Metric: Proper Space vs Coordinate Space

[Haorong Wu]

TL;DR

This basic concept in GR has always confused me. I am not sure is my understanding correct.

I imagine there is a isotropic space. Well, I would call it the proper space which will remain unchanged in any cases. And there is another space I call the coordinate space which will be distorted by gravitational field, i.e., metric.

a) Suppose there are two stationary points. Their coordinates will be given in the coordinate space. No matter what kind of gravitational field is imposed on them, their coordinate will remain unchanged. However, due to the metric, their projection to the proper space will be altered. So the proper length between them will be affected by the metric.

b) Consider a light path. Suppose a beam propagates along z axis in the coordinate space. Then it will remain on it. However, due to the gravitational field, the projection of z axis to the proper system may be a curve, so the light actually travel along a curve.

So could I treat the metric like some projection operation from the coordinate space to the proper space?

 

[Dale]

I have never heard of anything like your concept of proper space. Can you express your question in terms of the standard terms and concepts?

 

[Haorong Wu]

I have never heard of anything like your concept of proper space. Can you express your question in terms of the standard terms and concepts?

Like proper length, peroper volume element, proper velocity, etc., I am confused about the word "proper". In what way that these quantities are proper?

 

[Dale]

Like proper length, peroper volume element, proper velocity, etc., I am confused about the word "proper". In what way that these quantities are proper?

I have heard of all of those other terms and I understand what they mean. But I have never heard of proper space.

Do you have a reference where proper space is defined, or can you do so here (preferably mathematically).

 

[Ibix]

In what way that these quantities are proper?

"Proper" in this context shares its root with the word "property" - something that is your own. So proper time is the time experienced by one person/atom/whatever as measured by their own clock. It does not mean anything like its usual modern English meaning of "correct".

I'm not sure what you mean by the word in your post.

 

[Nugatory]

This basic concept in GR has always confused me. I am not sure is my understanding correct.

The metric is fundamental to GR, but that's because GR is built on differential geometry and the metric is fundamental to differential geometry. If you're confused by the role of the metric in GR, you may want to go back the basics and see what it does for differential geometry.

However, if you're already comfortable with the Minkowski geometry of special relativity (and if you aren't, you're missing an important prerequisite for GR) it may be easier to think of the metric as the definition of the distance (formally space-time interval) between any two nearby points in spacetime.

 

[Will Learn]

Hi.

I imagine there is a isotropic space. Well, I would call it the proper space...

It seems that you (Haorong) are not claiming there is a well defined notion of proper space but instead just asking if it is possible to imagine things this way.
Well, possibly yes. However, I don't think we (myself and the previous posters) would recommend that you try.

That's not the way the metric is understood or used in the main texts. It doesn't look like a small modification or personal aid to memory that would help you in your studies. Instead it seems like a big modification which would keep giving you problems when you try and match it up with the way things are used and described in the main texts. I quite like Nugatory's answer (above), re-examine what the main texts say about the metric before continuing to build your own mental image of this "proper space".

You used an example of light traveling in straight lines in your "co-ordinate space" but it would be difficult (impossible) to build a "co-ordinate space" where all of our (pre-relativity) ideas of motion apply to every object everywhere and at all times. It's far easier to modify your understanding of the laws of motion rather than build a space where they can continue to apply as you imagine. For example, don't assume there is a co-ordinate space where light travels in straight lines. Instead, change your understanding of the path light will take - it will take a geodesic path and the metric tells you how to find that path.

Hope that helps.

 

[Haorong Wu]

Thanks to @Dale , @Ibix , @Nugatory , @Will Learn .

Because I learned that we can built local inertial frame, which is isotropic in my mind, in a point, then I tried to imagine that the metric is a projection between coordinate and the inertial frame. Well, it is not a correct image. I will try harder to understand the metric. I guess I need solve more problems. I always get a better understanding through solving them.

Also, I know now proper means nothing to do with correction.

Thanks again!