Coordinates in GR: An Introduction & Question

[someGorilla]

Hello everyone!

I'm new on the forum (been browsing threads for some time though) and this post is both an introduction of myself and a first question.
I have a huge interest for physics but my working knowledge (having studied it in school, oh well some 15 years ago) is limited to classical physics, let's say up to the turn of the twentieth century, electromagnetism included. I've been reading here and there about more modern physics for some years, in a casual way, just to realize eventually that "popular" science books, including those written by serious researchers, articles on the Italian edition of Scientific American, or random readings on the web, only served to build misconceptions since they are at least unexact, and at most downright wrong. And since these sources systematically avoid working out the maths, my knowledge remained quite superficial. So I decided to start to actually learn it the hard way. With no schedules of course, it's just to satisfy my personal curiosity.

On to the question. The way I understand it, the four coordinates in GR don't necessarily correspond to one time coordinate and three space coordinates. I tend to think of them more as four independent parameters, defining a space on which the theory is set. They don't correspond to any physical notion of distance or time. Not always, at least, and surely not globally in a curved spacetime (unless distance and time are redefined accodingly). This is clear and not my question.
My question is more about the sociology of science perhaps: Why do we read so often statements formulated as if those "abstract" coordinates actually referred to something directly measurable? An example is the radius of the observable universe, or in general the descriptions of the expanding universe. If I think of how much gibberish I've read... like "the galaxies are not actually getting farther from each other, rather the space between them is expanding". It may well be shrinking, if you choose another coordinate system. The point is that the "space" in the "spacetime" of GR is not what you normally mean by "space". Or take the Schwarzschild radius of a black hole. Fair, it may be a useful quantity to compute, but it's not the radius of any sphere embedded in space: a radius measured along a timelike coordinate??
Am I right in finding that this fundamental aspect of GR is rarely emphasized? I started to study GR thinking of it as a theory of curved spacetime (right now with this: http://www.lightandmatter.com/genrel/ ) but I'm developping a somewhat different idea, a theory formulated on an abstract coordinate space which might locally correspond to space and time.

Ok this post is long enough, I'll wait for your comments!
And kudos for the wonderful site.

 

[PAllen]

While much of what you say is true, and even in SR some use if made of coordinates that are all lightlike (i.e. no coordinate is spatial or temporal), most commonly you do try to have coordinates that mean something physical, esp. for some class of special observers. For example, the r coordinate in Schwarzschild coordinates does have a clear physical meaning - the circumference around the central mass divided by 2π. This can be readily measured outside the event horizon, even if you can't readily measure distance to the center of a star, let alone a singularity. The time coordinate also has an operational definition.

Similarly, Kruskal coordinates are those that seek to have one time like, and 3 spatial coordinates, that preserve their character everywhere (across the horizon and up to - but not reaching - the singularity - which is normally considered not part of the manifold). The further defining feature of these coordinates is the light like geodesics are all straight lines so that light cones look exactly as they do in Minkowski coordinates in flat spacetime.

Careful authors should be spelling out the significance of the coordinates, if it matters (for many discussions, it doesn't matter).

An additional point is that the expression of the metric in chosen coordinates tells you directly the physical nature of those coordinates in some region (the physical nature of coordinates can vary from one spacetime locale to another).

 

[Naty1]

ok Gorilla, back at you with some long 'gibberish'!
Welcome to Physicsforums...

In GR, the 'best observations' are the ones you make locally...right where you are. Sounds like you have picked that up already. So carrying your own clock ticks proper time, no ambiguities there, and your ruler is also quite reliable right there. But even so inertial versus accelerated motion can be confusing. And what you observe locally will not in general be what another distant observer records at your location.

just to realize eventually that "popular" science books, including those written by serious researchers,.. only served to build misconceptions since they are at least unexact

Whoaa...Misconceptions are 99% in the mind of the reader rather than the authors...But ,sure, there must be some simplifications in introductory material. I'd urge to you to read a few of interest to you... here are several outstanding 'popular' books I know 'personally':

Warped Passages, Lisa Randall,
Black Holes and Time warps, Kip Thorne,
The Black Hole War, Leonard Susskind,
The Fabric of the Cosmos.

If you are mathematically inclined,
The Road to Reality, Roger Penrose,
Quantum Mechanics, Albert Messiah.I believe I read most or all of the first group summers before discovering Physicsforums... I felt right at home here, was able to understand many discussions, and was able to get answers to questions those books raised.

The way I understand it, the four coordinates in GR don't necessarily correspond to one time coordinate and three space coordinates. I tend to think of them more as four independent parameters, defining a space on which the theory is set. They don't correspond to any physical notion of distance or time. Not always, at least, and surely not globally in a curved spacetime (unless distance and time are redefined accodingly).

Space and time, or spacetime, are clear in relativity but perhaps different than everyday conceptions; They morph into each other at high relative speeds, and distance can be 'ambiguous' in GR as you point out but adopting conventions, like the FLRW metric and model for cosmology enables discussions based on a set of mutually agreed upon definitions, a common framework.

Why do we read so often statements formulated as if those "abstract" coordinates actually referred to something directly measurable?

because those abstract coordinates enable exact solutions to very difficult equations. You can make approximations, like perturbation theory, and reach approximate solutions, or in some cases use 'abstract' coordinates which offer their own insights.

If I think of how much gibberish I've read... like "the galaxies are not actually getting farther from each other, rather the space between them is expanding". It may well be shrinking, if you choose another coordinate system. The point is that the "space" in the "spacetime" of GR is not what you normally mean by "space". Or take the Schwarzschild radius of a black hole. Fair, it may be a useful quantity to compute, but it's not the radius of any sphere embedded in space: a radius measured along a timelike coordinate??

