Is Coordinate-Free Relativity the Key to Understanding General Relativity?

[Dale]

No. Dale used a coordinate system, and an origin. He is hiding information from me

I did not use a coordinate system nor an origin, neither explicitly nor implicitly.

I am not hiding anything. You are the one who claimed that the mere fact that a measurement was performed explicitly determines an origin. I thus provided you the information that you claimed was required.

If you wish to revise your claim, then I will be glad to provide as much detail as you claim is required.

Note, however, that there is more to a coordinate system than just an origin, so this is a much weaker claim than the claim that any measurement defines a unique coordinate system. However, since even this very weak claim is false I think it is instructive to pursue it.

 

[Dale]

However, even if I used an un-numbered ruler, I still had to count from one end to another of the screen. I used one end or the other as the origin. Or maybe I played a trick on you and counted both directions from the center then added. But that just makes the origin at the center.

So which is it? The simple fact is that any origin will do, and you will get the same measurement regardless of the origin. The distance measured does not depend in any way on the origin.

This is conceptually similar to Lorentz's aether. You assert that the coordinate system exists and is necessary even though it has no effect on any physical experiment and any choice is consistent with experimental results.

 

[JDoolin]

I am not hiding anything.

First of all, I did not mean to imply any sort of malice by saying you were hiding information. I neither asked nor expected you to provide this information. However, that does not change the fact that the information exists, (or existed, if you've forgotten it).

You are the one who claimed that the mere fact that a measurement was performed explicitly determines an origin. I thus provided you the information that you claimed was required.

If you wish to revise your claim, then I will be glad to provide as much detail as you claim is required.

You've made a strange proposal.

I say you have hidden information from me, but you say if I revise my claim, then you will provide detail to me? If my claim is incorrect, why don't you provide detail to me now, and show me that the claim is wrong?

But I don't really care about the hidden information, as long as a couple of assumptions hold.

(1) the space is not appreciably warped by gravitation where you're taking this measurement, and (2) the origin is stationary with respect to the thing you're measuring

The simple fact is that any origin will do, and you will get the same measurement regardless of the origin. The distance measured does not depend in any way on the origin.

That is only true in regards distances between points in Euclidean space. It is also true with events in Minkowski spacetime, using the space-time interval between events.

However, if you are measuring distance between objects and time between events in Minkowski spacetime, distances DO depend on the origin, because the origin has an intrinsic velocity.

 

[Dale]

You've made a strange proposal.

I say you have hidden information from me, but you say if I revise my claim, then you will provide detail to me? If my claim is incorrect, why don't you provide detail to me now, and show me that the claim is wrong?

The strangeness is inherent in your contradictory claims. First, you claim that the mere fact that a measurement is performed uniquely identifies an origin. Then second, when you have been given the information that a measurement was performed you claim that unspecified additional required information was withheld. The second claim contradicts the first.

However, I can describe in detail the measurement and then you can feel free to tell me what information in addition to the mere fact of the measurement is necessary to specify the origin.

A and B are two marks on a piece of paper lying on my desk. The marks are stationary wrt the paper but not located at any particularly special location or orientation wrt the paper, and the paper is resting on the top of the desk, but not particularly located in any special position or orientation wrt the desk. The desk is stationary wrt the house, etc. The acceleration due to gravity in my house can be taken to be approximately uniform at 9.8 m/s². The measuring device is an unmarked standard rod of 8.5" length composed of a piece of standard "letter paper" constructed according to the usual specifications for letter paper. I carefully placed the two appropriate corners of the rod on the marks and noted that the length matched. Thus, the distance from A to B was measured to be 8.5". The rod was not moving wrt A or B during the measurement.

(1) the space is not appreciably warped by gravitation where you're taking this measurement, and (2) the origin is stationary with respect to the thing you're measuring

1, gravitation is not an appreciable factor in my measurement
2, there is no origin so since it doesn't exist it is not stationary nor is it moving wrt the thing being measured

The simple fact is that any origin will do, and you will get the same measurement regardless of the origin. The distance measured does not depend in any way on the origin.

That is only true in regards distances between points in Euclidean space. It is also true with events in Minkowski spacetime, using the space-time interval between events.

Then you agree it is true with those caveats?

 

[JDoolin]

The strangeness is inherent in your contradictory claims. First, you claim that the mere fact that a measurement is performed uniquely identifies an origin. Then second, when you have been given the information that a measurement was performed you claim that unspecified additional required information was withheld. The second claim contradicts the first.

But there is a difference between performing a measurement and communicating the results of a measurement.

However, I can describe in detail the measurement and then you can feel free to tell me what information in addition to the mere fact of the measurement is necessary to specify the origin.

A and B are two marks on a piece of paper lying on my desk. The marks are stationary wrt the paper but not located at any particularly special location or orientation wrt the paper, and the paper is resting on the top of the desk, but not particularly located in any special position or orientation wrt the desk. The desk is stationary wrt the house, etc. The acceleration due to gravity in my house can be taken to be approximately uniform at 9.8 m/s². The measuring device is an unmarked standard rod of 8.5" length composed of a piece of standard "letter paper" constructed according to the usual specifications for letter paper. I carefully placed the two appropriate corners of the rod on the marks and noted that the length matched. Thus, the distance from A to B was measured to be 8.5". The rod was not moving wrt A or B during the measurement.

Very well done! Because the lengths matched, in fact, you did not have to decide which point you were measuring "from" and which point you were measuring "to". I admit this is one scenario that hadn't occurred to me.

But how would you modify this process if you needed to measure lengths of things that were not exactly 8.5" long?

Dalespam: "The simple fact is that any origin will do, and you will get the same measurement regardless of the origin. The distance measured does not depend in any way on the origin."

JDoolin: "That is only true in regards distances between points in Euclidean space. It is also true with events in Minkowski spacetime, using the space-time interval between events."

Dalespam: "Then you agree it is true with those caveats?"

Yes.

But if you get into general relativity; for instance, the Schwarzschild metric, even your choice of origin will affect measurement of distance, time, and space-time intervals.

 

[Dale]

Very well done! Because the lengths matched, in fact, you did not have to decide which point you were measuring "from" and which point you were measuring "to". I admit this is one scenario that hadn't occurred to me.

But how would you modify this process if you needed to measure lengths of things that were not exactly 8.5" long?

Get another standard rod that is as long as needed, or (more commonly) get a large number of very small standard rods and count how many are used.

Yes.

That is all we are saying. Those caveats are acceptable. The geometry we are interested in spacetime is the spacetime interval.

But if you get into general relativity; for instance, the Schwarzschild metric, even your choice of origin will affect measurement of distance, time, and space-time intervals.

Not really. The origin can be moved in time as desired without even changing the components of the metric tensor. And you can do a diffeomorphism to a coordinate system with any arbitrary origin. Such a transformation will cause the components of the metric to change, but all measurements of spacetime intervals will be unchanged. Since such quantities do not depend on the choice of coordinate system you can express them without reference to any coordinate system if you wish. That is the point of coordinate-free relativity.