Please explain General Relativity to me

[Jocko Homo]

I've always wanted to understand the Theory of General Relativity but I've never found anyone to properly explain it to me. I was hoping that someone here could explain the principles to me in fine detail.

I find the sheet of rubber with masses bending it to be an irritatingly unsatisfying analogy. It's obvious that what's happening in that scenario is that gravity (actual gravity) is pulling objects down sloped surfaces and doesn't really demonstrate how deformations of space-time can create acceleration or the equivalent, thereof. I can even imagine that, somehow, space-time is warped in some way to cause moving objects to not simply move in straight lines but I can't imagine how objects at rest relative to a gravitational mass begin to accelerate towards said mass.

I understand the equivalence of an accelerated frame of reference and being in a gravity well but I don't know how to build General Relativity out of that.

Is anyone willing to try to explain this to me? If it helps at all, I have a sound understanding of Special Relativity...

Thank you...

 

[Passionflower]

The best thing to do is to buy or borrow from the library a good introductory book on general relativity. A few postings on a forum will not be sufficient to explain it.

 

[Jocko Homo]

Are you sure? While I appreciate that it may be a difficult subject to try to explain, I have trouble believing the general principle can't be explained in a couple of posts...

For example, perhaps you can show me an example of how one dimensional space-time (one dimension for space and one for time) can be warped to produce a familiar gravity well? Such a diagram can conveniently be plotted in two dimensions, can it not?

At the very least, could you recommend a specific book? I've looked at various books in the book store and scraped the internet and couldn't find what I'm looking for...

Thank you...

 

[atyy]

http://www.aei.mpg.de/einsteinOnline/en/elementary/index.html

http://www.aei.mpg.de/einsteinOnline/en/spotlights/equivalence_principle/index.html

http://books.google.com/books?id=P_T0xxhDcsIC&dq=gravity+from+the+ground+up&source=gbs_navlinks_s

http://www.blau.itp.unibe.ch/lecturesGR.pdf

 

[Academic]

Are you sure? While I appreciate that it may be a difficult subject to try to explain, I have trouble believing the general principle can't be explained in a couple of posts...

For example, perhaps you can show me an example of how one dimensional space-time (one dimension for space and one for time) can be warped to produce a familiar gravity well? Such a diagram can conveniently be plotted in two dimensions, can it not?

At the very least, could you recommend a specific book? I've looked at various books in the book store and scraped the internet and couldn't find what I'm looking for...

Thank you...

The analogy you don't like, the rubber sheet, that is the two dimensional version you seek. Forget gravity since that is a confusion in the analogy. The sheet is held still and you push in the center with your finger. The harder you push is analogous to more mass there. Now think of the lines that are at the same height. There lines of same height, when looked at from above look like little circles inside larger circles and so forth. Does that make sense? So the finger pushing down on the sheet is representing the mass, the sheet is representing space-time and the circles of constant height are representing geodesics which are what like straight lines in general relativity's space time.

So there is the a field equation for general relativity. It relates how the mass warps the space time around it. When warping it changes the straight lines, kind of like how the finger warps the lines in a sheet. Those straight lines are the path objects take.

As always, with analogies its best to try to find the similarities but recognizing the differences can be useful too.

 

[skycastlefish]

Try visualizing space time as something physical. After all, physics is the study of the physical world. For example, without education, we would most likely think of the air as being nothing. But we now know that air is matter. Imagine being a great white shark shooting out of the south African sea in pursuit of a seal. To that shark the air must seem like a vacuum! Its relative definition of "nothing" is its standard H2O. If you're looking for a relateable visualization of general relativity, then imagine space and time as physical property, almost like a foam that can be compressed or stretched. The Earth is not necessarily pulling mass toward itself -- space time is pushing you toward mass! Althought Space and Time are variables, this doesn't mean the are flat like a rubber sheet. They exist from every reference frame we know.

