General Relativity- the Sun revolves around the Earth?

[Dale]

By this I am assuming you mean an Earth-centered, Earth-fixed system in which the Sun appears to orbit the Earth once per day and traces an analemma over the course of a year.

Yes, either that or an Earth-centered, non-rotating reference frame where the stars are fixed, or any other absurd coordinate system you might choose. The point is that GR works fine regardless of your coordinates, which is what the OP was asking.

No, it doesn't make it invalid. It just makes it a stupid choice.

Exactly.

 

[Dale]

The bottom line: size matters.
If a large mass and a small mass push off against each other then the small mass undergoes a proportionally larger change of velocity. This is not relative.

You have to be careful here. There are two separate concepts which embody the idea of a "change in velocity". One is called "proper acceleration" and is a coordinate independent concept (the covariant derivative of the tangent vector), it is the acceleration measured by an accelerometer. The other is called "coordinate acceleration" and is a coordinate dependent concept (the second time derivative of the position). Those two measures of acceleration need not be equal and, in fact, they are unequal in the presence of gravity.

In your scenario the small mass will have a greater proper acceleration, this is the acceleration which is not relative and it will be greater regardless of the coordinate system used. But it may have a smaller coordinate acceleration or even no coordinate acceleration, depending on the coordinate system chosen. There is no requirement that the coordinate system be such that a given object is at rest.

 

[Cleonis]

[...] if we cannot measure our direction, or speed, relative to space, how can we possibly determine who traveled the more "spatial distance?"

All motion, distance, direction, and speed is measured from something else. Therefore it is impossible to say who covered more "spatial distance."

I have selected these statements to quote, because I think they capture the core of your questions.

Note that your questions are very distant from the origin of this thread, which is about GR. Your questions are about first introduction to SR.

I will discuss SR only in this message. I intend it to be my final message in this thread. In retrospect I realize I needed to be reminded why it's not a good idea to discuss relativistic physics on internet.I have uploaded three images to physicsforums. Three spacetime diagrams representing the twin scenario. The three diagrams are for three respective coordinate systems.

1. co-moving with the stationary twin
2. co-moving with the away journey of the traveling twin
3. co-moving with the return journey of the traveling twin.

Of course the scenario can be diagrammed in any member of the equivalence class of inertial coordinate systems.

For special relativity the idea is to identify the things that are common to all diagrams. A lot of things, such as coordinate velocity, are frame dependent: on transformation they transform to another value.
But crucially some things are common to all diagrams, these aspects are thought of as inherent in the phenomena.

What is common to all diagrams is that the traveler covers more spatial distance than the stay-at-home twin. (The precise value in coordinate distance will be different from diagram to diagram, but it's always more for the traveler.)
You can map the twin scenario in any member of the equivalence class of inertial coordinate systems. When you evaluate how much difference in amount of elapsed proper time there will be from parting to rejoining everyone of those mappings will yield the same answer.
There is no individual assessment of distance traveled, you can only say something in comparison.

[...] if we cannot measure our direction, or speed, relative to space, how can we possibly determine who traveled the more "spatial distance?"

Specifically to your question:
Before special relativity the assumption was that it is possible to assign an absolute velocity vector to objects, a velocity with respect to the luminiferous ether. Obviously it was also assumed that the luminiferous ether is uniform, since any erratic thing cannot be a background reference.

Special relativity asserts that there is no such thing as assigning a velocity vector of motion with respect to space: the principle of relativity of inertial motion. However, special relativity does have the underlying assumption that space is uniform. Or, saying the same thing with other words, special relativity depends on the underlying assumption that when an object is in inertial motion it covers equal distances in equal intervals of time.

You have to separate those two concepts:
- You cannot assign a velocity vector representing motion with respect to space.
- Space is uniform: in inertial motion you cover equal distances in equal intervals of time.

(Of course, since SR works with spacetime rather than with space and time separately it's better to say that SR has as underlying assumption that spacetime is uniform.)

Without the underlying assumption of the uniformity of spacetime it would be impossible to formulate the invariance of the spacetime interval. Given the assumption that spacetime is uniform it is possible to make statements about the twins traveling different spatial distance from parting to rejoining. When the twins rejoin they may find that for one of them a smaller amount of proper time has elapsed. According to SR the twin with the least amount of elapsed proper time has traveled more spatial distance.About acceleration:

I often notice the differential aging of the twins being attributed to the acceleration. While the acceleration is necessary, thinking of it as the cause of the differential aging doesn't hold up: it leads to self-contradiction.

Some time ago I came across the following diagram that was uploaded in 2008:

https://www.physicsforums.com/attachment.php?attachmentid=14191&d=1212060478
This is from the thread https://www.physicsforums.com/showpost.php?p=1747855&postcount=4"

It's a triplet this time. C stays at home, A and B go on a journey. In the worldlines the red sections represent a phase of acceleration. A and B both experience the same acceleration for the same time, but A's total elapsed time is shorter than B's.

.

 

[Dale]

What is common to all diagrams is that the traveler covers more spatial distance than the stay-at-home twin.

Look at your diagrams. This is only true in the first diagram. In the 2nd and 3rd the spatial distance traveled by the two twins is equal.

 

[Cleonis]

In the 2nd and 3rd the spatial distance traveled by the two twins is equal.

Yeah.

I was completely focused on the aspect that in all diagrams the stay-at-home worldline is a continuous straight line, whereas the traveler's worldline always consists of multiple sections that are at an angle to each other.

Ah well.

 

[Dale]

I was completely focused on the aspect that in all diagrams the stay-at-home worldline is a continuous straight line, whereas the traveler's worldline always consists of multiple sections that are at an angle to each other.

That is indeed a correct observation. One twin's worldline forms two sides of a triangle and the other twin's worldline is a single side of the triangle. This observation analogous to the triangle inequality.

In Euclidean geometry the sum of the lengths of two sides of a triangle is always longer than the length of the third side. In Minkowski geometry the sum of the time of two sides of a timelike triangle is always shorter than the time of the third side.