# Incorporation of special relativity into general relativity

I am writing a paper about SR and GR. I have the following:

Special relativity deals with inertial reference frames and flat space. General
relativity deals with accelerating reference frames and curved space. Inertial
reference frames are only an approximation that applies in a region that is small
enough for the curvature of space to be negligible, so the motion is rectilinear
and the rate of change of velocity can be considered to be 0.

So how do we incorporate SR into GR on a more general level (curved space,
acceleration, circular motion)? Empirically, we should be able to do this by
taking the SR equations from flat space and expressing them in tensor form.
Tensors possess the property that if they have a certain value in flat space,
then they will have the same value in any other coordinate system.

.....

....

.....

Is my interpretation correct? "rate of change of velocity can be considered to be 0" bothers me.

I am trying to get a fundamental understanding without theheavy math.

Thanks

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## Answers and Replies

WannabeNewton
Science Advisor
Special relativity deals with inertial reference frames...

This is incorrect. SR can deal with any and all frames.

Empirically, we should be able to do this by taking the SR equations from flat space and expressing them in tensor form...

SR equations are already in tensor form.

I am trying to get a fundamental understanding without theheavy math.

Unless you're Feynman that isn't possible I'm afraid.

Nugatory
Mentor
I am writing a paper about SR and GR. I have the following:

Special relativity deals with [STRIKE]inertial reference frames and[/STRIKE] flat space. General
relativity deals with [STRIKE]accelerating reference frames and[/STRIKE] curved space....

So how do we incorporate SR into GR on a more general level

SR is already incorporated into GR. If you take the equations of GR, which work for all curvatures including zero, and set the curvature to zero what pops out is SR. That's what makes SR "special" and GR "general" - SR only works for the special case of zero curvature while GR works for the general case of any curvature.

Both forms of the theory are applicable to both accelerated and inertial frames; it's gravity that makes the difference, not acceleration. If gravitational effects are present the spacetime will be curved and we have to use GR. If they are absent or weak enough to ignore, the spacetime will be flat and we can use SR.

Nugatory
Mentor
Tensors possess the property that if they have a certain value in flat space,
then they will have the same value in any other coordinate system

No. Flat space isn't a coordinate system so speaking of "flat space or any other coordinate system" doesn't make sense.

An equation written using tensors will have the same form no matter what coordinate system we use, but even in flat space we can use different coordinate systems, such as Cartesian or polar.

An example: if I'm using Cartesian coordinates in flat space, I can write the dot-product of two vectors ##A\cdot{B}## as ##A_xB_x+A_yB_y+A_zB_z##, but if I'm using polar coordinates I need a completely different and more complicated formula. Or I can write the dot-product using tensorial notation and the formula will be the same no matter what coordinate system I'm using: ##A\cdot{B}=g(A,B)=g_{ij}A^iB^j## works for all coordinate systems.

...Special relativity deals with inertial reference frames and flat space. General relativity deals with accelerating reference frames and curved space....

I must admit the proposed answers surprised me a lot:

This is incorrect. SR can deal with any and all frames...

So far I believed that SR, in particular the Lorentz transformation, operates between two inertial reference frames: it “translates” the value of physical quantities expressed in relation to an IRF (typically the coordinates of an event) into the value of the same physical quantities (the coordinates of the same event) expressed in relation to another IRF which is, by definition, in constant relative motion in respect to the first IRF. Is this incorrect?

... Both forms of the theory are applicable to both accelerated and inertial frames; it's gravity that makes the difference, not acceleration. If gravitational effects are present the spacetime will be curved and we have to use GR. If they are absent or weak enough to ignore, the spacetime will be flat and we can use SR.

IMHO SR can deal with any physical scenario provided it analyzes the scenario in relation to an inertial reference frame: by inception SR descriptions of the world are related to an IRF, it being noted that this is nothing else than an arbitrary convention about which ones among non-accelerated objects are assumed to be at rest. This constraint being met, SR can deal with objects and observers which are in non-inertial motion in respect to the coordinate system of the selected IRF. Is this incorrect?