 

[Jocko Homo]

The analogy you don't like, the rubber sheet, that is the two dimensional version you seek. Forget gravity since that is a confusion in the analogy. The sheet is held still and you push in the center with your finger. The harder you push is analogous to more mass there. Now think of the lines that are at the same height. There lines of same height, when looked at from above look like little circles inside larger circles and so forth. Does that make sense? So the finger pushing down on the sheet is representing the mass, the sheet is representing space-time and the circles of constant height are representing geodesics which are what like straight lines in general relativity's space time.

Yes, I do understand the analogy but I find it unsatisfying...

Using your example, let's look at an elevation line of the rubber sheet. What does this contour represent? It's tempting to think it represents the connectivity of space around that mass so if a satellite is in orbit around the mass then this contour line is the path of the orbit! However, with just a little more thought, this analogy breaks down quickly. I know that if you give the object a greater initial velocity, then the object will spiral out of orbit... or if you give it too little velocity, it will fall into the mass. How is this represented by these contour lines? How do these lines represent the warping of space-time?

It seems to me that these analogies were meant to merely describe that space-time can warp and not exactly how gravity warps them...

In particular, I think it will be revealing how a warping of space-time can cause an initially static object to fall into a simple gravity well. Obviously the object that appears static to us must be in motion in regards to the warping of space-time since it's following a geodesic into the well but this implies that this so called "warping" is more than just a physical warping of space like that of a rubber sheet. It must be a warping of space and time to move the apparently static object. Perhaps the warping of time along with space is enough to move the static object and it need not be moving in regards to the warping?

I wish to understand the nature of this warping!

So there is the a field equation for general relativity. It relates how the mass warps the space time around it. When warping it changes the straight lines, kind of like how the finger warps the lines in a sheet. Those straight lines are the path objects take.

I'm guessing the domain of this field equation is space and maybe time? What's the range? Is it also space? ...and maybe time? How is this "warping?" How can this explain the acceleration of static bodies?

As always, with analogies its best to try to find the similarities but recognizing the differences can be useful too.

I'm having trouble recognizing anything! I think I'm seeing the differences... What are the similarities?

Thank you...

 

[espen180]

If you want to see how mass curves space-time, you will ahve to learn a bit of tensor calculus.

 

[Fredrik]

This isn't going to make any sense to you, but it should help you see why you will never get a satisfying explanation unless you study differential geometry.

I'm guessing the domain of this field equation is space and maybe time?

Yes, it's "spacetime", which is a smooth 4-dimensional Lorentzian manifold. The word "Lorentzian" indicates that the manifold is equipped with the type of metric tensor field that's appropriate when one dimension is to be interpreted as time. Einstein's equation describes the relationship between the metric tensor field and the stress-energy tensor field of matter. Using a local coordinate system, the stress-energy tensor field can be represented as 16 real-valued functions $T^{\mu\nu}$ on an open subset of spacetime, each one labeled by two indices ranging from 0 to 3. They satisfy they condition $T^{\mu\nu}=T^{\nu\mu}$, which brings the number of independent functions down to 10. $T^{\mu\nu}$ can be interpreted as "the flow of $\mu$-momentum across a hypersurface of constant $\nu$" (or is it the other way round...I'm too lazy to check). That means that $T^{00}$ is the density of matter, while the other components represents internal forces and stresses.

(In this context, the term "matter" includes such things as electromagnetic fields).

What's the range? Is it also space? ...and maybe time?

No, it's the vector bundle of type (2,0) tensors at all the different points of spacetime. The set of all (2,0) tensors at a point p in spacetime has the structure of a vector space. The union of all those vector spaces is a vector bundle. The fields in Einstein's equation are sections of that bundle. That means that the function (i.e. the field) takes a point p in spacetime to a point in the vector space of (2,0) tensors at p.

(Wald calls them (2,0) tensors, Lee calls them $$\begin{pmatrix}0\\ 2\end{pmatrix}$$ tensors).

Edit: Actually, that vector bundle is the codomain of the function, not the range.

How is this "warping?" How can this explain the acceleration of static bodies?