It's not gibberish, but subtle distinctions. For better or worse it also exists in the other major theory of our time, quantum mechanics...try explaing what THAT means...SHUT UP AND CALCULATE is one mantra meaning who cares how the mathematical details may be interpreted by different people, we can agree on the mathematics. Try explaining why you like your boyfirend or girlfriend...what makes them so SPECIAL??

Am I right in finding that this fundamental aspect of GR is rarely emphasized?

I'd readily agree that descriptions do not often take into account they are unique perspectives based on a particular set of coordinates...and of a particular set of mathematics. What we all learn in GR is that there are a variety of 'realities': Different observers make different observations. An example which I like is the Unruh effect in which an accelerating observer apparently measures a different local temperature than a nearby inertial observer. And perhaps as weird if not more so the ADS/CFT correspondence where a number of dimensions without gravity seems to be the same as another environment with an additional dimension plus gravity!

So it's maybe analogous to showing up for a 'big date' with a girl whose looks you think you know...BAM! all of a sudden her hair is styled different than you have ever seen, the color may be different, she has a Mike Tyson type tatoo on her face! She is as tall as you...oh,yeah, that's those shoes she is wearing!... Is this the same entity I saw yesterday?

edit: I see PAllen posted while I was composing...I make my own notes from maybe 10 or 12 regular people here... keeping their descriptions that 'click for me'...You might do the same if the time and effort is worth it...PAllen is one of that group.

 

[someGorilla]

Thank you both for your quick and detailed answers!

While much of what you say is true, and even in SR some use if made of coordinates that are all lightlike (i.e. no coordinate is spatial or temporal), most commonly you do try to have coordinates that mean something physical, esp. for some class of special observers. For example, the r coordinate in Schwarzschild coordinates does have a clear physical meaning - the circumference around the central mass divided by 2π. This can be readily measured outside the event horizon, even if you can't readily measure distance to the center of a star, let alone a singularity. The time coordinate also has an operational definition.

Ok. Of course it's better to use coordinates that have an immediate physical meaning!

Similarly, Kruskal coordinates are those that seek to have one time like, and 3 spatial coordinates, that preserve their character everywhere (across the horizon and up to - but not reaching - the singularity - which is normally considered not part of the manifold). The further defining feature of these coordinates is the light like geodesics are all straight lines so that light cones look exactly as they do in Minkowski coordinates in flat spacetime.

Careful authors should be spelling out the significance of the coordinates, if it matters (for many discussions, it doesn't matter).

Yes, I know about Kruskal coordinates. And about spelling out the significance of the coordinates, as you say

An additional point is that the expression of the metric in chosen coordinates tells you directly the physical nature of those coordinates in some region (the physical nature of coordinates can vary from one spacetime locale to another).

the metric tells it all. What I meant is exactly that the expression of the metric is necessary for the coordinates to make physical sense. I know this is banal; it's just a post-reflection on my learning path I guess.

 

[someGorilla]

In GR, the 'best observations' are the ones you make locally...right where you are. Sounds like you have picked that up already. So carrying your own clock ticks proper time, no ambiguities there, and your ruler is also quite reliable right there. But even so inertial versus accelerated motion can be confusing. And what you observe locally will not in general be what another distant observer records at your location.

Of course, this is clear to me.

Whoaa...Misconceptions are 99% in the mind of the reader rather than the authors...But ,sure, there must be some simplifications in introductory material. I'd urge to you to read a few of interest to you... here are several outstanding 'popular' books I know 'personally':

[...]

If you are mathematically inclined,
[...]

I am definitely mathematically inclined.
I agree about misconceptions being in the mind of the reader, but still they can be helped or prevented to a certain degree. And you know better than I do all the discussions on this forum where somebody thinks the universe must be finite "because it all started from a point" and I relate them to authors writing "when the universe was as big as a golf ball..."

because those abstract coordinates enable exact solutions to very difficult equations. You can make approximations, like perturbation theory, and reach approximate solutions, or in some cases use 'abstract' coordinates which offer their own insights.

Of course! I have nothing against "abstract" coordinates. I actually think that GR itself shows that even Newtonian coordinates are no less "abstract" than this, just maybe more easily defined operationally.

It's not gibberish, but subtle distinctions.

I was referring to how sometime things are presented to the ignorant reader. I'm more and more disliking sloppy divulgation, but this might be material for a thread in another section maybe.

For better or worse it also exists in the other major theory of our time, quantum mechanics...try explaing what THAT means...SHUT UP AND CALCULATE is one mantra meaning who cares how the mathematical details may be interpreted by different people, we can agree on the mathematics. Try explaining why you like your boyfirend or girlfriend...what makes them so SPECIAL??

I'm not trying to question the deep meaning of it all. I find it fascinating to think and discuss about it, but my original subject was much more down to Earth :)

I'd readily agree that descriptions do not often take into account they are unique perspectives based on a particular set of coordinates...and of a particular set of mathematics. What we all learn in GR is that there are a variety of 'realities': Different observers make different observations. An example which I like is the Unruh effect in which an accelerating observer apparently measures a different local temperature than a nearby inertial observer. And perhaps as weird if not more so the ADS/CFT correspondence where a number of dimensions without gravity seems to be the same as another environment with an additional dimension plus gravity!

I guess this has to do with the holographic universe theories? Hey hey, one thing at a time, I don't want to skip steps this time!

So it's maybe analogous to showing up for a 'big date' with a girl whose looks you think you know...BAM! all of a sudden her hair is styled different than you have ever seen, the color may be different, she has a Mike Tyson type tatoo on her face! She is as tall as you...oh,yeah, that's those shoese she is wearing!... Is this the same entity I saw yesterday?

You do have a talent for graphic examples.