Many threads in this SR forum refer to IRFs and never mention "flat space" at all. May be “flat space” refers to the linearity of the Lorentz transformation in respect to space and time quantities? In which case the expression "flat space-time" is equivalent to stating that SR deals with "inertial reference frames" as suggested by the OP... May be I missed something essential.

pervect
Staff Emeritus
Science Advisor
If I might suggest, a better theme for the paper would be the expansion of special relativity into general relativity, rather than the incorporation of SR into GR. This is how it happened historically, and it makes a lot more sense to present it that way.

In GR, to quote Misner, Thorne, Wheeler in "Gravitation", with regards to GR: "The geometry of space-time is locally Lorentzian. This is in contrast to SR, where the geometry of space-time is globally Lorentzian.

If this seems confusing, an analogy with a plane may help.

The geometry of a plane is globally Euclidean. The plane has the ordinary flat Euclidean spatially geometry everywhere.

The surface of a sphere is locally Euclidean. The surface of a sphere doesn't have an ordinary flat Euclidean spatial geometry - for instance, Euclid's parallel postulate doesn't hold.

However, the geometry of the surface of a sphere is locally Euclidean. If you take a small region of the sphere (like, say, your local town), the curvature effects are insignificant and the geometry of the small region , is "locally Euclidean", even though it's not globally Euclidean (because the sphere isn't flat).

WannabeNewton
Science Advisor
So far I believed that SR, in particular the Lorentz transformation, operates between two inertial reference frames: it “translates” the value of physical quantities expressed in relation to an IRF (typically the coordinates of an event) into the value of the same physical quantities (the coordinates of the same event) expressed in relation to another IRF which is, by definition, in constant relative motion in respect to the first IRF. Is this incorrect?

No but nothing in your statement implies that SR cannot deal with non-inertial reference frames.

IMHO SR can deal with any physical scenario provided it analyzes the scenario in relation to an inertial reference frame: by inception SR descriptions of the world are related to an IRF, it being noted that this is nothing else than an arbitrary convention about which ones among non-accelerated objects are assumed to be at rest. This constraint being met, SR can deal with objects and observers which are in non-inertial motion in respect to the coordinate system of the selected IRF. Is this incorrect?

Yes very, very much so. There are quite a few misconceptions hiding in there that only a good textbook can eradicate. This kind of mindset is a result of poor pedagogy in typical introductory SR books. I would recommend reading the first six chapters of MTW. Better yet check out this book: https://www.physicsforums.com/showthread.php?t=730724

Nugatory
Mentor
Maybe “flat space” refers to the linearity of the Lorentz transformation in respect to space and time quantities? In which case the expression "flat space-time" is equivalent to stating that SR deals with "inertial reference frames" as suggested by the OP

"Flat" means "not curved", which is to say free of significant gravitational influences. This is different from"inertial". You can look to the Rindler solution for an example of a non-inertial frame in flat space-time; and to a free-falling observer in a gravitational field for an example of a (locally) inertial frame in curved space-time.

DrGreg
Science Advisor
Gold Member
It seems to me that relativity divides into three parts, rather than the two (SR & GR) that everyone refers to.

Loosely speaking they are:
• inertial frames (or to be more precise, Minkowski coordinates) without gravitation (flat spacetime)
• non-inertial coordinates without gravitation (flat spacetime)
• non-inertial coordinates with gravitation (curved spacetime)
Everyone agrees that (A) is special relativity and (C) is general relativity, but there has been some confusion over (B). The modern consensus among professional relativists is that (B) is part of special relativity, but I don't think that was always the case historically, and I think many beginners in relativity today still think that (B) is part of general relativity. I've noticed, for example, that some Wikipedia articles about relativity topics state that they are restricting themselves to SR only, when the truth is they are restricting themselves to what I've called topic (A).

If you have no interest in gravity then you never need to go beyond (A), as everything in SR can be analysed using inertial frames.

The point of (B) is really to act as stepping-stone from (A) to (C). (B) is more difficult than (A) but still easier than (C), and almost everything you learn in (B) can be modified to work in (C). In fact, in my opinion, the jump from (A) to (B) is more difficult than the jump from (B) to (C).

Bill_K
Science Advisor
cf you have no interest in gravity then you never need to go beyond (A), as everything in SR can be analysed using inertial frames. The point of (B) is really to act as stepping-stone from (A) to (C).
Wow. Thank you for reminding me to take my pressure pill. Nothing could be further from the truth, DrGreg. HallsofIvy
Science Advisor
Homework Helper
Tensors possess the property that if they have a certain value in flat space, then they will have the same value in any other coordinate system.
No, tensors have the property that if a tensor equation is true in a given coordinate system, then it is true in any other coordinate system in that same space.