That's sort of a mischaracterization of what it does. It tells you which curves in spacetime we should consider "straight". The technical term is "geodesic". Acceleration is defined relative to those geodesics. The first step is to define non-accelerating motion as motion represented by a geodesic, and then you define acceleration as a measure of the deviation from geodesic motion, and "force" by F=ma. This is why gravity isn't a force in special relativity. Gravity just determines the metric tensor field, which tells us which curves are geodesics. The only way to do geodesic motion is to be in free fall, so right now, the normal force from the chair you're sitting on is accelerating you in the "up" direction, preventing you from doing geodesic (i.e. non-accelerating) motion.

I'm not even going to try to explain what this has to do with curvature. That stuff is even harder. This book has the answer.

 

[Jocko Homo]

This isn't going to make any sense to you, but it should help you see why you will never get a satisfying explanation unless you study differential geometry.

This is fair and I thank you for the effort!

Yes, it's "spacetime", which is a smooth 4-dimensional Lorentzian manifold. The word "Lorentzian" indicates that the manifold is equipped with the type of metric tensor field that's appropriate when one dimension is to be interpreted as time. Einstein's equation describes the relationship between the metric tensor field and the stress-energy tensor field of matter. Using a local coordinate system, the stress-energy tensor field can be represented as 16 real-valued functions $T^{\mu\nu}$ on an open subset of spacetime, each one labeled by two indices ranging from 0 to 3. They satisfy they condition $T^{\mu\nu}=T^{\nu\mu}$, which brings the number of independent functions down to 10. $T^{\mu\nu}$ can be interpreted as "the flow of $\mu$-momentum across a hypersurface of constant $\nu$" (or is it the other way round...I'm too lazy to check). That means that $T^{00}$ is the density of matter, while the other components represents internal forces and stresses.

(In this context, the term "matter" includes such things as electromagnetic fields).

Can you give me a more concrete but simplified example of this? Perhaps a 2 dimensional space-time (one dimension of space and one for time) example? In this case, you should end up with 4 real valued functions, right? I don't know how many would be "independent" in this case...

How would you calculate the acceleration of a massless static object next to a point mass gravity well?

No, it's the vector bundle of type (2,0) tensors at all the different points of spacetime. The set of all (2,0) tensors at a point p in spacetime has the structure of a vector space. The union of all those vector spaces is a vector bundle. The fields in Einstein's equation are sections of that bundle. That means that the function (i.e. the field) takes a point p in spacetime to a point in the vector space of (2,0) tensors at p.

(Wald calls them (2,0) tensors, Lee calls them $$\begin{pmatrix}0\\ 2\end{pmatrix}$$ tensors).

Okay, so every point in space and time maps to... a point in the "vector space of (2,0) tensors at p?"

I'm guessing that, for a point mass, this field equation is a function of nothing but the mass at the point and its geometric symmetry?

That's sort of a mischaracterization of what it does. It tells you which curves in spacetime we should consider "straight". The technical term is "geodesic". Acceleration is defined relative to those geodesics. The first step is to define non-accelerating motion as motion represented by a geodesic, and then you define acceleration as a measure of the deviation from geodesic motion, and "force" by F=ma. This is why gravity isn't a force in special relativity. Gravity just determines the metric tensor field, which tells us which curves are geodesics. The only way to do geodesic motion is to be in free fall, so right now, the normal force from the chair you're sitting on is accelerating you in the "up" direction, preventing you from doing geodesic (i.e. non-accelerating) motion.

Actually, this makes a lot of sense. I've never heard General Relativity characterized quite like this before. Thank you...

Okay, so Einstein's field equation are used to determine what it means to be "straight." Given an initial velocity, you use this field equation to determine a path that the object will take baring any external forces. This is consistent with the traditional notion that acceleration due to gravity is independent of the weight of the object...

