Beginner's Guide to GR: Questions & Answers

[Chenkel]

TL;DR Summary

I just saw a video on YouTube talk about the basics of GR, and I have a few questions that hopefully the kind community of physics forums might be able to answer.

Hello everyone!

I was just watching a video, and to my surprise I was able to understand a little about GR (General Relativity). I'm looking to confirm my understanding, and also ask some basic questions.

From what I understand in Newtonian gravity the space time is considered to be uniform, contiguous, and orthogonal. From Newton's perspective a force has to be acting on the object to make it move in a curve around Earth.

Under GR, instead of the velocity vector changing for the orbiting body, the basis vectors for the coordinate system of the rotating body are changing. This is because you can create an orbiting body by warping space, and leaving the velocity vector unchanging. This effectively eliminates the Newtonian force of gravity, while maintaining a orbiting body while it moves in a straight line in curved space-time.

I have a few questions:

What kind of basis vectors are we looking at for the local coordinate system of the orbiting body?

Is the Newtonian gravitational force a psuedo force?

Why is GR more accurate than a Newtonian gravitational model?

Is there a relatively easy way I can understand the GR equations with an intermediate understanding of calculus?

Does GR explain anything at the quantum level?

Also what would your advice be for someone just getting the hang of the theory?

I am making this post out of inspiration, and I feel I admittedly have not done a lot of research, so my apologies if I made any assumptions that make it difficult to enlighten my understanding.

If you would like to reply and discuss the topics I've laid out, or add something, please feel free to, I am looking forward to what you have to say.

Let me know what you think, thank you!

 

[Orodruin]

Hello everyone!

Hi!

I was just watching a video, and to my surprise I was able to understand a little about GR (General Relativity). I'm looking to confirm my understanding, and also ask some basic questions.

What video? Please provide the link when you are asking specific questions about something you read or saw.

Under GR, instead of the velocity vector changing for the orbiting body, the basis vectors for the coordinate system of the rotating body are changing.

No. You can always pick any basis vectors you want. Regardless of Newtonian physics or relativity. The coordinate system is a convention and should not affect the physics.

This is because you can create an orbiting body by warping space, and leaving the velocity vector unchanging. This effectively eliminates the Newtonian force of gravity, while maintaining a orbiting body while it moves in a straight line in curved space-time.

Close enough, but what is curved is spacetime, not just space. Without seeing the video it is impossible to tell if this is a fault of the video or if you misunderstood it.

I have a few questions:

What kind of basis vectors are we looking at for the local coordinate system of the orbiting body?

See above. The basis is irrelevant.

Is the Newtonian gravitational force a psuedo force?

Not in Newtonian gravity. It does not exist in GR.

Why is GR more accurate than a Newtonian gravitational model?

This is a philosophical question, not a physical one. When two competing theories exist, which one is better is ultimately decided by experimentation and onservation. GR describes the world we live in more accurately. That’s it.

Is there a relatively easy way I can understand the GR equations with an intermediate understanding of calculus?

It depends on what you mean by ”GR equations”. There are of course some basic reaults in terms of formulas that you will understand, but the full mathematical framework requires differential geometry. This is usually included (at least the basics) in GR textbooks, but prerequisites typically include a good understanding of linear algebra and multivariable calculus.

Does GR explain anything at the quantum level?

No. It is a classical theory.

Also what would your advice be for someone just getting the hang of the theory?

Get a good grip of the prerequisite mathematics and then take a course on GR. If this is not an option, at least follow a standard textbook and ask questions you may have here.

 

[Ibix]

What kind of basis vectors are we looking at for the local coordinate system of the orbiting body?

Basis vectors are like north and east on the Earth - you can travel without knowing them and you can give directions without knowing them. And you can use different ones - the Museum of Modern Art in New York used to sell New York Compasses that were labelled Uptown, Downtown, Eastside and Westside, pointing along the four cardinal directions of New York's street grid. In New York it's quite convenient to give directions in that basis, but you can specify compass bearings.

So there's no one basis for anything. There is often one that makes the maths easy (well, less difficult), but which one that is probably depends on why you want to know as well as the physical form of the system.

Summary: I just saw a video on YouTube talk about the basics of GR, and I have a few questions that hopefully the kind community of physics forums might be able to answer.

Is there a relatively easy way I can understand the GR equations with an intermediate understanding of calculus?

Download Sean Carroll's GR notes and read chapters 1 and 2. If you can follow the second one you're in business. If not, you need to brush up your calculus. If you haven't studied Special Relativity I'd recommend doing that first, since all GR texts will assume you're reasonably confident with SR.

 

[Chenkel]

What video? Please provide the link when you are asking specific questions about something you read or saw.

