A coordinate-free understanding of the space-time manifold

I study dynamics and continuum mechanics. Over the years I've gotten used to the coordinate-free, or geometric, way of thinking. A velocity vector, for example, is a tensor. It is the same object when you describe it from any coordinate system/frame/set of basis vectors, despite the fact that you have many non-unique triplets, or 3 by 1 matrices, of numbers to describe it. You must know the basis in addition to the 3 numbers to make sense of it.

A particle traveling on a sphere occupies points of a manifold, $$S^2$$. This manifold does not have coordinates on it until you need to introduce them through an atlas of charts to describe where real points are. But the manifold itself exists and comprises all the possible points that the particle might occupy.

I am trying to use the same coordinate-free perspective to understand the space-time manifold, but it seems that I can't get away from it, and descriptions of relativity always seem to necessarily introduce coordinates to describe "two observers." A spatial manifold is a collection of places. A time manifold is a collection of instances. The endpoints of a path in a space manifold yields a distance. The endpoints of a path in a time manifold yields an interval. In the classical point of view, distances and intervals are scalar invariants, no matter your frame, no matter your time zone.

I am starting simple for the space-time manifold. I see that in the simple example of 1-D space, the space-time manifold is a flat two-dimensional cone. An element of the space-time manifold is called an event. A path in the space-time manifold is called a world-line. My problem is that it seems that you cannot think of an event without introducing coordinates. It is a time and a place, but those things are now necessarily relative to an observer, which necessarily introduces a coordinate system. In contrast, I can think of a place in a spatial manifold without introducing a coordinate system. A tree or a house occupies a certain place on the manifold.

My question is: Is it even possible to think of the space-time manifold in a coordinate-free way, much like we can for tensors as operators or packages of physical information, and spatial manifolds as collections of places?

• "Is it even possible to think of the space-time manifold in a coordinate-free way" Of course, and it works the same way for manifolds with a pseudo-Riemannian metric as it does for other manifolds. Sure, an observer assigns coordinates to events, but events are points on the spacetime manifold independent of any choice of coordinate system.
– d_b
Jul 10 '21 at 18:59

Just like you can say "x is the place where this tree is" or "y is the place where my house is" you could also say that "event z is when supernova A exploded" or something. Hence you have your "coordinate free way" of representing spacetime, to stay with your example.

Practical physics is always done in specific coordinates. Coordinate-free representations are used to economically formulate recipes to obtain such coordinate representations. It's a lucky property of our universe that this is so successful.

That being said, you seem to mix up the light cone with flat spacetime. Flat spacetime can be represented as $$R^4$$ equipped with an constant indefinite metric tensor of signature $$(1,-1,-1,-1)$$ (or reverse). Just as you can write equations in coordinate free representation in general relativity, you can do so in special relativity. If you choose curvilinear coordinates in special relativity, the equation of a straight line (inertial movement) becomes the geodesic equation $$\frac{d^2 x}{d\tau^2}=\Gamma\left(\frac{dx}{d\tau},\frac{dx}{d\tau}\right)$$

The only difference is that the manifold is flat, that is the Einstein tensor computed from the metric (or the Christoffel symbol $$\Gamma$$) is equal to zero.

• Thanks. It leads me to a follow up question. Are events necessarily.... events? For example, if I pick an empty place in deep, dark, space at a specific time, is that considered an event? Or does a supernova need to be exploding in that space to qualify as an event, for example? Something needs to be radiating light? If I pick an arbitrary point of the space-time manifold, what does that represent? I would like to understand space-time as fixed, with different observers simply being coordinate transformations, but I often see people deforming and warping the space itself to explain things.
– Evan
Jul 10 '21 at 19:15
• As far as I know, an event is just a point in spacetime. So, just as you can have empty space without a house or a tree, you can have an event where/when no supernova explodes. But of course, completely empty space makes not sense, physically. You, as the experimenter, would always have to relate to something tangible. For example, 10 years after supernova A exploded, 1 million lightyears in the direction of the center of the milkyway, there may not be much happening in empty space. Jul 10 '21 at 19:21
• If you pick a point on a manifold, you have to specify how you pick it, and that goes over coordinates, i.e. putting metersticks next to each other, and counting the ticks of a clock. Jul 10 '21 at 19:23
• The book Gravitation of Misner, Thorne and Wheeler has a great deal of coordinate-free stuff in it and they carefully distinghuish it from descriptions with coordinates. At the latest when they give a much simpler recipe to calculate curvature from the metric by just using forms one should be fully convinced of the coordinate free fomalism. Jul 10 '21 at 19:52