If you change coordinates, even in flat space, a tensor will change "value". But if A= B in a given coordinate system, A= B in any other coordinate system. But A and B may have changed value as long as they changed in the same way.

WannabeNewton
Science Advisor
If you have no interest in gravity then you never need to go beyond (A), as everything in SR can be analysed using inertial frames.

I have to disagree slightly. This is sort of like saying there's no point in learning about non-inertial frames in Newtonian mechanics. In both SR and Newtonian mechanics non-inertial frames play very important roles both conceptually and computationally. Why would I want to expend unneeded effort and time in deriving the Thomas precession of a gyroscope in circular orbit by using a background global inertial frame when I could do it very swiftly and more elegantly using the rest frame of the gyroscope or in deriving the departure from idealness of a uniformly accelerating light clock by using a background global inertial frame when I could derive it in a couple of lines using the rest frame of either of the two mirrors composing the light clock?

DrGreg
Science Advisor
Gold Member
If you have no interest in gravity then you never need to go beyond (A), as everything in SR can be analysed using inertial frames.
I have to disagree slightly. This is sort of like saying there's no point in learning about non-inertial frames in Newtonian mechanics. In both SR and Newtonian mechanics non-inertial frames play very important roles both conceptually and computationally. Why would I want to expend unneeded effort and time in deriving the Thomas precession of a gyroscope in circular orbit by using a background global inertial frame when I could do it very swiftly and more elegantly using the rest frame of the gyroscope or in deriving the departure from idealness of a uniformly accelerating light clock by using a background global inertial frame when I could derive it in a couple of lines using the rest frame of either of the two mirrors composing the light clock?
You make a fair point and I guess I oversimplified the situation. I agree there are some situations where non-inertial coordinates provide a more elegant solution. But I think I'm right to say you could, nevertheless, restrict yourself to considering only inertial coordinates if you wanted to. It's swings and roundabouts. Non-inertial coordinates can certainly simplify some problems but you have the overhead of learning to use them first.

So, yes, the situation isn't quite as black-and-white as my last post suggested. I retract my sentence quoted above.

PeroK
Science Advisor
Homework Helper
Gold Member
2020 Award
I am writing a paper about SR and GR.

It looks like your plan of asking the physicists about it has backfired somewhat!

Perhaps a sort of meta-relativity is at work: in the same way that different observers will disagree about the time and position of events, physicists will disagree about the structure of the theory. As simultaneity is relative in spacetime, the boundary between SR and GR depends on the reference frame of the observer.

pervect
Staff Emeritus
Science Advisor
It looks like your plan of asking the physicists about it has backfired somewhat!

Perhaps a sort of meta-relativity is at work: in the same way that different observers will disagree about the time and position of events, physicists will disagree about the structure of the theory. As simultaneity is relative in spacetime, the boundary between SR and GR depends on the reference frame of the observer.

That's more than a bit exaggerated, in my opinion. The only direct argument I see is one about semantics. There are often arguments over semantics , even among non-physicists (English Majors, for example).

I'd hate to see people avoid noticing the broad agreement about the basics (such as the fact that GR grew out of SR, and not the reverse) by focusing their attention on minor issues. Of course I don't have any control over what other people focus on, learn, or do not learn.

It seems to me that the reason something like this can cause the confusions of the OP is the way SR is often taught. Some very interesting physical effects, e.g. time dilation or length contraction, are derived in a very simple way. Then the concept of the interval and Lorentz transformations are presented, almost as an afterthought. I think it's easy for newcomers to not see the fundamental shift that SR introduces: the concept of physics in (time + 3D space) is replaced with 4D physics. It's easy to miss that inertial frames are nice because they provide a convenient way to globally slice up spacetime into space + time, and demonstrate things like time dilation, but there's nothing fundamental about them.