No. You can always pick any basis vectors you want. Regardless of Newtonian physics or relativity. The coordinate system is a convention and should not affect the physics.

So if the basis vectors are not changing, how does space-time warp? I believe based on your post I'm correct that the orbiting body moves in a straight line, but relative to an observer on Earth it seems to rotate around the center of the earth, so something seems to be changing relative to an unchanging velocity; I'm curious what that would be, is there a laymen's way of understanding it?

Also just from an intuition basis I was initially thinking that in order to satisfy Kepler's equal area law, when the planet is farther away, the change in position using an unchanging velocity will be smaller relative to a change in position nearer the massive body; I'm wondering how I can think of Kepler's laws in a way that is favorable to GR.

Get a good grip of the prerequisite mathematics and then take a course on GR. If this is not an option, at least follow a standard textbook and ask questions you may have here.

My way of learning physics is what you might consider a little unconventional, I'll pick things up here and there, but I think one could dedicate a lifetime to understanding, so I need to pick my battles when it comes to any subject. Anything that makes my way of viewing the universe more elegant is something that is usually meaningful to me, so that's the reason GR caught my eye. Is it possible to get a solid understand from researching a variety of articles, and videos from the internet? I am not anti textbook, it's just the textbooks that I have read are sometimes more interested in covering ground rather than exploring key concept; the way I study math books is to skip most parts I am familiar with and familiarize myself with what is intuitive and at the same time unknown to me.

It depends on what you mean by ”GR equations”. There are of course some basic reaults in terms of formulas that you will understand, but the full mathematical framework requires differential geometry. This is usually included (at least the basics) in GR textbooks, but prerequisites typically include a good understanding of linear algebra and multivariable calculus.

Hopefully my understanding of math will be good enough, I am by no means an expert, but I have a pretty good mathematical intuition about vectors, and rates of change.

Is differential geometry the understanding of geometry in terms rates of change? If so, I think I could wrap my head around that, it should require a basic understanding of calculus and geometry?

Basis vectors are like north and east on the Earth - you can travel without knowing them and you can give directions without knowing them. And you can use different ones - the Museum of Modern Art in New York used to sell New York Compasses that were labelled Uptown, Downtown, Eastside and Westside, pointing along the four cardinal directions of New York's street grid. In New York it's quite convenient to give directions in that basis, but you can specify compass bearings.

So there's no one basis for anything. There is often one that makes the maths easy (well, less difficult), but which one that is probably depends on why you want to know as well as the physical form of the system.

Download Sean Carroll's GR notes and read chapters 1 and 2. If you can follow the second one you're in business. If not, you need to brush up your calculus. If you haven't studied Special Relativity I'd recommend doing that first, since all GR texts will assume you're reasonably confident with SR.

I'm a little confused how there is not many coordinate systems in relativity, it seems I can not imagine any other way for space-time to warp and objects to move non uniformly relative to each other with unchanging velocity.

Hopefully SR and GR prove to be rewarding in my goals of making a little sense out of the universe; are the prerequisites for SR the same as GR, can both subjects be studied in a similar way?

Thank you for the feedback.

 

[PeroK]

So if the basis vectors are not changing, how does space-time warp?

Spacetime has a geometry based on the mass distribution (more precisely stress-energy distribution). You then describe that geometry in whatever way is simplest, most useful or most appropriate.

I believe based on your post I'm correct that the orbiting body moves in a straight line

More accurately it moves on a timelike geodesic (in spacetime). If you want to call a timelike geodesic a "straight line", then you're in good company. But, don't then confuse it with a straight line through Euclidean space.

I'm wondering how I can think of Kepler's laws in a way that is favorable to GR.

By first deriving the Newtonian theory as an approximation for the solar system.

Is it possible to get a solid understand from researching a variety of articles, and videos from the internet?

No, it isn't. You'll have to put in the hard work with months of study (if not a year or two).

I am not anti textbook, it's just the textbooks that I have read are sometimes more interested in covering ground rather than exploring key concept; the way I study math books is to skip most parts I am familiar with and familiarize myself with what is intuitive and at the same time unknown to me.

Then you'll never learn anything worth learning. Intuition is BS when faced with hard mathematics.

Hopefully my understanding of math will be good enough, I am by no means an expert, but I have a pretty good mathematical intuition about vectors, and rates of change.

I doubt it. I'll bet I'm better than you at maths and I had to work really hard to learn GR.

Is differential geometry the understanding of geometry in terms rates of change?

Not really. It shouldn't be hard to find a description of differential geometry online.

I'm a little confused how there is not many coordinate systems in relativity, it seems I can not imagine any other way for space-time to warp and objects to move non uniformly relative to each other with unchanging velocity.

This makes no sense and shows how little you really grasp about GR.

