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I started on my own learning about GR and SR two months ago, and I still do not have clear if it is possible or not. The following example was explained to me by someone who affirmed: "SR applies only on inertial reference frames":

Let's imagine we have two different reference frames : A' and A. Reference frame (RF) A' is moving with constant velocity (v), meanwhile RF A has no velocity (A' moves relative to A with constant v).

RF A' has a wire underneath and RF A has an aerial above. When both interact, clocks start running in both RFs (clock A' and clock A) and a light ray emerges (from the wire-aerial interaction and with the same velocity vector direction RF A' has).

Then we agree distance can be determined from both RFs.

i.e. : x = x' + vt'

Then I asked myself: why would not be correct consider the case where A' is an accelerated RF and distance is determined from RF A (i.e.) as x = x' + at'?

My doubts about if "SR applies only on inertial reference frames" sentence was true increased when I checked out more sources and they affirmed accelerated reference frames were possible in SR.

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3 Answers 3

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The claim that a certain physical theory "applies only in inertial reference frames" is not even logically possible. "Physical theories" describe physical quantities, which by definition are independent of one's reference frame or choice of coordinates. At most, one could claim that "many of the equations found in standard textbooks on SR only apply to inertial references frames," which is indeed the case.

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    $\begingroup$ A common claim related to "SR only applies in inertial reference frames" is "SR only applies to particles that do not experience any net force" - i.e. particles moving at constant velocity in inertial reference frames. This claim is at least logically possible, but is incorrect. Four-fources and four-accelerations are perfectly valid and well-understood concepts in SR. $\endgroup$
    – tparker
    Aug 4, 2017 at 2:30
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    $\begingroup$ Okey, so definitely it is possible to work with accelerated reference frames in SR. $\endgroup$
    – JD_PM
    Aug 4, 2017 at 14:33
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    $\begingroup$ @JD_PM Yep, that's right. $\endgroup$
    – tparker
    Aug 4, 2017 at 17:43
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It is possible to use accelerated reference frames in special relativity. It is more advanced than many undergraduate texts cover. But, see for example chapter 7 of "Special Relativity", A.P. French, CRC Press, 1968. There it is shown that the direction of the acceleration is not necessarily equal to the direction of the force applied to a moving body. This is done using accelerated reference frames.

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There is something called The Clock Hypothesis that lets you deal with accelerated frames by assuming that you can break an accelerated frame into infinitesimal inertial frames, each of them measuring the proper time. Some authors like Goldstein in his Classical Mechanics book refer to it like an "stratagem" to deal with non inertial frames so it's not one of the postulates, but that's the usual way to deal with them on SR.

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    $\begingroup$ I don't see how the clock hypothesis relates to the question. It simply states that the line element has the form usually assumed, with no additional explicit dependence on acceleration. $\endgroup$
    – user4552
    Aug 4, 2017 at 4:05
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    $\begingroup$ Isn't that assumption that the line element has no explicit dependence on acceleration what actually let's you define infinitesimal intertial frames along the path and ultimately deal with the fact that the frame is not inertial? $\endgroup$
    – David Leonardo Ramos
    Aug 4, 2017 at 11:46
  • $\begingroup$ link, talking about non inertial frames on SR. $\endgroup$
    – David Leonardo Ramos
    Aug 4, 2017 at 12:04

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