, the only sort of congruence I usually use, rather than a more abstract case.

 

[PeterDonis]

in this case we sepcify the tangent vector of a curve for instance $\partial t / \partial \tau, \partial x / \partial \tau, \partial y / \partial \tau, \partial z / \partial \tau$, to find the curve itself, $t(\tau), x(\tau), y(\tau), z(\tau)$.

It's worth noting that, if all you have is a tangent vector at an event, that is not sufficient to specify a unique curve. You have to add some additional information.

For example, if you specify that the curve is a geodesic, then the tangent vector at an event does uniquely specify the curve.

Or, if you specify that the tangent vector field is a Killing vector field, then Killing's equation provides enough additional information to uniquely specify a curve given a tangent vector at an event.

Basically, these are two different ways of specifying the path curvature of the curve (geodesic = path curvature is zero, Killing = path curvature is a function of the norm of the Killing vector), which is enough additional information to uniquely specify it.

 

[PAllen]

I’ll propose some definitions that I use. A family of paths is simply a set of paths each identified by 3 parameters. If one further requires that every point in the manifold lies on exactly one path in the family, and that every point on every path has a tangent vector (this is a minimal smoothness requirement), then you have a congruence. The associated vector field is simply all of the tangent vectors. Personally, I find it more useful to reverse the standard definition in this way, in most situations, than to specify a vector field first.

 

[PeterDonis]

A family of paths is simply a set of paths each identified by 3 parameters.

Is there a standard technical definition of "family" in a textbook or paper? I've never seen one. The only technical term I've seen a standard definition for in this connection is "congruence".

 

[PeterDonis]

In the literature, it seems that the former is favoured over the latter.

Can you give any examples from the literature of the use of "family" as a technical term (as opposed to just an informal word meaning "some bunch of worldlines")?

 

[PAllen]

Is there a standard technical definition of "family" in a textbook or paper? I've never seen one. The only technical term I've seen a standard definition for in this connection is "congruence".

No, that's why I identified it as a personal definition. However, it is inspired by many areas of math that talk in terms of a "k parameter family of X". At least if I want to refer to a family of world lines, for example, I would simply give my definition. A definition cannot be wrong really, though it would be perverse if it contradicts a standard one. In this case, you and I are both unaware of a standard one.

 

[PeterDonis]

that's why I identified it as a personal definition

I understand, but the OP's question doesn't seem to be about personal definitions. It seems to be about standard technical definitions. The OP can clarify if I am mistaken.