Hopefully SR and GR prove to be rewarding in my goals of making a little sense out of the universe; are the prerequisites for SR the same as GR, can both subjects be studied in a similar way?

SR is not too hard to learn. It took me about 3 months. GR is a different beast altogether: you need considerable mathematical maturity and a firm grasp of how mathematical physics is done. There are no short cuts or royal road to GR.

 

[Orodruin]

So if the basis vectors are not changing, how does space-time warp?

The basis changing has nothing to do with the space being curved or not. For example, the base vectors of Euclidean space in polar coordinates depend on the position yet it is still the same old flat Euclidean space. What determines whether a space is flat or not is its geometry, ie, how angles add up in triangles and other geometrical shapes, how the area of a circle relates to its circumference, etc.

Also just from an intuition basis I was initially thinking that in order to satisfy Kepler's equal area law, when the planet is farther away, the change in position using an unchanging velocity will be smaller relative to a change in position nearer the massive body; I'm wondering how I can think of Kepler's laws in a way that is favorable to GR.

You should not try to understand GR from intuition. Intuition is based on previous experience and people in general have very little previous experience with curved spacetime. In order to think intuitively about GR, you first need to do the work to familiarize yourself with GR.

My way of learning physics is what you might consider a little unconventional, I'll pick things up here and there, but I think one could dedicate a lifetime to understanding, so I need to pick my battles when it comes to any subject. Anything that makes my way of viewing the universe more elegant is something that is usually meaningful to me, so that's the reason GR caught my eye. Is it possible to get a solid understand from researching a variety of articles, and videos from the internet? I am not anti textbook, it's just the textbooks that I have read are sometimes more interested in covering ground rather than exploring key concept; the way I study math books is to skip most parts I am familiar with and familiarize myself with what is intuitive and at the same time unknown to me.

This will generally not work for the reasons given above. To have any sort of intuition for GR, you need to work at building that intuition first.

I'm a little confused how there is not many coordinate systems in relativity, it seems I can not imagine any other way for space-time to warp and objects to move non uniformly relative to each other with unchanging velocity.

Huh? There are as many coordinate systems as you want. It is in fact an underlying foundation that you can pick any coordinate system that you fancy, the theory will be the same. It may however be easier to express in particular coordinate systems.

 

[Ibix]

I have a pretty good mathematical intuition about vectors, and rates of change.

Me too. It took me a long time to unlearn all that in order to understand how vectors work in GR. They have to be defined in a much more formal way in order to be usable in curved spacetime.

I don't want to put you off studying GR, but you do need to be realistic. You need a solid systematic study of the basics because many things in it are counter to our everyday intuition, and you need a solid foundation of maths on which to build a more general intuition. Once you get the basics down you can more or less ignore cosmological solutions if you're only interested in black holes (or whatever), but if you don't build that foundation you'll just end up even more confused than you are at the moment.

 

[PAllen]

I’ll share my personal experience with getting started in GR without a course. After self studying calculus sufficient to get a 5 on the BC AP calculus exam (this is US specific, roughly corresponding to a standard 3 semester calculus sequence), I wanted to learn SR and GR. At the time, the standard college text was Peter Bergmann’s 1942 text. Though a relatively short text, I spent 1.5 years working at going through the book doing all the exercises, consulting many other books as needed when I realized I was missing some background. I believe I came away with no more understanding than if I had taken a 1 semester college course in GR. That is, self study (done right) in my experience, is slower than taking courses.

 

[Nugatory]

As well as Sean Carroll’s full lecture notes (recommended by @Orodruin above, and I strongly agree) you might also try his light-weight intro: https://preposterousuniverse.com/wp-content/uploads/2015/08/grtinypdf.pdf

It’s no substitute for the real thing, but it is very helpful as an overview, letting you know why concepts are being introduced and why they matter.

 

[Ibix]

As well as Sean Carroll’s full lecture notes (recommended by @Orodruin above, and I strongly agree)

Ahem!

 

[Nugatory]

Ahem!

Oops -sorry about that.

 

[Daverz]

Summary: I just saw a video on YouTube talk about the basics of GR

There are some good formal GR courses on YT that you could peak at to get an idea of the math and physics involved.

Like this one from MIT:



It seems to be midway undergraduate/graduate in level. He uses Schutz for a lot of his derivations, which is a good choice.

Here's a nice collection of physics lectures:

https://www.youtube.com/c/SchoolofTheoreticalPhysicsAppliedMathematics/playlists

 

[Chenkel]

Thank you for all the replies.

GR doesn't look like an easy subject, but it definitely looks like you can make it harder than it has to be if you don't have the right approach.

 

[Vanadium 50]

looks like you can make it harder than it has to be if you don't have the right approach.

That is very often the case. Not subject-specific.