I can see how changing the meaning of "straight" might be characterized as a warping of space-time but that seems to me to be a very poor analogy... except that this analogy is taken even farther than this! I've heard people say that, according to General Relativity, it's possible to create wormholes connecting otherwise separate regions of space. That really does sound like warping space! What's up with claims like that?

I'm also a little confused about the notion of "free fall." Straight line movement, in any sense, requires movement. While I'm sitting in my chair, am I moving? I understand that I'm being accelerated upwards by my chair but why am I moving downwards? Even if that's a straight path, why would I be moving in that path if it weren't for my chair?

I think what I'm getting at is that, in the absence of mass, the geodesic would be a point. Why are geodesics in the presence of mass always lines?

I'm not even going to try to explain what this has to do with curvature. That stuff is even harder. This book has the answer.

Where'd the geometry go?

Thank you very much for your input!

 

[haushofer]

They satisfy they condition $T^{\mu\nu}=T^{\nu\mu}$, which brings the number of independent functions down to 10. $T^{\mu\nu}$ can be interpreted as "the flow of $\mu$-momentum across a hypersurface of constant $\nu$" (or is it the other way round...I'm too lazy to check).

The EM-tensor is symmetric in both indices, so that shouldn't matter.

 

[stevenb]

Is anyone willing to try to explain this to me? If it helps at all, I have a sound understanding of Special Relativity...

Thank you...

Do a search on YouTube for "Leonard Susskind General Relativity Lectures". There are 12 lectures that are a very good introduction to the topic. GR is one of those subjects that looks daunting in forsight and almost trivial in hindsight. That is, for the basic intuitive understanding, not the actual practical calculations. However, a fair amount of work is needed to make the journey to see that vista.

Personally, I don't think there is any substitute for taking on the tensor level understanding of the subject. Some subjects can't be adequately summarized effectively from a layman's book or a one hour documentary on the science channel.

Someone here once posted a nice paper on an intuitive interpretation of GR, which I've attached because I can't find exactly where I found it.

 

[Fredrik]

Can you give me a more concrete but simplified example of this?

Not sure what specific detail in that quote you want an example of. Regarding the number of independent components, just think of a 4×4 matrix. If A_ij=A_ji, then the components below the diagonal are the same as the ones above it, so only 10 of the 16 are different.

Perhaps a 2 dimensional space-time (one dimension of space and one for time) example? In this case, you should end up with 4 real valued functions, right? I don't know how many would be "independent" in this case...

Yes, and 3.

Okay, so every point in space and time maps to... a point in the "vector space of (2,0) tensors at p?"

Perhaps the word "point" is confusing. I just mean a member of that space, i.e. a (2,0) tensor at p.

I'm guessing that, for a point mass, this field equation is a function of nothing but the mass at the point and its geometric symmetry?

I don't think there is a solution where mass is concentrated at a point, but the Schwarzschild solution describes spacetime outside a spherical distribution of mass (with no electric charge and no rotation). Only the total mass contributes to the metric. And yes, the symmetries make makes this solution relatively simple. The corresponding solution for a rotating distribution of mass wasn't found until 48 years later.

I can see how changing the meaning of "straight" might be characterized as a warping of space-time but that seems to me to be a very poor analogy... except that this analogy is taken even farther than this! I've heard people say that, according to General Relativity, it's possible to create wormholes connecting otherwise separate regions of space. That really does sound like warping space! What's up with claims like that?

I have only read half the book I linked to, but the chapters near the end seem to explain exactly how the curvature tensor is related to more intuitive notions of curvature. I can only mention such things as e.g. that the interior angles of a triangle usually don't add up to pi in a curved space.

While I'm sitting in my chair, am I moving?

There's no coordinate-independent way to define "not moving". Your existence defines a curve in spacetime. That curve can be mapped to a curve in $\mathbb R^4$ by a coordinate system. If the coordinate system maps the curve that represents your existence to a line that's parallel to the time axis, then you're not moving in that coordinate system.

A geodesic is "straight" in a coordinate-independent sense, but a coordinate system can certainly take a geodesic to a strange-looking curve in $\mathbb R^4$.

