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## Main Question or Discussion Point

What in the mathematics of the derivation of special relativity limits the model to inertial frames? How is an inertial frame defined in the context of the derivation?

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What in the mathematics of the derivation of special relativity limits the model to inertial frames? How is an inertial frame defined in the context of the derivation?

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Ibix

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Inertial frames are by far the easiest to work with though. And I gather that some older texts do consider non-inertial frames to be in the domain of GR. Modern ones don't, though. Or shouldn't.

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I'm not aware that there IS any such thing as "the derivation of special relativity". Special Relativity is a theory (not an equation) based on two postulates, the first of which (The Principle of Relativity) is that it is talking about things in uniform motion relative to each other (and this generally means inertial frames of reference although as ibix states, it COULD be that two objects are both accelerating but not relative to each other)What in the mathematics of the derivation of special relativity limits the model to inertial frames? How is an inertial frame defined in the context of the derivation?

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Ibix

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What is derived are the Lorentz transformations, and those are defined relative to inertial frames - exactly as the "Galilean transformations" of classical mechanics. Very likely your question is therefore more basic, and belongs in the classical physics forum. Can you answer the question what in the mathematics of the derivation of classical relativity limits the model to inertial frames? How is an inertial frame defined in the context of that derivation?

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Nugatory

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Do you have a source for such a derivation?

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Mister T

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An inertial reference frame is one in which objects at rest remain at rest and objects in motion continue to move in a straight line atspeeteady speed.

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A tensor treatment of special relativity can handle non-inertial frames just fine. So there isn't any real limitation of special relativity itself that prevents it from being applied to non-inertial frames. The limit is knowing tensor mathematics which is needed to handle arbitrary coordinates and to understand the way physical qualities transform under arbitrary coordinate mappings.

Jackson's textbook on electrodynamics, for instance, is a graduate level textbook about electromagnetism that would have the necessary math. However, I'm not sure if Jackson treats accelerating frames specifically - I really don't recall. The basic point is that the mathematical treatment that can handle arbitrary coordinate systems can handle the particular case of "accelerated frames" just fine.

Rindler's book (Relativity, Special and General - or a similar title) is about special and general relativity does have a treatment of accelerating frames.

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It is also true that, if you write SR in terms of generally covariant tensors, it looks very close to GR already. Nevertheless there is a difference between the intertial forces due to using a non-inertial reference frame and the presence of a "true" gravitational field: If you have a non-inertial reference frame in GR of course you can always introduce a global inertial reference frame, and this is the case if the curvature tensor vanishes identically everywhere in the entire spacetime. If a true gravitational field is present in GR, the curvature tensor is no longer identically 0, and you cannot introduce a global inertial reference frame.

The equivalence principle however assumes that you can always introduce a local inertial reference frame at each (regular) point in spacetime. I'd say that's the precise meaning of the equivalence principle and not the usually envoked heuristic arguments to argue why in GR spacetime is a Lorentzian manifold, i.e., a pseudo-Riemannian manifold with a fundamental form of signature (1,3) or equivalentliy (3,1).

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Mister T

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That's a good point. Newton's Laws (within their limits of validity) are valid only in inertial reference frames. Certainly one can use Newton's Second Law to describe motion in non-inertial frames, but that involves introducing forces that violate Newton's Third Law.This is often misunderstood. Even in some (minor) textbooks you can read that SR can't handle non-inertial frames, which is of course wrong. With the same right you can argue that you can't handle non-inertial frames in Newtonian physics, which is of course also wrong.

Isn't part of the confusion of Special Relativity's ability to handle non-inertial frames historical? What I mean is that didn't Einstein, after developing Special Relativity, immediately set to work on what became General Relativity and in the process introduce the formalism of non-inertial frames? Of do I have it wrong?

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The history of Einstein's thought about a relativistic theory of gravity is very interesting indeed. As far as I know, Einstein started as soon as 1907 to think about how to describe gravity relativistically. At this time he wrote a review article about SR, and the only other then known fundamental force than electromagnetis was gravity. So for Einstein it was very natural to think about this question. Legend has it that he came to the idea to make the (weak) equivalence principle the right starting point to study the problem is that in a Berlin newspaper he read about a man who fell from a high roof and he asked himself what he might feel concerning gravity. This brought him to the idea that for a homogeneous gravitational field he can reinterpret the force as an inertial force in a gravitation free environment but described in a non-inertial frame of reference. Then he started to think about non-inertial frames in special relativity and soon discovered that an observer in such a non-inertial frame considers space as non-Euclidean. It took him about 10 years of hard work with progress and setbacks on the way to come to the elegant solution that gravity can be described by a pseudo-Riemannian 4D spacetime manifold with the meaning of the equivalence principle that you can always introduce local inertial reference frames at any point in spacetime (but no global inertial frames if true gravitational fields are present due to the presence of any kind of energy, momentum, and stress distributions due to matter and radiation).

