# Why does special relativity exclude gravity?

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1. Apr 13, 2014

### luinthoron

Hello,

I was unable to find a similar thread, so I would like to ask about this myself. I have several textbooks on SR and GR at my disposal but none of them gave me the answer to my question. I remeber from undergraduate course that SR brakes down if we want to include gravitation. I feel that I need to properly understand why before I continue my study of relativity. I think it might somehow go against the relativity principle but I'd really like to see some conrete reasoning behind it. Could you shed some light on it or refer me to a book that treats this topic thoroughly?

Thank you for your (space)time.

2. Apr 13, 2014

### WannabeNewton

3. Apr 13, 2014

### Staff: Mentor

In special relativity two objects which are moving inertially (meaning on-board accelerometers read 0) and are initially at rest with respect to one another will always maintain a constant separation. In the presence of gravity two objects which are moving inertially and are initially at rest with respect to one another may approach and even collide. The structure of SR simply cannot predict that.

4. Apr 13, 2014

### luinthoron

Although I find the discussion in the posted thread interesting, I fail to see the immediate connection with my question. I don't find answers including tensors very helpful at this point. I understand the mathematics and basic principles but the larger physical pictures eludes me.

5. Apr 13, 2014

### WannabeNewton

Well the issue is mostly mathematical-there is nothing deep going on here so if you understand the mathematics then you can just write down the Lagrangian density for a classical field theory of gravity on a flat background (e.g. a scalar or vector theory) and attempt to match the theory to known predictions of gravity. The point is you can have gravitational field theories defined on a flat background, nothing in SR prevents that. They just fail to predict all of the observed properties of the gravitational field. This is all discussed extensively in chapter 3 of "Gravitation: Foundations and Frontiers"-Padmanabhan.

6. Apr 13, 2014

### Geometry_dude

In principle you could take Minkowski spacetime $(\mathbb{R}^4, \eta)$ along with the canonical global chart and just introduce a (covariant) force via $F= \dot p$. Then you could find the flow(=general solution) on the tangent bundle (=velocity phase space) and be happy with your solution. This is how you treat forces in special relativity.

However, what Einstein realized is that treating gravity like this is not physically meaningful - you would feel the acceleration. Imagine you were somewhere in outer space and gravitationally attracted to some massive body. Even if it is very massive, you should not feel the acceleration (neglecting tidal "forces"): You're just falling! However, if an actual force acts on you, you definitely feel its acceleration. For this reason, we needed something else.

Last edited: Apr 13, 2014
7. Apr 13, 2014

### pervect

Staff Emeritus
In special relativity, one assumes that the Lorentz interval can be put into the form:

ds^2 = -dt^2 + dx^2 + dy^2 + dz^2

This implies that space-time is flat. In GR, the expression for the Lorentz interval is different, and it is not assumed that space-time is flat.

The popular notions of "flat" and "curved" aren't very precise, but the precise defintions (such as flatness being the vanishing of the Riemann tensor) are rather technical. It's probalby sufficient to think of a plane as being flat, and the surface of a sphere as being non-flat, for this level of explanation. To be slightly less technical than the Riemann definition (but more precise than saying curved), we can say that we are talking about "intrinsic" curvature, something that can be defined given a local neighborhood of points, and a way of measuring the distances between them, without reference to any particular embedding. See the Wiki on intrinsic curvature, for more.

8. Apr 13, 2014

### haruna

In unison with Mach's principle any coordinate change of flat spacetime resulting in space-time interval other than:

ds^2 = -dt^2 + dx^2 + dy^2 + dz^2 must produce tidal forces ( see Newton and rotating bucket ).

Tidal forces are an immediate outcome of curved space-times, hence GR.

9. Apr 13, 2014

### WannabeNewton

This is entirely incorrect. First of all Mach's principle is not an established principle of physics. Secondly, GR violates Mach's principle in more ways than one. Lastly, and most importantly, coordinate transformations do not introduce tidal forces. I can transform the Minkowski metric into a coordinate system rigidly rotating with uniform angular velocity about a preferred axis and there will still be no tidal forces. Newton's rotating bucket was a demonstration of the fact that a circulating fluid will feel centrifugal forces that pull it up at the edges hence rotation must be absolute-this has nothing to do with tidal forces. Coordinate transformations are gauge transformations of the gravitational field which tidal forces (i.e. space-time curvature) are gauge invariant of.

Again, you can write down the Lagrangian for a field theory of gravity on a flat static background. Nothing in SR prevents you from doing this. A weak gravitational wave in vacuum is in fact a symmetric tensor field propagating on a flat static background with dynamics governed by a simple wave-equation. The problem with gravitational field theories on a flat static background is they don't fit all the observed effects of gravity; one can find the motivation for the description of gravity as a dynamical curved background in chapter 2 of "General Relativity"-Straumann.

10. Apr 14, 2014

### Wes Tausend

I think DaleSpam gave you the most concise and simple answer in post #3, and it is a classic.

Fundamental principles of all theories, SR and GR included, try to boil down the essence of comparing physical properties, and their initial definition must be considered very carefully. GR does not, "somehow go against the relativity principle" of SR, if that is what you are thinking. They are combined to work together.

