B General Relativity and the curvature of space: more space or less than flat?

[Dale]

move to that new direction as starting point and draw another light cone

The new light cone will still be at a 45 degree angle, just like the first one and just like the following one.

 

[Martian2020]

The new light cone will still be at a 45 degree angle, just like the first one and just like the following one.

Just so you see my addition: is it correct?
Added:
I think I've found a solution:
1. I imagine a plane.
2. Then scatter some events on it randomly. Fine as of now.
3. Add causal relationships. Then each two events of causal relationship can represent direction of time and space is at right angle. And with that coordinates check if other causal relationships are within light cones from cause to the result. Check for all pairs of events. If violated, then my set of events represent non-causal universe. If not then this is example of SR causal universe.

 

[Dale]

is it correct?

I am not sure. I don’t really understand what you are saying.

In any inertial frame if $ds^2=-dt^2+dx^2>0$ then the two events are not causally related. Otherwise the one that occurs earlier could in principle cause the one that happens later.

 

[Nugatory]

Then move to that new direction as starting point and draw another light cone and choose 3rd "observer" moving within the cone to the right. That way I can make full circle. And my 3rd observer would move in relation to 1st one outside of light cone for 1st one. That is as I know should not be in SR. I got puzzled how can I have single spacetime...

Draw the worldline of an object (Object A) that is at rest in whatever frame you’re using; it will be a vertical line going straight up the page. Next draw the worldline of an object (Object B) moving at .9c to the right relative to A; this will be a line that slants up and to the right, rising ten units for every nine units sideways. It’s easiest if you have the two worldlines intersect at the origin.

Pick a point, any point, on B’s worldline. Say that B emits a flash of light off to the left and off to the right at that point. Draw the path of those two flashes; they will of course form a lightcone because that’s the definition of a lightcone. Do remember that the slant of B’s worldline is irrelevant - one flash is moving to left at speed c and the other is moving to the right at speed c, no matter what B is doing. That is, all lightcones are always at a 45 degree angle.

Next draw the worldline of a third object, C, which is moving to the right at speed .9c relative to B and passes B at the moment the light flashes. But we’re still working in the frame in which A is at rest, so the slope of C’s worldline will be determined by C’s speed relative to A - you will have to use the relativistic velocity addition formula and you’ll find that the line is still inside the lightcone, a bit less than 45 degrees from the vertical.

Finally, start over with a fresh sheet of graph paper and draw the same exact situation, except using the frame in which B is at rest. A’s worldline now slants off and to the left, B’s worldline is straight up, and C’s worldline slants off to the right. You’ve just drawn the same spacetime in two different ways, with the angles on the paper distorted in different ways when you make a different choice of which set of x,t axes intersect at a 90 degree angle.

 

[PeroK]

I have one "observer" with direction toward future, then movement of second observer at angle of slightly less than light cone to the right.

If not then this is example of SR causal universe.

There is only one flat universe. In the same way that there is only one 3D Euclidean space. SR changed the way we understand the nature of spacetime, but there is still only one 4D flat spacetime.

Although much of SR is presented using "observers", the theory is really one of flat spacetime and inertial reference frames (related by Lorentz Transformations). Especially when studying the geometry of spacetime you do not need observers.

The "observer" in SR is often over-emphasised - to the point where some students wrongly believe it's all about the light signals received by each and every observer.

If we imagine an IRF, then we are doing something quite abstract - we are imagining that we can assign coordinates to every event in spacetime. This is much less physical than imagining what a single observer "sees". It's much better to think of an IFR as a grid (1D, 2D or 3D) of equally spaced observers, all at rest with respect to each other and all with synchronised clocks. All events are measured locally, by the nearest observer and the overall picture can only be pieced together by collating all those local observations/measurements.

And, as I mentioned previously, when you start studying GR, you must make a further abstraction to a coordinate-independent or reference-frame independent approach to curved spacetime. In particular, an observer does not have a global view of spacetime (only a local view, where SR applies).

 

[Martian2020]

Do remember that the slant of B’s worldline is irrelevant - one flash is moving to left at speed c and the other is moving to the right at speed c, no matter what B is doing. That is, all lightcones are always at a 45 degree angle.

Thanx, but I do know how to draw Minkowski diagram. I used the name to imply I have 1d of space, flat sheet of paper representing flat spacetime. I want to draw events on spacetime w/out initial IRF to start with, as @PeroK said:
"One conceptual hurdle that you need not cross when you study SR is how to think about spacetime without invoking a suitable global IRF. You can learn SR and use the comfort blanket of always thinking about things in terms of IRF's. In GR you have no such luxury and you must confront the coordinate-independence of spacetime."
As I understood the quotation above, there is no global IRF to always draw cones at 45 angle with. But straight lines on 2d sheet represent list of event related by causality - object is at some point because it was in adjustant point just a moment ago in time and is moving inertially. If we draw light cones at 45% from some path, due to symmetry we should do the same for other line on spacetime. Otherwise there is one global time direction on spacetime. Maybe so, but then how to find it? I'm not certain what to do next.

 

[Nugatory]

I want to draw events on spacetime w/out initial IRF to start with

When you draw a Minkowski diagram, you’re plotting the $x,t$ coordinates of various events on a piece of graph paper, so you need some frame to assign these coordinate values. You can choose some non-inertial frame (that is, one in which an object whose position coordinate is constant is not moving inertially) to assign these values and then you’ll get a diagram drawn using that non-inertial frame. One example would be the Rindler diagram you get by using the Rindler X and T coordinates for your axes, and labeling events by their Rindler coordinates. (Rindler coordinates are one in which we consider an accelerating object to be at rest and the rest of the universe to be accelerating away from it in the opposite direction).

