Does Special Relativity Deal with Non-Inertial Frames?

[Logic314]

In several special relativity textbooks, I have read that special relativity only deals with observations made in inertial frames, and that it makes no predictions about observations made in non-inertial frames, and that only general relativity deals with non-inertial frames through the equivalence principle. But this does not make sense to me, for the following reason:

General relativity definitely deals with gravity, and the principle of equivalence links gravity to inertial (a.k.a. fictitious) forces in non-inertial reference frames. But how one derives basic general-theoretic results (such as gravitational time dilation) is by first considering what happens in non-inertial frames, and then using the principle of equivalence to link it to gravity. But what does one use to do the first step of figuring out what happens in accelerating frames (before using the equivalence principle)? By using special relativity of course (and a few reasonable assumptions). One does not need to know the equivalence principle or even know that gravity exists to be able to figure out the laws of physics in accelerating frames.

Special relativity seems (at least to me) to be quite adequate in predicting measurements made in accelerating frames where gravity plays no role. For example, consider two synchronized clocks at rest and separated by a distance L in a given inertial frame. Now, let an observer starting at rest (with respect to this frame) accelerate (in the direction of the line connecting the two clocks) with a constant proper acceleration a for some proper time t, after which the observer stops accelerating, now in a new inertial frame that moves at a constant speed with respect to the two clocks.

We know from special relativity that in this new frame, the "rear" clock is ahead of the "front" clock by a constant time. Therefore, to get ahead of the front clock, the rear clock had to be ticking faster than the front clock relative to the observer during the time of acceleration. It seems reasonable that one can (with perhaps only a few reasonable assumptions) deduce a special-theoretic formula for how much faster the rear clock had to be ticking compared to the front clock (and whether this factor is constant or changes with time during the observer's acceleration).

Therefore, it at least seems to me that special relativity can predict the laws of kinematics and dynamics in all reference frames where gravity plays no role, not just inertial frames.

So I am wondering whether there is something missing in my reasoning, or whether the texts that claim that special relativity is unequipped to deal with non-inertial frames are wrong.

 

[PeterDonis]

In several special relativity textbooks, I have read that special relativity only deals with observations made in inertial frames

If you've read that in a texbook, it was probably an old one. (Some specific examples might help.) Modern textbooks recognize that SR can deal perfectly well with non-inertial frames, as long as spacetime is flat.

I am wondering whether there is something missing in my reasoning, or whether the texts that claim that special relativity is unequipped to deal with non-inertial frames are wrong.

Your reasoning is fine. Any textbooks that say that are wrong.

 

[Dale]

Special relativity seems (at least to me) to be quite adequate in predicting measurements made in accelerating frames where gravity plays no role.

This is correct. The line between SR and GR is gravity, not acceleration.

The grain of truth that makes it a common mistake is that the usual postulates are phrased in terms of inertial frames

 

[FactChecker]

That misconception caught me. The "Twins Paradox" is a well-known example. Intuitively, an inertial frame can independently detect the acceleration of the twin who is not following a straight space-time line during acceleration. So acceleration is not relative, even though it is the derivative of velocity, which is relative (if no acceleration). And a Minkowski diagram can be used to give the correct answer.

 

[haushofer]

If special relativity couldn't deal with non-inertial frames, why would classical mechanics with Galilean relativity be able to do it? Do we need something more then just classical mechanics and Galilean relativity to explain e.g. non-inertial forces?Of course not.

The calculations to transform to accelerating observers, how the equations of motion change etc. etc. are exactly the same in both special relativity and classical mechanics, except for one tiny thing: you just use different coordinate transformations and equations of motion. The sophisticated physicist would say that the difference would be the underlying Lie-algebras/groups: Poincaré v.s. Bargmann.

 

[haushofer]

This is correct. The line between SR and GR is gravity, not acceleration.

The grain of truth that makes it a common mistake is that the usual postulates are phrased in terms of inertial frames

Yes. We should start a petition to all textbookwriters on relativity: put more emphasis on accelerated observers! ;)

 

[stevendaryl]

Yes. We should start a petition to all textbookwriters on relativity: put more emphasis on accelerated observers! ;)

In my opinion, it's more interesting (and useful, since it is good preparation for General Relativity) to learn about noninertial coordinate systems.

In the case of inertial observers, you can sort of associate an inertial coordinate system with an observer. But in the case of accelerated observers, unless the acceleration is of a particularly simple form (such as constant proper acceleration, or circular motion) then there is no obvious unique coordinate system associated with the observer. Except locally.

It's a little presumptuous to second-guess one of the greatest physicists ever, but I think Einstein was a little misled by SR's focus on observers. General Relativity really isn't a generalization to accelerated observers.

 

[FactChecker]

Yes. We should start a petition to all textbookwriters on relativity: put more emphasis on accelerated observers! ;)

It is definitely worth a few comments since the misconceptions are so common.

 

[geordief]

Is there somewhere where I might read a little about how Special Relativity deals with non inertial frameworks?

Not in too much detail as I don't have that ability.

 

[Nugatory]

Is there somewhere where I might read a little about how Special Relativity deals with non inertial frameworks?