The other post answers are good treatises on the associated math to form a precise blueprint, but detailed math is not imperative to recognise some important observations and deductions. Some characteristics can be roughly "drawn out on a napkin" (decribed) to get the basic idea. So, in simple terms, what are the differences, besides what DaleSpam said?

First of all, SR deals in velocities in a straight line, or linear direction. Starting with a straight line, what happens as speed increases? Einstein pondered this and came up with a simple, but non-intuitive, logical theory based on other ideas and measurements. This thinking helped simplify an otherwise complicated decription and get that down pat before anything more than straight lines were tackled. I think SR is harder to describe and accept, than the changes that came later by merely curving the proven straight SR relativity lines.

GR tries to explain what happens when bodies, or light, travel in a curve, relative to one another, such as the all important orbits of bodies, or gravitational bending of light. It just so happens Einstein noticed that gravity is an "attraction" acceleration, just like the momentum acceleration of highway curves in an automobile. SR, designed for simple straight lines, cannot adequately describe curves, attractions, or accelerations, if at all. Einstein basically deduced that if orbits follow a curved road, so should light. By experimentation humans found light does follow curved roads. Voila! Einstein's GR theory worked; mass causes space-roads to curve. The coverted SR-math of straight lines could now be successfully applied to gravity-curves of space at various velocities.

I had the best luck reading science books by Isaac Asimov to gain basic understanding. Good luck.

Wes
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11. Apr 14, 2014

### WannabeNewton

SR can more than adequately describe accelerations; it most certainly is not restricted to uniform motion.

12. Apr 14, 2014

### Staff: Mentor

SR handles these just fine. Many people don't realize this because introductory SR texts usually don't cover these problems; the math is appreciably more challenging without contributing any additional understanding of the basic principles so the texts don't go there.

You might want to google for "Rindler coordinates" to find a complete SR treatment of linear acceleration, and "Ehrenfest paradox" will get you started on circular motion and radial acceleration.

The essential difference between SR and GR is that SR limits itself to situations in which spacetime is flat (described by the Minkowski metric) and GR does not. Either way, we can handle any path through spacetime, whether curved or straight, accelerated or not.

13. Apr 16, 2014

### Staff: Mentor

I know Wannabe knows this, but not only is it inconsistent but actually tells you how to fix it up. You will find this approach in Ohanian:
https://www.amazon.com/Gravitation-Spacetime-Hans-C-Ohanian/dp/1107012945

Thanks
Bill

Last edited by a moderator: May 6, 2017
14. Apr 16, 2014

### HomogenousCow

Keep in mind, that QFT is done in an SR compatible framework.
In that light, it kind of makes no sense that accelerations cannot be handled in SR.
The reason fully dynamical motion is not often taught in those algebra-only SR books is because they are best treated in a lagrangian formulation.
The equations of motion in SR (for a particle in a scalar field) are more complicated than those in classical mechanics and contain (four) velocity dependant terms.

15. Apr 16, 2014

### Wes Tausend

Hi Nugatory,

I've taken my time to think it over.

Maybe we can make lemonade out of a lemon here. The OP asks a question and some basic answers seem unnecessarily complicated and one does not. Another naive student (myself) has raised his hand and volunteered a slightly expanded basic answer that is not quite right. Confusion reigns amongst students with widely differing backgrounds. The class can still benefit.

A green student might naively ask, if SR is so comprehensive, why do we even need GR?

Both Rindler (born 1924) coordinates and Ehrenfest paradox(1909) came along after SR but the Ehrenfest paradox before GR. There is little doubt that Einstein was aware of Rindler-like Minkowski space since Minkowski was his former instructor, and that influenced SR. But, keeping in mind the fallibilty of my sources, it seems the following is true.
This indicates to me that Einstein had a very simple mental picture of SR. This Occams Razor view is what a naive student may best start with. What was it?

The beauty of DaleSpams simple, classic answer to the OP in #3 was in that it was axiomatic. It is easy for all to see as the difference of SR vs GR is self-evident as long as, in part, "In the presence of gravity two objects which are moving inertially and are initially at rest with respect to one another may approach and even collide." In keeping it simple, DaleSpam doesn't even expand enough to define gravity as an attraction, so he still allows that any collision caused by acceleration could be gravity, something dear to my heart and the Equivalence Principle.

As Asimov has taught us in The Relativity of Wrong and other wisdoms, it is the weakness of axioms that fool us and the very weakness of axioms is the weakness of human intuition. Intuition never fails us entirely, but may be later supplanted by better intuition. It is the root of SR vs GR that the OP seeks, that foundation we all here continue to seek. What is the simple essence of our world in the fewest words?

Linear is not quite the same as flat and I understand SR does allow for at least acceleration in a straight line, or perhaps flat spacetime acceleration. How can we reword the error(s) in my two short paragraphs up above to remain simple, yet be truthful? In other words, in the spirit of learning, yet keeping it simple, can you explain in pure logical concise prose, why we need GR if SR is so adequate?

Thanks,
Wes
...