But straight lines on 2d sheet represent list of event related by causality

Only if you’ve chosen coordinates such that when you plot the path of a flash of light, you get straight lines moving up and out at a 45 degree angle. The causality relationships at an event are determined by the paths of hypothetical light signals passing through that point; whether these paths follow straight lines is completely a matter of the coordinates you’re using.

 

[Martian2020]

When you draw a Minkowski diagram, you’re plotting the $x,t$ coordinates ...
Only if you’ve chosen coordinates ...

I'm not sure why @PeroK liked this post. It is correct, as far as I can see for Minkowski. But in my post before I tried to expain I wanted to do what he said:

you must confront the coordinate-independence of spacetime.

I want to confront spacetime w/out coordinates, as @PeroK challenged me to do. Please help me to, if you can, explain your point w/out coordinates.
Coordinates are necessary so say point (1,3) etc. So say go from point A to point/event B we don't need coordinates, as far as I understand.

 

[Nugatory]

want to confront spacetime w/out coordinates, as @PeroK challenged me to do. Please help me to, if you can, explain your point w/out coordinates.

You can confront spacetime without coordinates, but you can't draw spacetime diagrams without them. When you draw a diagram you are putting marks on a piece of paper, and the coordinates tell you where the marks go.

Confronting spacetime without coordinates doesn't mean you never use coordinates, it means that you have to learn to distinguish the things that are true no matter what cordinates we choose ("invariants" in the lingo) like the fact that causal relationships are determined by the path that a hypothetical flash of light would follow through spacetime ("lightlike geodesics" in the lingo) from the things that are coordinate dependent like the 45-degree straight line lightcones that appear in Minkowski diagrams.

 

[Martian2020]

You can confront spacetime without coordinates, but you can't draw spacetime diagrams without them. When you draw a diagram you are putting marks on a piece of paper, and the coordinates tell you where the marks go.

Confronting spacetime without coordinates doesn't mean you never use coordinates, it means that you have to learn to distinguish the things that are true no matter what cordinates we choose ("invariants" in the lingo) like the fact that causal relationships are determined by the path that a hypothetical flash of light would follow through spacetime ("lightlike geodesics" in the lingo) from the things that are coordinate dependent like the 45-degree straight line lightcones that appear in Minkowski diagrams.

Thank you. I think I'm understanding better now. Could you please clarify: invariants are e.g. causal relationships, that is clear. Are paths that a hypothetical flash of light would follow through spacetime invariants (I say paths, because light can go many directions, correct?)?

 

[jbriggs444]

Thank you. I think I'm understanding better now. Could you please clarify: invariants are e.g. causal relationships, that is clear. Are paths that a hypothetical flash of light would follow through spacetime invariants (I say paths, because light can go many directions, correct?)?

I would say that the path that a flash of light follows is invariant, yes. The coordinates used to describe the set of events on that path will vary from one coordinate system to the next. But the set of events on the path is the same.

However, the angle that a particular light pulse takes from its launching point or the angle at which it arrives at its detection point can vary depending on one's choice of reference frame (e.g. stellar abberation). The path is still an invariant. The angle that it takes when projected onto a spacelike snapshot is not.

 

[PeroK]

Thank you. I think I'm understanding better now. Could you please clarify: invariants are e.g. causal relationships, that is clear. Are paths that a hypothetical flash of light would follow through spacetime invariants (I say paths, because light can go many directions, correct?)?

A path through spacetime is not really what we mean by invariant. That's a defined set of points in the spacetime manifold. It's not easy to describe that path until you have chosen a coordinate system, but (and this is the key point), the path exists and is well-defined without being given a coordinate description.

Generally there are two types of path: timelike (followed by massive particles) and null (followed by light). And, there are general timelike and null paths and geodesic timelike and null paths, which are the natural paths that particles and light follow through spacetime. Massive particles can, of course, be forced off geodesic paths, but the path remains timelike. I'm not sure there's any way to force a light onto a null non-geodesic path(?)

There are clearly an infinitude of possible paths, but each particle or light ray can only take one path through spacetime (its worldline).

An invariant is something you calculate, like the length of a spacetime path between two events. Null paths have zero length in all coordinate systems and timelike paths have the same non-zero length in all coordinate systems. So, it's the length of the spacetime path that is invariant.

We don't really talk about the path itself being invariant.

 

[PeterDonis]

A path through spacetime is not really what we mean by invariant.

It can be. You say:

the path exists and is well-defined without being given a coordinate description

That's what "invariant" means, so yes, a path through spacetime would be an invariant.

What it would not be is what I would call a "local" invariant, i.e., an invariant defined at a single spacetime point. As you say, it's a set of spacetime points. But that set of points is the same no matter what coordinates you choose.

I'm not sure there's any way to force a light onto a null non-geodesic path(?)

There is: a waveguide or fiber optic cable are examples of things that can do this.

it's the length of the spacetime path that is invariant.

We don't really talk about the path itself being invariant.

It's true that the term "invariant" is more likely to be used to describe the arc length along the path than the path itself. However, I don't think that means it's wrong to describe the path itself as invariant; it's just a less common use of the term.