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I would say that, for both SR and Newtonian physics, the laws of motion, as expressed in terms of coordinates, have the simplest form if the coordinates are inertial, Cartesian. If you write down the equations of motion in spherical coordinates, I would say that you're still dealing with Newtonian physics, even though they don't have the same form as for Cartesian coordinates. The business about "forces that violate Newton's Third Law" just means that you have misidentified what the "forces" are.That's a good point. Newton's Laws (within their limits of validity) are valid only in inertial reference frames. Certainly one can use Newton's Second Law to describe motion in non-inertial frames, but that involves introducing forces that violate Newton's Third Law.

With the hindsight given by studying General Relativity, I would say that the "correct" way to formulate Newton's laws is in terms of 4-vectors. If you do that, then the Newtonian equations of motion, as well as the third law, are true in any coordinate system, inertial or not:

[itex]m \frac{dV}{dt} = F[/itex]

With this 4-vector formulation, the "inertial forces" are seen as not forces at all, but as connection coefficients (essentially, the derivatives of the basis vectors).

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The following might help - or not, I'm not sure. But it's worth saying, I hope. In Newtonian physics, the math of transforming to an accelerated frame yields Newtonian physics plus additional "fictitious forces".That's a good point. Newton's Laws (within their limits of validity) are valid only in inertial reference frames. Certainly one can use Newton's Second Law to describe motion in non-inertial frames, but that involves introducing forces that violate Newton's Third Law.

Isn't part of the confusion of Special Relativity's ability to handle non-inertial frames historical? What I mean is that didn't Einstein, after developing Special Relativity, immediately set to work on what became General Relativity and in the process introduce the formalism of non-inertial frames? Of do I have it wrong?

Historically, Einstein realized early on that this wouldn't be the case for special relativity. In particular, we notice effects that we can call "gravitational time dilation" in accelerated frames. The usual argument here is the doppler shift argument. A signal emitted from the stern of an accelerated space-ship will be red-shifted when it reaches the bow, because while the light is travelling, the space-ship is accelerating. Going the other way, the signal is blue-shifted. The philosohical issue is how to combine this with the principle of equivalence. If we have a pair of identical clocks, for specificity imagine atomic clocks, the clocks must "tick" at the same rate as the signals they send out. The signals get doppler shifted, and we are forced to conclude that the rate at which the clocks tick exactly matches the signals, and since the signals are doppler shifted, we wind up concluding that the clocks themselves speed up or slowing down depending on their position when we adopt an accelerating frame.

This isn't just theory, by the way. Tests with the Mossbauer effect show the phenomenon is real, gamma rays emitted from a lower sorce won't "resonate" with the upper source in a gravitational field.

This argument shows the need for a paradigm shift. It's clear that "inertial forces" simply can't explain how clocks tick at different rates depending on their position, we need something more.

I'm not aware of any substitute here for simply doing the math. The two textbook treatments of accelerated frames I'm aware of ([URL='https://www.amazon.com/dp/0716703440/?tag=pfamazon01-20&tag=pfamazon01-20 and MTW's - see links for details) both use tensors. Some of the basic issues can be outlined with simple algebra, as in our example of two clocks at the bow and stern of the acclerating rocket exchanging signals, but usually such treatments are not full enough to give a complete picture of how the accelerated frame works. They are good enough to show that the "inertial force" model of accelerated frames will not be sufficiently general in special relativity, however.

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I think such questions often arise when one is trying to learn special relativity. Unfortunately, I don't think they can be fully answered until after one has learned SR on its own terms in the simpler case of a non-accelerated frame first. After that, one needs a high degree of abstraction - and a willingness to do the math. It would probably be possible to get somewhere without tensors on ones own if one has the neessary ability to do error-free algebra (either on ones own or with modern symbolic algebra packages), and the justified confidence to believe one own's results. But, if one wants the support of the literature, of reading what's been written about the topic, one has the not-insignificant task of learning about tensors to be able to follow the existing treatments.

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DrGreg

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Presumably a typo, but that should be ## \frac{DV}{d\tau}##, where ##\tau## is proper time.[itex]m \frac{dV}{dt} = F[/itex]

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No, it's not a typo. I was talking about Newtonian physics, where [itex]t[/itex] is universal.Presumably a typo, but that should be ## \frac{DV}{d\tau}##, where ##\tau## is proper time.

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The weird thing is using 4-vectors for Newtonian physics, but you have to have a time component to the vector in order for an accelerated coordinate system to be just a coordinate transformation, and in order for "g"-forces to be connection coefficients.No, it's not a typo. I was talking about Newtonian physics, where [itex]t[/itex] is universal.

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