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So I'm still reading up on Einstein's theory of special relativity and what it means in regards to space-time and motion.

One key feature and cornerstone being that there exists no such thing as a privileged reference frame. All reference frames are equal. Essentially there is no cosmic or metaphysical grid with set coordinates that we change each time we move.

For example, shoot a bullet into a magic portal leading to an infinite empty void, and for all intents and purposes the bullet will have no way of knowing if it is even in motion.

A rocket travels to a planet, and for all intents and purposes the planet can equally be said to be traveling to the rocket. With the only difference being our own subjective judgment on how best to phrase it.

Assuming my examples aren't incorrect and I haven't misunderstood what the reference frame idea means. What would a privileged reference frame look like?

What would it mean or look like if in the rocket/planet example, either the rocket or planet had a reference frame that was more legitimate than the other?

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    $\begingroup$ Do you know about the frame in which the cosmic microwave background radiation is isotropic? $\endgroup$
    – Ghoster
    Jul 5, 2023 at 17:50
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    $\begingroup$ Not a formal answer, but can you imagine the lines people draw in pictures to show that something is moving? Like in this image? In a preferred reference frame, everything that was moving might have these lines around them to show that they were moving, and one could conclusively say that objects with bigger lines were moving faster. $\endgroup$ Jul 5, 2023 at 18:29
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    $\begingroup$ This is not special relativity - it's Galilean relativity, first enounced nearly 400 years ago. $\endgroup$ Jul 5, 2023 at 20:38
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    $\begingroup$ @StéphaneRollandin I think that comment requires some more explanation. Before the discovery of Maxwell's equations, all known laws of physics were invariant under Galilean transformations. Newton would say that there were no preferred frames of reference in classical mechanics, appealing to Galilean relativity. $\endgroup$
    – Andrew
    Jul 5, 2023 at 22:25
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    $\begingroup$ A universe with preferntial referencr frame already exists and physicists can't see it rifht before their eyes. It's called the speed of light, a hard limit imposed on all of reality AS WE KNOW it by one man and now all his followers believe it despite knowing the theory already breaks down such as in quantum singularities. Time travel paradoxes or ftl causal paradoxes ONLY happen BECAUSE we give light speed PREFERENTIAL TREATMENT. No causal paradox would occur if we instead gave preferntial reference to the FASTEST speed among a set of our choosing regardless of speed (not infinite) $\endgroup$ Aug 12, 2023 at 8:00

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If there were a privileged frame, one or more key physicals factors—such as the speed of light—would have a special value in it. The idea of there being no privileged frame is most associated with special relativity and the principle that the speed of light is the same relative to any frame. You could imagine a universe in which light moved at a fixed speed relative to some special frame, so that in any other frame moving relative to the special one, the speed of light would take a different value.

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    $\begingroup$ The idea of there being no privileged frame is associated with Galilean relativity just as much as it's associated with Einsteinian. It's what the term "relativity" refers to, and Einstein certainly didn't invent that. $\endgroup$
    – Arthur
    Jul 6, 2023 at 19:52
  • $\begingroup$ There is a privileged frame in SR. It's callrd the speed of light. Astonjshjng how physicists CANT see the obvious. SR already breaks down in black holes which is why physicists are desperate for a better theory but are forced to rectify relativity with black holes and you get alternate universe behind the singularity. Laughable. FTL travel causing backwards time travel or causal paradoxes? Sure. If you make light a PRIVILEGED reference frame in the set. Shift the reference frame to give preference to the FASTEST reference regardless of speed, and ALL causal paradoxes IMMEADIATELY DISAPPEAR. $\endgroup$ Aug 12, 2023 at 8:03
  • $\begingroup$ There is a privileged frame in SR. It's callrd the speed of light. Astonjshjng how physicists CANT see the obvious. SR already breaks down in black holes which is why physicists are desperate for a better theory but are forced to rectify relativity with black holes and you get alternate universe behind the singularity. Laughable. FTL travel causing backwards time travel or causal paradoxes? Sure. If you make light a PRIVILEGED reference frame in the set. Shift the reference frame to give preference to the FASTEST reference regardless of speed, and ALL causal paradoxes IMMEADIATELY DISAPPEAR. $\endgroup$ Aug 12, 2023 at 8:04
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Both special and general relativity are built on the postulate there are no locally privileged reference frames. What this means is the following: if you design an experiment which is sufficiently limited in spatial and temporal extent - loosely, one which can be done in a small spatial region and which takes a sufficiently short amount of time - then that experiment will yield the same result in any inertial frame.

However, if we permit ourselves to do relativity on more interesting spacetime manifolds, then it is possible to find globally privileged reference frames. These frames cannot be distinguished from other local reference frames by local experiments, but can be distinguished on the basis of global ones which explore a larger region of the spacetime.

Here is a concrete example. In the following, I set $c=1$. Consider the spacetime manifold $\mathcal M \simeq \mathbb R\times \mathrm S^1$. This would correspond to a universe which is spatially finite - if you keep traveling in either direction, you will ultimately return to your starting point.