Googling for "Rindler coordinates" will get you started. These are coordinates in which a uniformly accelerating observer is "at rest" at the origin and the rest of the universe is accelerating backwards. This may be the mathematically simplest non-inertial frame that also corresponds to an interesting physical situation. Another interesting frame is the one in which a rotating observer is "at rest" while the rest of the universe rotates about them - that one is generally harder to work with.

The general concept is fairly straightforward: You have your $x$, $y$, $z$ and $t$ coordinates which define an inertial frame in the flat spacetime of special relativity; define new coordinates using functions of these and you have defined a new frame. For example, if you use the Lorentz transformations as the functions that give the new coordinates as functions of the old, you will find that you have defined a new inertial frame moving at speed $v$ relative to the first. If you use some other functions (and if certain other mathematical niceties are respected) you can define a non-inertial frame.

Although the general concept is fairly straightforward, in practice the math tends to get messy quickly, which is why most introductory texts don't cover applications to non-inertial frames... and this in turn leads to the very widespread misunderstanding that it can't be done.

 

[geordief]

Thanks Nugatory.I will do that.

 

[pervect]

In my opinion, it's more interesting (and useful, since it is good preparation for General Relativity) to learn about noninertial coordinate systems.

I mostly agree, but I think that is already being done to some extent.

A full treatment of generalized coordinate systems such as non-inertial coordinate systems leads (IMO) to tensors.

Tensors are an important tool that can be introduced in several contexts, typically the context are special relativity and electromagnetism. Once some familiarity with tensors and tensor methods has been obtained, these techniques provide a natural language for the description of General relativity. However, this is a graduate level treatment.

There is a competing effort to make General relativity understandable at an undergraduate level, without tensors, I'm not sure how successful these efforts have been.

In line with this non-tensor effort, some non-tensor treatments of accelerating observers in special relativity could be useful. And I think such treatments are mostly lacking. But I'm not sure if such non-tensor treatments could really be used to describe a frame of reference.

IMO, the whole idea of a "frame of reference" is usually based on the concept of basis vectors. And tensors are the natural approach to describing basis vectors, and how they are managed in generalized coordinates. But this whole approach is at the graduate-level, and it may be unrealistic to expect everyone to be able to use it, even if it's probably the most powerful approach.

 

[PAllen]

I don’t know what is current practice, but when I started a physics major at a top university, most freshman physics majors were prepared to skip calculus and were prepared for differential geometry by sophomore year. This was in 1971.

 

[vanhees71]

Yes. We should start a petition to all textbookwriters on relativity: put more emphasis on accelerated observers! ;)

Well, first of all hit them on their fingers when they introduce ideas about "relativistic masses" or non-scalar notions of temperature (well, I'd be a bit shy to hit Planck and Pauli on their fingers, but in this case, it would be well deserved! ;-)). Maybe, a good idea is to read

W. Rindler, Relativity, special, general, cosmological, 2nd ed., Oxford University Press (2006)

 

[vanhees71]

I don’t know what is current practice, but when I started a physics major at a top university, most freshman physics majors were prepared to skip calculus and were prepared for differential geometry by sophomore year. This was in 1971.

Well, in 1971... I think nowadays we have to start with high-school math to begin with (at least here in Germany). What's called "math" at high school is pretty much a shame, and it's not even clear what you can expect the incoming freshmen students to know concerning math. I always ask the students, whether they know from high school what I think they should know, because I extrapolate from what I knew when I came from high school (geting my "Abitur" in 1990), and quite often they tell me that they don't know what I think they should at least have seen at high school. E.g., there are some students who already know complex numbers, many don't; all students know basic one-variable calculus of real functions, but whether or not they can take derivatives of functions or even do the standard integrals using substitution, integration by parts etc. is not guaranteed. I'm very sure, it's not the students' fault, but a failed educational system. A lot of this desaster started (at least in Germany) with the "PISA shock", triggering a lot of confusing political acitivity trying to "reform" the high-school teaching strategies, using more or less sensical dicactical ideas. One of the worst ideas is the socalled "competence concept". Rather than teaching "knowledge" one has to somehow transfer "competences". However, this is an euphemism for teaching math as collection of recipies to solve some standard problems without really understanding the math behind it. In this way math becomes a set of instructions to gain some "competences" by rote learning the solution of standard problems, which for me is the opposite to a good math education, which first of all should convey the fun of thinking yourself and understand the concepts to attack a problem in a creative way. The result is precisely what we see in the decline of the pre-knowledge of our incoming freshmen students!

 

[Aleberto69]

This is correct. The line between SR and GR is gravity, not acceleration.

The grain of truth that makes it a common mistake is that the usual postulates are phrased in terms of inertial frames

Hello,
Apologise entering in the midlle of the discussion, however I tought that it would have been more appropriate than asking similar things in a new thread.
Well, I'm very interested on the topics on SR applying in flat spacetime and the misconception that it doesn't apply for not IRF.
However I'm struggleing a lot applying the right concepts, namely for the reason and the point that Dale raised.

