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

[Dale]

But if you use the distance formula distance = \left |x_b-x_a \right |, then x_b and x_a must be defined from some other point (the origin). ...

Yes, you can arbitrarily designate your origin, but that does not mean you can make the measurement without choosing an origin at all.

Since you seem to be aware that the distance is completely independent of choice of origin then I don't see why you think it must be defined at all. I made the measurement without defining a coordinate system and choosing an origin.

That's correct. You could even choose some weird logarithmic scale if you wanted. But again, similarly, you can arbitrarily designate any unit length you wish, but that does not mean you can make the measurement without choosing a unit length at all.

I never said it did. But the fact is that making a measurement with a chosen unit of length does not constrain nor inform your choice of coordinates in any way.

You seem to be under the misapprehension that a coordinate system is the same as making a measurement that results in a number. That is incorrect, all measurements are invariant under arbitrary coordinate transforms. So the existence of a measurement does not require nor inform you as to any coordinate system. Nature does not come equipped with a required set of coordinates, regardless of what measurements you may make.

You should know the definition of a coordinate system by now. Can you unambiguously define a unique coordinate system from the fact that the distance between A and B is 6"?

A coordinate system need only have dimension high enough to measure whatever quantities you are interested in. If I ask you for the diagonal length of your computer screen, we only need a one dimensional coordinate system. If I ask you the dimensions of the computer screen, we need a two-dimensional coordinate system. If I ask you where your left eye is in relation to the computer screen, we need a three-dimensional coordinate system. If I allow for the fact that your eyes may be moving at a relative velocity wih the computer screen, and I ask the same question, we need a four-dimensional coordinate system.

You don't need a coordinate system for any of those measurements.

I cannot imagine any way to measure a distance without specifying a "from" point and defining some scale with which to mark off the distance.

I (almost*) agree, but that is irrelevant. The "from" point need not be the origin and the scale to mark off the distance need not correspond to any coordinates.

*The almost is that in order to measure the distance between A and B you do not need to identify one as "from" and the other as "to". The measurement of distance is invariant under that choice.

 

[JDoolin]

You can describe the distance between two points without defining a coordinate system.

But in order to measure the distance between two points, you must define an origin and unit length, which is the same as defining a coordinate system.

 

[JDoolin]

That is incorrect, all measurements are invariant under arbitrary coordinate transforms.

The angles between time-like paths are not invariant under velocity transformation.

I went ahead and made an animation so that you can see what I mean.

The angle between the two marked paths is 90 degrees, but from the perspective of the dot in the middle, the angle between the particle paths is 180 degrees.

To claim that "all measurements are invariant under arbitrary coordinate transforms" is one of those true, but misleading statements. The measurement of the angle depends on the velocity of the protractor that measures the angle. Sure, no matter what reference frame you're in that protractor will measure the same angle, but some observers will note that the protractor is distorted, and the measurement is actually incorrect. Just because the measurement is invariant, but the actual observation is very different.


And I want to reiterate what I said in my previous post:

While it is possible to describe things without defining an origin, it is impossible to measure things without explicitly defining an origin and unit length, and it is impossible to visualize anything without implicitly defining an origin.

Even my ability to describe things without defining a coordinate system: ( "My monitor is 20 bloots across" ) actually conveys no useful information, until I define what a bloot is.

 

[Dale]

To claim that "all measurements are invariant under arbitrary coordinate transforms" is one of those true, but misleading statements. The measurement of the angle depends on the velocity of the protractor that measures the angle.

In what way is that at all misleading? I said exactly what I meant (and what I said was true), and you understood exactly what I meant (and acknowledged its truth).

And I want to reiterate what I said in my previous post:

While it is possible to describe things without defining an origin, it is impossible to measure things without explicitly defining an origin and unit length, and it is impossible to visualize anything without implicitly defining an origin.

OK, I have measured the distance between A and B to be 8.5". Where is the origin?

 

[Ben Niehoff]

But in order to measure the distance between two points, you must define an origin and unit length, which is the same as defining a coordinate system.

No it isn't. A coordinate system is "A continuous map from some open subset U of the manifold M into R^n, where n is the dimension of M". Defining "an origin and a unit length" does not give you a continuous map from U to R^n.

 

[JDoolin]

In what way is that at all misleading? I said exactly what I meant (and what I said was true), and you understood exactly what I meant (and acknowledged its truth).

Measurements, as a rule, are observer dependent, but any given measurement is observer independent, because the reference frame of the observation device is already determined.

To say "all measurements are invariant under arbitrary coordinate transforms" is misleading because it does not specify whether you mean "measurements in general" or "any given measurement." It strongly suggests you mean "measurements in general" which would make the statement false.

I only acknowledged the statement's truth based on one possible interpretation.

OK, I have measured the distance between A and B to be 8.5". Where is the origin?

I don't know. All I can guarantee is that when you made the measurement, you referenced your measurement from some origin, and in order to visualize that distance, I must reference it from some origin.

