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I am so much confused about several things of the special relativity.

First, in the flat spacetime, are the available frames of reference all inertial frames? I think that because inertial frames are defined as frames without gravity and the flat spacetime is not curved, the frames of reference for the flat spacetime must all be inertial. Am I correct?

Second, are all inertial frames of reference connected by Lorentz transformations? I cannot find a way to connect an inertial frame 'translated' by a fixed amount from another inertial frame by Lorentz transformations. If there exist inertial frames that are not connected by Lorentz transformations, then is it possible to guarantee that physics law does not depend on coordinates?

Third, in special relativity, the force and acceleration are handled in a given inertial frame. However, then, how should I deal with the frame of the particle that is accelerating in the structure of special relativity? The accelerating particle's frame is certainly not inertial...

I am extremely confused and frustrated due to the questions above. Could anyone please help me...?

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  • $\begingroup$ If you involve coordinate systems for which the origins don't coincide at $\vec{x}^{S} = \vec{x}^{S'} = \vec{0}$ when $t^{S} = t^{S'} = 0$ then the transformation law is much nastier. So people generally don't. You can always apply simple spacial and temporal shifts later if needed. $\endgroup$
    – dmckee --- ex-moderator kitten
    Commented Feb 27, 2018 at 22:26
  • $\begingroup$ Do boost tramsformations leave the origin fixed?? One frame is moving wrt to another. So I think they certainly the origin moves... $\endgroup$
    – Keith
    Commented Feb 28, 2018 at 2:42
  • $\begingroup$ "when $t^{S} = t^{S'} = 0$" Not at other times. $\endgroup$
    – dmckee --- ex-moderator kitten
    Commented Feb 28, 2018 at 6:09
  • $\begingroup$ The origin is x=y=z=t=0 - this is Minkowski spacetime, not just space. $\endgroup$
    – NickD
    Commented Feb 28, 2018 at 14:06

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are the available frames of reference all inertial frames?

No, as you mention in your third question, a frame accelerated wrt an inertial frame is not inertial.

are all inertial frames of reference connected by Lorentz transformations?

It depends on what you mean by Lorentz transformations. The Lorentz group is usually defined as the set of isometries in Minkowski spacetime that leave the origin fixed, so spatial rotations around the origin and Lorentz boosts are included, but translations are not. The Lorentz group is a subgroup of the Poincare group which does include translations. But that's the same thing with Galilean relativity: there's the translations and then there's the rotations around the origin and the (Galilean) boosts. See Lorentz Group. Not sure that clarifies it: if not, please add more details to your question.

Third, in special relativity, the force and acceleration are handled in a given inertial frame. However, then, how should I deal with the frame of the particle that is accelerating in the structure of special relativity? The accelerating particle's frame is certainly not inertial...

Yes, indeed - but I'm not sure what the difficulty is here. Maybe add more detail to this part of your question? Possibly with a specific example?

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  • $\begingroup$ If accelerating frames are allowed in special relativity. Don't they imply the existence of gravity by the equivalenxe principle and therefore that the spacetime is curved? However, special relativity is about the flat spacetime. So I cannot see how to solve this discrepancy. $\endgroup$
    – Keith
    Commented Feb 28, 2018 at 2:19
  • $\begingroup$ Also do boost transformations leave the origin fixed?? One frame is moving wrt another. So I think certainly the origin moves... $\endgroup$
    – Keith
    Commented Feb 28, 2018 at 2:32
  • $\begingroup$ And do you mean by the second answer that the Lorentz group connects all inertial frames with the fixed origin? $\endgroup$
    – Keith
    Commented Feb 28, 2018 at 2:46
  • $\begingroup$ I am really curious. I am waiting for your answers. $\endgroup$
    – Keith
    Commented Feb 28, 2018 at 3:05
  • $\begingroup$ @Keith wrote "Don't they imply the existence of gravity by the equivalenxe principle". In flat spacetime, there exist global inertial reference frames. In a curved spacetime, there are in general only local inertial reference frames (there is a flat spacetime 'tangent' to each event in a curved spacetime). This difference is crucial to understanding gravitation. Accelerated reference frames in a flat spacetime do not describe gravitation. $\endgroup$
    – Alfred Centauri
    Commented Feb 28, 2018 at 3:30

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