How would you calculate the acceleration of a massless static object next to a point mass gravity well?

If the object is held at fixed position coordinates, then its coordinate velocity and coordinate acceleration in that coordinate system are both zero.

I understand that I'm being accelerated upwards by my chair but why am I moving downwards? Even if that's a straight path, why would I be moving in that path if it weren't for my chair?

That's an axiom of the theory, and not something that it can explain. Perhaps some other theory will explain it some day, like GR explained the mysterious action at a distance in Newton's theory, but that would of course give us a new set of unexplained axioms.

I think what I'm getting at is that, in the absence of mass, the geodesic would be a point. Why are geodesics in the presence of mass always lines?

A universe without mass, or rather a universe where mass doesn't have any effect on the geometry, is exactly what special relativity is describing. Your existence is a curve in that theory too. (You exist right now...and now too. Those are two different points on that curve). It just happens to be parallel to the time axis of the coordinate system you'd like to use.

 

[stevmg]

I'm also a little confused about the notion of "free fall." Straight line movement, in any sense, requires movement. While I'm sitting in my chair, am I moving? I understand that I'm being accelerated upwards by my chair but why am I moving downwards? Even if that's a straight path, why would I be moving in that path if it weren't for my chair?

The simplistic way that I understand this and I am probably the biggest novice on this forum is that every point in 4D spacetime is moving. Each point has a worldline of some sort. At every instant a point is "moving" along its worldline, even if one does nothing to it (time does not stand still.). If mass distorts spacetime so that it is curved, the worldline is curved (?along the geodesic?) and a curved worldline means acceleration (or deceleration) in the 3D space coordinates no matter what frame of reference is used. Acceleration therefore implies "gravity" of the 3D world. There is no place in the universe that there is no distortion so there will always be gravity (curved worldlines.)

I got this interpretation from the book General Relativity From A to B by Robert Geroch. In it he does state that curved worldlines means acceleration. There are no places in our universe where there is no distortions, ergo, curved worldines for every point. The rest of what I wrote comes from my digesting what he said. In previous posts, some of the PF Mentors did say that I was kind of right. As far as tensor applications discussed above in earlier posts I have no clue.

 

[Jocko Homo]

Can you give me a more concrete but simplified example of this?

Not sure what specific detail in that quote you want an example of. Regarding the number of independent components, just think of a 4×4 matrix. If A_ij=A_ji, then the components below the diagonal are the same as the ones above it, so only 10 of the 16 are different.

Perhaps a 2 dimensional space-time (one dimension of space and one for time) example? In this case, you should end up with 4 real valued functions, right? I don't know how many would be "independent" in this case...

Yes, and 3.

I don't think there is a solution where mass is concentrated at a point, but the Schwarzschild solution describes spacetime outside a spherical distribution of mass (with no electric charge and no rotation). Only the total mass contributes to the metric. And yes, the symmetries make makes this solution relatively simple. The corresponding solution for a rotating distribution of mass wasn't found until 48 years later.

If it's not too much trouble and I will understand if it is, I was hoping for a concrete but simplified example of the mathematics describing General Relativity.

For example, in the 2 dimensional space-time case, what would the three functions for A_ij be for the case of a point mass (or spherical mass if point masses are not allowed) with mass m and distance x from the mass? Would these functions be meaningless to me without some background in differential geometry or could you qualitatively describe what they mean?

While I'm sitting in my chair, am I moving?

There's no coordinate-independent way to define "not moving". Your existence defines a curve in spacetime. That curve can be mapped to a curve in $\mathbb R^4$ by a coordinate system. If the coordinate system maps the curve that represents your existence to a line that's parallel to the time axis, then you're not moving in that coordinate system.

A geodesic is "straight" in a coordinate-independent sense, but a coordinate system can certainly take a geodesic to a strange-looking curve in $\mathbb R^4$.

I understand that I'm being accelerated upwards by my chair but why am I moving downwards? Even if that's a straight path, why would I be moving in that path if it weren't for my chair?

