relativity - Nature of spacetime 4-vector and tangent space?

An entry level confusion about spacetime. I understand that a 4-vector describes a point or event in spacetime. But I've also read (Bertschinger, 1999) that re spacetime "we are discussing tangent vectors that lie in the tangent space of the manifold at each point". If a point/event is described by a single 4-vector, what are all these tangent vectors that lie on the same point? Do they have different co-ordinates to the 'point/event' 4-vector? Could I also ask how a 4-vector 'contains' any sense of direction (I'm thinking here of a vector having direction and magnitude)?

Answer

Your confusion comes from the difference between special and general relativity. In special relativity, the space-time manifold is assumed to carry the structure of 4-dimensional Minkowski space, which has the nice property that it is canonically identified with its own tangent space at the origin (since it is a vector space). So in special relativity you can speak of a space-time event as a 4-vector, and you can also speak of global Lorentz transformations (by doing the local Lorentz transformation in the tangent space and propagating the transformation to the whole space-time using the canonical identification).

In general relativity, however, the space-time is allowed to be an arbitrary Lorentzian manifold (what we do is break the global Lorentz symmetry of Minkowski space and require it to only hold infinitesimally, i.e. on the tangent space), and you don't have a canonical way of identifying the entire space-time with the tangent space of a fixed point. Therefore you cannot speak of a space-time event (now just a point in your space-time manifold) as a 4-vector!

Edit Let me try to make the difference between an affine and non-affine space more apparent.

Let us start by considering a two dimensional manifold. In fact, we'll just compare the usual flat plane and the sphere.

On the plane, we can fix an arbitrary point and call it the origin. Starting from this origin, pick a direction, and call it the $x$ direction. Consider the following two paths.

1. First you go 1 unit in the $x$ direction. Turn left for 90 degrees. travel another unit.


2. First you face the $x$ direction. Now turn left for 90 degrees, travel one unit. Turn right now for 90 degrees and travel another unit.

These two operations take you to the same place.

On the sphere, however, you can again fix an arbitrary point and call it the "origin", and a direction which we call "$x$". Now consider

1. First you go a quarter the way around the sphere in the $x$ direction. Turn left for 90 degrees, travel another quarter of the circumference.


2. First you face $x$ direction, turn left for 90 degrees, travel a quarter of the circumference. Now turn right and travel a quarter of the circumference.

(Try to trace this out on a tennis ball or on an orange.) You don't end up at the same place!

Now, going back to the plane, if instead you start by going "face $x$. Turn left for 45 degrees. Travel for $\sqrt{2}$ units." You will again end up at the same place as the previous two tries.

But on the sphere, if you start by "face $x$. Turn left for 45 degrees. Travel for $\sqrt{2}$ times a quarter circumference." You will end up somewhere else entirely again compared to the previous two tries.

What does this mean? On a flat plane, from the above demonstration, after fixing an origin and a direction $x$, we can uniquely describe every point on the plane as "certain units in the $x$ direction, and then certain units to the left of that". And it doesn't matter at all in which orders you zigzag to the destination. This identifies the plane with a vector space. That is, you can say that "the displacement from point $p$ to point $q$ is the vector $\vec{v}$, and the displacement from point $q$ to point $r$ is the vector $\vec{w}$, so the displacement from the point $p$ to point $r$ is the sum of vectors $\vec{v} + \vec{w} = \vec{w} + \vec{v}$". So after fixing a point as the origin, we can describe all other points as a vector displacement relative to this origin, and add and subtract the vectors as appropriate.

On the surface of the sphere (a curved manifold), however, the above example shows that the intuition we learned from studying classical mechanics about using vector addition for the "displacement vector" cannot hold for the positions of points. So you shouldn't think about space-time events in general relativity (which is a point on some possibly curved manifold), as vectors relative to a fixed origin. (Because with vectors you would be tempted to add them and so forth.)