To me it's almost, except for the signature difference, like teaching 3D Euclidean geometry by focusing on Cartesian coordinates then looking at various ways of slicing up the 3D space into xy-planes, say by rotating the original coordinates about the origin. Then focusing on how the z-component of vectors change and how the length of the vector projected into the xy-planes change and one changes the z-axis/xy-planes. Then later briefly talking about vectors as 3D geometrical objects, "by the way the length of a vector is independent of the particular way one chooses the z-axis".

I wouldn't necessarily change the order of presentation in SR, but I would spend more time emphasizing the 4D end result. I've seen a lot of people miss that. Not only would the students' understanding of SR be enhanced, it would make the migration to GR concepts trivial (obviously still a lot to learn about curvature, parallel transport, stress-energy, ...).

As a quick example of former, consider the twin paradox. Once one understands the consequences of the minus sign in the metric in relation to the length of a world line and can relate the length of a world line with the time elapsed on a clock that travels on that world line, the paradox is trivially resolved. The role of acceleration is also clear, i.e. making the line not straight.

stevendaryl
Staff Emeritus
Science Advisor
It seems to me that relativity divides into three parts, rather than the two (SR & GR) that everyone refers to.

Loosely speaking they are:
• inertial frames (or to be more precise, Minkowski coordinates) without gravitation (flat spacetime)
• non-inertial coordinates without gravitation (flat spacetime)
• non-inertial coordinates with gravitation (curved spacetime)

I would actually say that there are 4 parts, rather than 3:

• inertial Cartesian coordinates in flat spacetime.
• non-inertial coordinates in flat spacetime.
• non-inertial coordinates in a fixed curved spacetime.
• dynamic spacetime.

There are two ways that GR goes beyond SR. One is allowing curved spacetime, and the second is allowing the curvature itself be dynamic. In a lot of applications of GR only involved curved spacetime that is fixed; that is, we take into account the effect of spacetime curvature on the equations of motion, but we don't take into account of matter and energy on spacetime curvature.

Nugatory
Mentor
I would actually say that there are 4 parts, rather than 3:

• inertial Cartesian coordinates in flat spacetime.
• non-inertial coordinates in flat spacetime.
• non-inertial coordinates in a fixed curved spacetime.
• dynamic spacetime.
I don't buy the difference between the first two: it's like suggesting that classical mechanics done in polar coordinates for planetary orbits is somehow different than classical mechanics done in Cartesian coordinates on the surface of the earth. I don't buy the step to the third one because its skips over a very important case: locally inertial coordinates in a curved spacetime.

Putting all of this together, I think it's a mistake to try categorizing physical theories according to the coordinate systems we choose use when applying the theories; trying to do so obscures the underlying physics. For example:

There are two ways that GR goes beyond SR. One is allowing curved spacetime, and the second is allowing the curvature itself be dynamic. In a lot of applications of GR only involved curved spacetime that is fixed; that is, we take into account the effect of spacetime curvature on the equations of motion, but we don't take into account of matter and energy on spacetime curvature.

That I will buy, and happily. But note that you're invoking no coordinate systems in describing either of these ways of extending beyond the flat and static spacetime of SR; both non-flat and non-static are attributes of the spacetime, not any particular coordinate system.

stevendaryl
Staff Emeritus
Science Advisor
I don't buy the difference between the first two: it's like suggesting that classical mechanics done in polar coordinates for planetary orbits is somehow different than classical mechanics done in Cartesian coordinates on the surface of the earth.

I'm not saying that it's different physics involved, but it requires different tools and skills.

I don't buy the step to the third one because its skips over a very important case: locally inertial coordinates in a curved spacetime.

Hmm. If you are only dealing with local phenomena, then there is no difference between flat spacetime and curved spacetime. Spacetime curvature only comes into play when you are dealing with a large enough region of spacetime that you can't use locally inertial coordinates to describe the whole region.

Putting all of this together, I think it's a mistake to try categorizing physical theories according to the coordinate systems we choose use when applying the theories; trying to do so obscures the underlying physics.

Mastering GR is not just a matter of learning a physical theory. It's also a matter of learning the mathematical tools for working with that theory.

WannabeNewton
Science Advisor
Hmm. If you are only dealing with local phenomena, then there is no difference between flat spacetime and curved spacetime. Spacetime curvature only comes into play when you are dealing with a large enough region of spacetime that you can't use locally inertial coordinates to describe the whole region.