That's an axiom of the theory, and not something that it can explain. Perhaps some other theory will explain it some day, like GR explained the mysterious action at a distance in Newton's theory, but that would of course give us a new set of unexplained axioms.

I think what I'm getting at is that, in the absence of mass, the geodesic would be a point. Why are geodesics in the presence of mass always lines?

A universe without mass, or rather a universe where mass doesn't have any effect on the geometry, is exactly what special relativity is describing. Your existence is a curve in that theory too. (You exist right now...and now too. Those are two different points on that curve). It just happens to be parallel to the time axis of the coordinate system you'd like to use.

I think I get it! Please tell me if you agree with my characterization of what you're saying...

The reason why things are always moving in a General Relativitic sense is because even if we're not moving through space, we are always moving through time! Because masses warp space-time (space and time), translations in time can be transformed into translations in space, hence the appearance of acceleration...

This might also explain why time dilates in a gravity well. Because translations in time are being transformed into translations in space, we're not moving through time as quickly...

Is this approaching reality at all?

Thank you very much!

 

[stevmg]

I hope you are right (post 15) because if you are then I was right, and that doesn't happen too often.

Anybody out there - what does Schwarzschild mean? It is a name but it means something.

Also, what does "going viral" mean? I hear it on TV, etc and I can't even Google it.

 

[Jocko Homo]

Anybody out there - what does Schwarzschild mean? It is a name but it means something.

Karl Schwarzschild was an astronomer and contemporary to Albert Einstein. He's famous for, among other things, deriving the Schwarzschild radius. Given an amount of mass, its Schwarzschild radius is the radius of a sphere of that mass that will collapse into a singularity. Coincidentally, that's the radius for which the escape velocity from the surface of the sphere is the speed of light (at least it was the last time I checked but I'm too lazy to double check it now)...

Also, what does "going viral" mean? I hear it on TV, etc and I can't even Google it.

To "go viral" is to become very popular by word of mouth...

 

[Mentz114]

Jocko Homo,
to go from special relativity, where inertial worldlines are straight, to general relativity where they may be bent requires adopting a geometry that can describe intrinsically curved spacetime. Einstein used Riemannian geometry to find the quantity that characterises the approach of two worldlines by taking the second covariant derivative of the positions wrt time ( ie acceleration) and found that a rank-4 tensor accounts for the acceleration. It is the Riemann tensor, of course. So one could say that the gravitational field which causes the inertial worldlines to be 'differently straight' is entirely described by the Riemann tensor.

To get to grips with this you need to understand tensors and Riemann geometry. Most people find the concept of covariant and contravariant ( raised and lowered indexes) difficult to grasp at first but it's essential. The reason is that the only things that are neither contravariant nor covariant are scalars formed from contractions of tensors. They are also invariant under change of coordinates, a necessary feature of measurabe physical quantities.

 

[stevmg]

Jocko Homo -

Thanks for the fill-in on Schwarzschild ("Black Sign") and "going viral."

It seems that the simplistic and intuitive post with regards to GR you have already done - curved world lines means acceleration.

All the other explanations are going into it deeper than that (sort of like describing the tress and missing the forest.)

 

[Jocko Homo]

Karl Schwarzschild was an astronomer and contemporary to Albert Einstein. He's famous for, among other things, deriving the Schwarzschild radius. Given an amount of mass, its Schwarzschild radius is the radius of a sphere of that mass that will collapse into a singularity. Coincidentally, that's the radius for which the escape velocity from the surface of the sphere is the speed of light (at least it was the last time I checked but I'm too lazy to double check it now)...

I left out an important detail which makes the utterly uninteresting statement above much more interesting...

The Schwarzschild radius of a mass just happens to be the radius of a spherical mass where the escape velocity from its surface is the speed of light according to Newtonian gravity. I'm still too lazy to confirm this so I'll leave it as an exercise to the reader to determine whether I'm correct on this and I apologize if I'm not...