 

[JDoolin]

You can describe the distance between two points without defining a coordinate system.

But in order to measure the distance between two points, you must define an origin and unit length, which is the same as defining a coordinate system.

No it isn't. A coordinate system is "A continuous map from some open subset U of the manifold M into R^n, where n is the dimension of M". Defining "an origin and a unit length" does not give you a continuous map from U to R^n.

But if you are giving a distance, you already have a continuous map between the two points.

Also, the unit length cannot be described at a single point. In a unit displacement vector, there is a continuous mapping of some one-dimensional space from 0 to 1.

 

[Ben Niehoff]

Measurements, as a rule, are observer dependent, but any given measurement is observer independent, because the reference frame of the observation device is already determined.

You seem to be using a strange definition of "measurement" that requires observers to be intentionally naive. Why in the world would anyone try to measure the angle between AB and AC with a moving protractor?

All geometric quantities are invariant under coordinate transformations. In Euclidean space, geometric quantities include angles and distances. An angle is always measured between two lines at the point they intersect. A distance is always measured between two points along the line that connects them. In Minkowski space, geometric quantities include angles, distances, and relative velocities. Relative velocity is really just the "angle" between two worldlines.

I've used the term "relative velocity", but you should note that ALL geometric quantities are already "relative". An angle is always an angle between two lines. One cannot say "The angle of line AB is 30 degrees", that makes no sense. Likewise, a distance is always a distance between two points.

I don't know. All I can guarantee is that when you made the measurement, you referenced your measurement from some origin, and in order to visualize that distance, I must reference it from some origin.

You realize that your inability to answer this question unambiguously proves that Dale is in fact not using a coordinate system?

But if you are giving a distance, you already have a continuous map between the two points.

You'll have to explain. A continuous map from what space into what space?

In a unit displacement vector, there is a continuous mapping of some one-dimensional space from 0 to 1.

This statement makes no sense. You don't seem to be using the words "vector" and "mapping" correctly. Furthermore, I have not once made any mention of vectors, so it's irrelevant anyway.

 

[JDoolin]

You seem to be using a strange definition of "measurement" that requires observers to be intentionally naive. Why in the world would anyone try to measure the angle between AB and AC with a moving protractor?

Are you saying that measuring an angle with a moving protractor is not valid?

In any case, as an observer performing an experiment, you may not have the option of measuring the system in a comoving frame. For instance, if you wish to measure the temperature of air passing by at 100,000 miles per hour, you can't simply place your thermometer in and hope to get the result. In all likelihood, your thermometer will disintegrate.

If you want to measure the shape of a body passing through our solar system at 90% of the speed of light, you don't have the option to run and catch up and place the protractor on the surface.

It's not a matter of naivete. It's a matter of what is convenient and possible.

All geometric quantities are invariant under coordinate transformations. In Euclidean space, geometric quantities include angles and distances. An angle is always measured between two lines at the point they intersect. A distance is always measured between two points along the line that connects them. In Minkowski space, geometric quantities include angles, distances, and relative velocities. Relative velocity is really just the "angle" between two worldlines.

Sure.

I've used the term "relative velocity", but you should note that ALL geometric quantities are already "relative". An angle is always an angle between two lines. One cannot say "The angle of line AB is 30 degrees", that makes no sense. Likewise, a distance is always a distance between two points.

Right, but you still need one vector, and a continuity of positions in between the first vector and the second vector.

You realize that your inability to answer this question unambiguously proves that Dale is in fact not using a coordinate system?

No. Dale used a coordinate system, and an origin. He is hiding information from me, and he is under no obligation to tell me that information, but that does not mean that the information does not exist.

I just measured my own computer screen was 13 and 1/8 inches across. You don't know whether I measured from the left to the right, or whether I was using a yard-stick or a ruler. But you do know something about how a length is measured, and you know that I must have placed an object near the screen, most likely that has a zero-point on it.

And I placed that zero-point somewhere in order to measure the screen.

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.

or hey, maybe I played a really crazy trick on you, counting off random little 1/16 inch segments until they were all marked. So now I've converted this vector quantity into a scalar quantity. Have I now succeeded in describing a distance without having an origin?

I don't think so. Because a distance is not made up of discontinuous chunks of ruler. It's made of consecutive chunks of ruler and the continuous space in-between the atoms.

You'll have to explain. A continuous map from what space into what space?

This statement makes no sense. You don't seem to be using the words "vector" and "mapping" correctly. Furthermore, I have not once made any mention of vectors, so it's irrelevant anyway.

Right, you said unit length, I said unit vector. I brought up the term vector to distinguish between "displacement" and other types of vectors, such as force, velocity, acceleration. These quantities are also vectors, but can exist at a single point.

Either way, a unit length, or a unit displacement vector requires space in one dimension to define.

 

[Dale]

I don't know.

Why don't you know? If the mere act of making a measurement explicitly defines a unique origin (as you have claimed) then you should know.

 

[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.