Concretely, define the spacetime $$\mathcal M = \left\{(t,x,y) \in \mathbb R^3 \ \bigg| \ x^2+y^2 = R^2\right\}$$ which inherits its metric from an ambient Lorentzian metric in $\mathbb R^3$: $$g_{\mathbb R^3} = -\mathrm dt^2 + \mathrm dx^2 + \mathrm dy^2$$ If we coordinatize $\mathcal M$ with coordinates $(t,\theta)$ where $(x,y)=\big(R\cos(\theta),R\sin(\theta)\big)$, $\theta\in(-\pi,\pi)$, then the metric on $\mathcal M$ takes the form $$g_{\mathcal M} = -\mathrm dt^2 + R^2 \mathrm d\theta^2$$


If we choose a spatial coordinate $u \equiv R\theta$, then the metric takes the form $$g_\mathcal M = -\mathrm dt^2 + \mathrm du^2$$

which is precisely the form of the Minkowski metric on $\mathbb R^2$. The difference here is that the coordinate $u$ does not cover the entire spacetime; it becomes ill-defined as $\theta\rightarrow \pm \pi, u \rightarrow \pm R\pi$.

This coordinate frame is mathematically privileged for the following reason. From our study of Minkowski space, we know that a boost must take the form $$\matrix{t' = \gamma(t- vu)\\u' = \gamma(u-vt)}$$ Due to the singularity in the previous coordinate chart, the primed coordinates become singular whenever $u' = vt' \pm R\pi/\gamma$. In other words, the singularity becomes "tilted" in the $(t',u')$ plane.

Physically, the original $(t,u)$ frame is privileged because it is the unique frame such that if you send out a light signal in opposite directions, they will both return to their starting points at the same time. Additionally, this frame serves as a privileged frame for a twin paradox experiment.

Consider two inertial observers - Alice, with worldline $\lambda \mapsto (t,R,0)$, and the Bob, with worldline $\lambda \mapsto (\lambda,R\cos(\omega\lambda),R\sin(\omega\lambda)$. These worldlines intersect at $\lambda=0$ and $\lambda = 2\pi/\omega$. Neither Alice nor Bob has accelerated, but Alice has aged by an amount $$\Delta \tau_{Alice} = \int_0^{2\pi/\omega} \mathrm d\lambda = 2\pi/\omega$$ while Bob has aged by an amount $$\Delta \tau_{Bob} = \int_0^{2\pi/\omega} \mathrm d\lambda \sqrt{1 - \omega^2 R^2} = \frac{2\pi}{\omega}\sqrt{1-\omega^2R^2} < \Delta \tau_{Alice}$$

Here we have a twin paradox whose resolution is opposite to the familiar one - the situation here is symmetric in the sense that neither Alice nor Bob undergo any acceleration, and no local experiment could distinguish between their respective frames. However, there is a global anisotropy in the spacetime, such that Alice's frame is (globally) privileged.


To recap: If we model spacetime as the cylinder $\mathcal M$ rather than the plane $\mathbb R^2$, we can construct inertial frames with coordinates $(t,u)$, in which the metric takes the form $g_\mathcal M = -\mathrm dt^2 + \mathrm du^2$ which looks just like the familiar Minkowski metric. This means that if you restrict your attention to a sufficiently small region of spacetime, any experiment will yield the same results as it would in ordinary SR, and in particular will not depend on the choice of reference frame.

However, if we design a global experiment - like sending light signals opposite ways around the entire spatial universe, or allowing a twin to fly around the universe at constant velocity until they return to their starting point - then we can identify a globally privileged reference frame.

In our specific example, we can make the statement that any experiment which involves something wrapping around the spacetime cylinder (physically, traveling in the same direction until you come back to your starting point) can be used to distinguish one reference frame from another. However, if an experiment can be performed in such a way that nothing ever wraps all the way around, then it will have the same outcome in any inertial frame.

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Simple example: an electric charge which takes a minimal value in some frame and a larger value in all other frames.

Another: if a human stands still in the special frame, they feel ok. If they try to stand still in any other frame, they feel nauseous or dizzy.

You can think of many such examples.

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A preferred reference frame is one in which the laws of physics are observably different/simpler from a different frame. No inertial frame is preferred with respect to any other inertial frame since the physics is indistinguishable in all inertial frames, but inertial frames are preferred with respect to non-inertial frames because they do not require fictitious bookkeeping forces with no real physical cause to explain the observed motion.

A frame that would be preferred with respect to an inertial frame would need to make the physics both different and simpler, but this is a catch-22, since inertial physics is already as "simple" as possible with all forces explained by direct physical causes. Changing the physics away from what's observed in an inertial frame can only introduce concepts that aren't present in an inertial frame, thereby making it more complex and not actually "preferred". A frame that was priviliged with respect to an inertial frame would need to explain something the inertial frame doesn't, but there is nothing left unexplained in the inertial frame in the first place.