This certainly isn't true and the misunderstanding is in part a result of a cursory use of the term "local" by standard physics books and in part due to an incorrect use of locally inertial coordinates. This can be summed up by saying that "locally inertial coordinates" and the "local equivalence of SR and GR" are only valid at any given event in space-time and only for first-order measurements. Obviously a second-order measurement at a single event, such as that of the Riemann curvature at said event, will yield a non-vanishing value in a curved space-time but a vanishing value in flat space-time. There is a wealth of difference in local phenomena between flat space-time and curved space-time mainly due to second-order measurements and effects.

Putting all of this together, I think it's a mistake to try categorizing physical theories according to the coordinate systems we choose use when applying the theories; trying to do so obscures the underlying physics.

Exactly. I couldn't have said it better myself.

atyy
Science Advisor
The way SR is incorporated into GR is via the principle of equivalence. This is the comma to semicolon rule, or the principle of minimal coupling. There are various different statements of the equivalence principle, and not all are equivalent to each other.

Good discussions of the equivalence principle, and their limitations, are found in

Blandford and Thorne, section 24.7
http://www.pma.caltech.edu/Courses/ph136/yr2011/1024.1.K.pdf

Casola, Liberati, Sonego
http://arxiv.org/abs/1310.7426

stevendaryl
Staff Emeritus
Science Advisor
This certainly isn't true and the misunderstanding is in part a result of a cursory use of the term "local" by standard physics books and in part due to an incorrect use of locally inertial coordinates.

It's not a misunderstanding. I was just being insufficiently precise.

This can be summed up by saying that "locally inertial coordinates" and the "local equivalence of SR and GR" are only valid at any given event in space-time and only for first-order measurements. Obviously a second-order measurement at a single event, such as that of the Riemann curvature at said event, will yield a non-vanishing value in a curved space-time but a vanishing value in flat space-time. There is a wealth of difference in local phenomena between flat space-time and curved space-time mainly due to second-order measurements and effects.

But is it really true that you can measure second-order effects in an arbitrarily small region of spacetime? For example, there is a difference between a flat plane and a section of a sphere that is measurable: If you draw a triangle on a plane, the angles add up to 180. But on the surface of a sphere, the angles add up to greater than 180 degrees. So that's a measurable difference.

However, when you take into account finite accuracy for distant and angle measurements, the distinction can become unobservable. In a small enough region, the angles of a triangle will be so close to 180 that the difference would be unmeasurable.

So I'm a little skeptical of your claim that second order effects are important, even in a small region of spacetime.

PAllen
Science Advisor
It's not a misunderstanding. I was just being insufficiently precise.

But is it really true that you can measure second-order effects in an arbitrarily small region of spacetime? For example, there is a difference between a flat plane and a section of a sphere that is measurable: If you draw a triangle on a plane, the angles add up to 180. But on the surface of a sphere, the angles add up to greater than 180 degrees. So that's a measurable difference.

However, when you take into account finite accuracy for distant and angle measurements, the distinction can become unobservable. In a small enough region, the angles of a triangle will be so close to 180 that the difference would be unmeasurable.

So I'm a little skeptical of your claim that second order effects are important, even in a small region of spacetime.

A cute example pointed out by Ohanian is the ratio of longer to shorter axis of a free falling drop of perfect fluid. In tidal gravity, this ratio is constant in the limit as drop size goes to zero, and measures one particular curvature scalar.

WannabeNewton
Science Advisor
But is it really true that you can measure second-order effects in an arbitrarily small region of spacetime?

Just measure the rate of change of the quantities in the kinematical decomposition ##\nabla \xi^{\flat} = \omega + \theta - \xi^{\flat}\otimes a^{\flat}## of a future directed unit time-like vector field ##\xi##. There are well known standard formulas relating the rate of change of these quantities to the Weyl and Ricci curvature. PAllen gave an example of such a measurement.

I would recommend reading "Classical Measurements in Curved Space-Times"-Bini and Felice.

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stevendaryl
Staff Emeritus
Science Advisor
A cute example pointed out by Ohanian is the ratio of longer to shorter axis of a free falling drop of perfect fluid. In tidal gravity, this ratio is constant in the limit as drop size goes to zero, and measures one particular curvature scalar.

Thanks, that's a good answer.