16. Apr 16, 2014

### Staff: Mentor

SR covers all the situations in which space-time is flat, but only those situations. Thus, we need GR to cover situations in which space-time is not flat. That includes all situations in which gravitational influences are significantly different from one point to another.

There is a formal mathematical definition of "flat", but you asked for an intuitive one . Here it is (but be aware that from a mathematical point of view it is somewhat bogus): Pick three points in space-time. Draw straight lines ("geodesics", in the lingo) between them to form a triangle. Do the three interior angles of this triangle add up to 180 degrees? If so, you're in a flat space-time (or at least a good approximation of flat).

Last edited: Apr 16, 2014
17. Apr 16, 2014

### WannabeNewton

Unfortunately your post is filled with a lot of unnecessary rhetoric so I'm just going to reply to the above. The simple answer is SR can deal with any and all forms of acceleration be it linear or rotation or combinations thereof. The difference between SR and GR is simply that SR is a theory of a flat static background whereas GR is a theory of a dynamical curved background codifying the notion that nature doesn't provide us with a fixed space-time geometry a priori but rather all the energy-momentum sources in a given system cause the space-time geometry to propagate and in return makes space-time geometry dependent the notions of rotation and acceleration.

GR is obviously much more than a theory of gravity. It is what Wheeler called a theory of "geometrodynamics", one that just also happens to accurately predict observational gravitational effects. To put it conversely, there is nothing in the underlying theoretical framework of SR that prevents one from establishing a relativistic theory of gravity on its static flat background. What weeds them out is experiment. However historically speaking Einstein wasn't led by experiment to GR but rather by purely theoretical considerations partly on the basis of aesthetics (and who can blame him? He gave us what many physicists consider to be the most beautiful theory of physics). In other words we don't need GR, it is just the theory that manages to consistently and accurate agree with experiment through an extremely elegant framework so we adopt it as the proper classical theory of gravity.

18. Apr 16, 2014

### Staff: Mentor

There is little doubt that Einstein was NOT aware of the Minkowski geometric formulation of SR when he developed and published SR in 1905. It's not in the 1905 paper, and it is all over the GR publications a decade later - because it was discovered in between.

Just about all of the mathematical apparatus of modern special relativity was developed after the fact, after other physicists recognized the value of Einstein's insight and started building on his original formulation - this is why you'll occasionally hear Einstein's formulation of SR described as "old-fashioned" or even "obsolete". Much of the modern treatment of SR didn't appear until after the discovery of GR, when people went back to reformulate SR in a way that made it clearer that it was the zero-curvature special case of a more general theory that worked for all values of curvatures including zero.

Last edited: Apr 16, 2014
19. Apr 16, 2014

### Wes Tausend

Can light curve in SR? I would think not, but flat mathematical graphs may/must display a curve. According to WannabeNewton, I must be wrong about light curving within SR, if SR can encompass gravity.

Ok, that makes sense, the triangular enclosed angles on a globe (sphere), nor any other curved surface, do not add up to 180. So... the curve of which we speak is merely the surface of a graph to calculate coordinates, not necessarily the path light must take (and is observed to), which is/forms the real dimension(s) of space?

I first worded it NOT, then decided that it is quite possible some discussion took place in class that might say, for instance, "...and we could add more dimensions if we wish". Even though Einstein made no use of it in SR, he might have heard it before and tucked it in a dusty bin. Pretty speculative I know, but after you so graciously introduced it, I felt I could not positively claim he couldn't have previously heard it in an earlier Minkowski timeline. I used to love little trivia tidbits offered by instructors. It offered additional usefulness for bothering to learn the subject material.

Very good insight! Thanks. I don't always trust pure math (basically a shorthand) as it can, and does let us down at times. Einsteins descriptive thought experiments are more invaluable than numbers to understanding in this way.

It is clear you gave some time and thought to your reply(s). I hope the OP and other readers have gotten as much out of this discussion as I have.

Nugatory, thank you so much for your time and the reply.

Wes
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20. Apr 16, 2014

### Wes Tausend

You have reinforced Nugatory's post with a few more and differently said words and added a cause and effect geometry loop between matter (mass) and energy. Thank you for your effort.

In respect to the rhetoric, my meaning is to emphasize the importance of employing basic pictoral intuition in understanding truth. Numbers and math are useless to reaching understanding without it. To a neophyte, the common suggestion to manipulate higher math does not paint much of a picture, and can deceive men altogether. I much appreciate your worded descriptions.

Asimov once showed how only high school algebra was required to derive E=mc², and that is a beautiful equation. Asimov attempted no such thing with GR that I know of, and the sorrow is that GR is only beautiful to those learned few that can read it.

This eludes me as to how:
"...there is nothing in the underlying theoretical framework of SR that prevents one from establishing a relativistic theory of gravity on its static flat background.

Unrelated, I once thought I could see a form of gravity in SR. I believe it's a fact that whenever one sees V² in a formula, it designates an acceleration. Since E= mc², and c is not just a constant, but a velocity, one could attempt to read the formula as "energy is related to mass by an acceleration". I know this is somehow not proper, but it was appealing at the time.

Thank you for your time and a very thoughtful response.

Wes
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