I struggle to find a better answer to describe what something that doesn't exist looks like. It's like asking what a polynomial with degree lower than 0 looks like, or what a triangle with 4 sides looks like.

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  • $\begingroup$ There is a privileged frame in SR. It's callrd the speed of light. Astonjshjng how physicists CANT see the obvious. SR already breaks down in black holes which is why physicists are desperate for a better theory but are forced to rectify relativity with black holes and you get alternate universe behind the singularity. Laughable. FTL travel causing backwards time travel or causal paradoxes? Sure. If you make light a PRIVILEGED reference frame in the set. Shift the reference frame to give preference to the FASTEST reference regardless of speed, and ALL causal paradoxes IMMEADIATELY DISAPPEAR. $\endgroup$ Aug 12, 2023 at 8:08
  • $\begingroup$ Take an FTL ship which in space time diagrams can cause paradoxes (if you include multiple reference frames of different velocities and keep shifting between frames, with light being at 45 degree angles), now, REMOVE the preferntial reference of light. Remove light all together. Put ftl ship in a rest frame with space time angles 90 degrees apart. Give the FASTEST speed in the set PREFERENCE just as we do with light, now watch as ALL FTL causal paradoxes disappear. Problem solved. Light speed limit is hilariously given ABSOLUTE PRIVILEGE yet apparently relativity has no preferential frames lol $\endgroup$ Aug 12, 2023 at 8:11
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Ask if the speed of light is the same in all directions. If the answer were yes in some frames but not in others, you'd probably want to call the "yes" frames preferred.

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  • $\begingroup$ So by your definition there are many preferred frames, namely the inertial ones? $\endgroup$ Jul 6, 2023 at 8:24
  • $\begingroup$ @JannikPitt By any reasonable definition there are going to be at least an $O(3)$ worth of preferred frames. $\endgroup$
    – WillO
    Jul 6, 2023 at 11:46
  • $\begingroup$ There is a privileged frame in SR. It's callrd the speed of light. Astonjshjng how physicists CANT see the obvious. SR already breaks down in black holes which is why physicists are desperate for a better theory but are forced to rectify relativity with black holes and you get alternate universe behind the singularity. Laughable. FTL travel causing backwards time travel or causal paradoxes? Sure. If you make light a PRIVILEGED reference frame in the set. Shift the reference frame to give preference to the FASTEST reference regardless of speed, and ALL causal paradoxes IMMEADIATELY DISAPPEAR. $\endgroup$ Aug 12, 2023 at 8:11
  • $\begingroup$ By including LIGHT ALWAYS in space time diagrams. Anything faster than light WILL ALWAYS cause paradoxes an issues. Black. Holes supposedly cause relativity to break down also, but black holes exist just fine. Space expands ftl just fine. Entangled pairs communicate ftl just fine. These things don't cause paradoxes or break reality. The only time causal reality breaks down in space time diagrams is when we FORCE include light at 45 degree angles and make it a hard limit. Instead,give preference to the fastest speed even if ftl, or, make thr fastest speed the rest frame. All paradoxes disappear $\endgroup$ Aug 12, 2023 at 8:14
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I'm surprised there is no mention of the Michelson Morley experiment. There was a concept of a "luminiferous aether", which could flow through space. It was thought that this aether might have a certain inertia, such that light flowing in the same direction as the aether would travel faster than light flowing perpendicular to the direction of the aether (a bit like a boat travelling in a moving stream). A reference frame which was moving at the same speed as the aether would be "special" or privileged in some sense, and you could simplify the physics by adjusting to the frame of reference in which the aether is fixed, and then adjust back to a given reference frame.

In context of the time period, this is almost certainly what Einstein was referencing when he discussed "privileged reference frames".

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  • $\begingroup$ There is a privileged frame in SR. It's callrd the speed of light. Astonjshjng how physicists CANT see the obvious. SR already breaks down in black holes which is why physicists are desperate for a better theory but are forced to rectify relativity with black holes and you get alternate universe behind the singularity. Laughable. FTL travel causing backwards time travel or causal paradoxes? Sure. If you make light a PRIVILEGED reference frame in the set. Shift the reference frame to give preference to the FASTEST reference regardless of speed, and ALL causal paradoxes IMMEADIATELY DISAPPEAR. $\endgroup$ Aug 12, 2023 at 8:11
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I guess there were, in antiquity, common ideas that involved a special frame (note that I can't really say when and by whom these were held, but they existed and are examples of special frames):

  • Everything falls towards the center of the earth (not due to the mass of the earth, but due to a basic feature of space, where earth sits in a special place).
  • Everything that moves comes to rest (relative to earth)-- again, not due to frction but due to the structure of space, where earth is at a special speed.
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What would a privileged reference frame look like?

It's impossible to answer the question specifically; it could look like anything.

When we say that there are no privileged reference frames, we mean that there aren't any reference frames that have any unique, special properties. If there were any privileged reference frames, that would mean that there exists at least one reference frame which has at least one unique, special property. Any kind of unique and special property would be sufficient to make a privileged reference frame.

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