Could someone provide then the rigth formulation of the SR postulates not limiting them to IRF?
Coud someone provide the espression for Lorentz transformation in the general case of a not costant velocity of one frame respect to another one?(e.g one frame is IRF and the second one is moving with some non constant velocity $\vec v(t)$ relatively to it)

 

[stevendaryl]

Could someone provide then the rigth formulation of the SR postulates not limiting them to IRF?

This might be unsatisfying, but if you have a description of physics in inertial coordinates, you can get a description in noninertial coordinates by performing a coordinate transformation. It's exactly analogous to the case with Newtonian physics. The most straightforward way of interpreting Newton's 3 laws of motion are using Cartesian coordinates, because in those coordinates, an object that has no forces acting on it travels so that it has zero coordinate acceleration. But obviously, if you have a description of physics in terms of the Cartesian coordinates $x,y,z$, then you can transform to a coordinate system $r, \theta, \phi$ using the transformation:

$x = r sin(\theta) cos(\phi)$
$y = r sin(\theta) sin(\phi)$
$z = r cos(\theta)$

You don't need any additional laws of physics to describe motion in spherical coordinates, you just need calculus.

The same thing is true in Special Relativity. The most straight-forward way to describe Einstein's theory of Special Relativity uses inertial coordinates $x,y,z,t$. But if you have a description in terms of those coordinates, all it takes is calculus to get a description in terms of noninertial coordinates $X,Y, Z,T$ defined by the transformations:

$x = X cosh(\frac{gT}{c})$
$t = \frac{X}{c} sinh(\frac{gT}{c})$
$Y = y$
$Z = z$

 

[stevendaryl]

This might be unsatisfying, but if you have a description of physics in inertial coordinates, you can get a description in noninertial coordinates by performing a coordinate transformation.

Alternatively, you can say that SR is the special case of General Relativity in which the curvature is zero everywhere. (There might need to be another condition, such as spacetime being simply-connected)

 

[Aleberto69]

Hi stevendaryl,
thanks for your reply however I haven't understood how it answers my questions:
How are the SR postulates better formulated including NIFRs?
Lorentz transformation for NIRFs?

Furthermore I do not understand your transforming formulas.
What is "g" representing?

 

[PeterDonis]

Could someone provide then the rigth formulation of the SR postulates not limiting them to IRF?

If you use the definition @stevendaryl gave in post #18, that SR is the special case of GR where the curvature of spacetime is zero everywhere, that statement does not require any choice of frames at all. Another way of saying the same thing that might be more intuitively recognizable is to say that SR applies when tidal gravity is zero.

Coud someone provide the espression for Lorentz transformation in the general case of a not costant velocity of one frame respect to another one?

There isn't one. Lorentz transformations only work between inertial frames moving at constant velocity relative to one another. For other frames, you need more general coordinate transformations, and there is no simple rule for how those work; it depends on the particular choice of coordinates. The only thing all such transformations have in common is that they are diffeomorphisms.

 

[stevendaryl]

Hi stevendaryl,
thanks for your reply however I haven't understood how it answers my questions:
How are the SR postulates better formulated including NIFRs?
Lorentz transformation for NIRFs?

Lorentz transformations are just coordinate transformations. To generalize them, you just use any coordinate transformation, whatsoever.

As far as the postulates of SR, as I said, General Relativity plus the additional assumption that spacetime has zero curvature everywhere and is simply connected. Instead of "light has speed c in every inertial reference frame", it's "light travels along null geodesics".

Furthermore I do not understand your transforming formulas.
What is "g" representing?

$g$ is just a constant of dimensions "meter/second²".

In the coordinate system $X,Y, Z, T$:

1. An observer at constant $X, Y, Z$ will "feel" a pseudo-gravitational field that varies as a function of $X$. $\frac{c^2}{X}$ is the proper acceleration required to keep at a constant value of $X$ (think of it as the force required to keep an object from falling in a gravitational field).
2. A standard clock at constant position $X$ will show elapsed time $\tau$ given by: $\Delta \tau = \frac{gX}{c^2} \Delta T$. So clocks with larger values of $X$ run faster.
3. There is a kind of "event horizon" at the position $X = 0$. No object can remain at rest at this location (it would require infinite acceleration) and it is impossible for objects beyond the event horizon to ever cross to the side with $X > 0$.
4. If you drop an object, it will "fall" toward the event horizon at $X=0$, but it will take an infinite amount of coordinate time $T$ to reach it. (On the other hand, a clock dropped will reach the event horizon in a finite amount of elapsed time. The clock rate $\frac{\Delta \tau}{\Delta T}$ gets slower and slower as the clock approaches the horizon.

All of these properties of the noninertial coordinate system just follow from SR and calculus.

 

[Dale]

Could someone provide then the rigth formulation of the SR postulates not limiting them to IRF?

I warn you in advance that this may not be particularly helpful, but the clearest way to state the physics of SR without reference to inertial frames is as follows:

Spacetime is a flat pseudo-Riemannian manifold of signature (-+++)

This statement says nothing about inertial reference frames, but from this statement you can derive the usual postulates and the Lorentz transform. Also, from this statement you can take any specific non-inertial reference frame and describe physics directly in that frame without reference to any other frame.