Is Event Sequencing Relative in the Theory of Manifolds?

[yossell]

What if they intersect to the right of the wordline? (i..e. their extensions intersect)

I don't know what you're asking for here. But by changing the direction of acceleration, you can get an intersection on either `side' of the world line.

Choose a frame, let rocket and person begin stationary in this frame, with the human remaining and having been stationary in this frame. Human having just died at (0, 0, 0, 0), while the rocket is at (0, X, 0, 0) (coordinates of the form (t, x, y, z).

Let rocket quickly accelerate away in the positive x direction, reaching speed v at time t (in stationary frame) in positive x direction and then maintaining v. Its lines of simultaneity are now tilted, and simply by letting X and v be big enough and t small enough, arbitrary events of coordinate (-T, 0, 0, 0) intersect its simultaneity line. For some of these events, the human is not a corpse.

 

[starthaus]

I don't know what you're asking for here. But by changing the direction of acceleration, you can get an intersection on either `side' of the world line.

Choose a frame, let rocket and person begin stationary in this frame, with the human remaining and having been stationary in this frame. Human having just died at (0, 0, 0, 0), while the rocket is at (0, X, 0, 0) (coordinates of the form (t, x, y, z).

Let rocket quickly accelerate away in the positive x direction, reaching speed v at time t (in stationary frame) in positive x direction and then maintaining v. Its lines of simultaneity are now tilted,

Yes, they start at 0 degrees wrt x and they tilt continously (with an asymptote angle of $$\pi/4$$ )as the rocket accelerates. Yet, starting point of each line of simultaneity also moves since the rocket is tracing the wordline.

and simply by letting X and v be big enough and t small enough, arbitrary events of coordinate (-T, 0, 0, 0) intersect its simultaneity line. For some of these events, the human is not a corpse.

Try proving this mathematically.

 

[yossell]

Yes, they start at 0 degrees wrt x and they tilt continously (with an asymptote angle of )as the rocket accelerates. Yet, starting point of each line of simultaneity also moves since the rocket is tracing the wordline.

But although the line of simultaneity has moved on, the lines are indefinitely extended in the x and -x directions. The equations mx + c and m'x + c' always have a solution if m and m' are distinct, no matter how big c - c' is.

Try proving this mathematically

Ok - I'll give it a shot - but what will you allow me? Can I work in natural units where lightrays equation is t = x and t = -x, and where objects with velocity v has a simultaneity line of 1/v? Or does this need to be established too?

 

[Al68]

Do you have a reference for that?

Sure: http://www.bartleby.com/173/11.html. That's just a reference for the lorentz transformations. I assume you know how to use them.

Assuming that the ship starts a (short duration) decel simultaneous with the 2020 Presidential election in the inertial frame of the ship prior to decel, it will be 2012 on Earth simultaneous with the decel in the inertial frame of the ship after decel.

Notice that Earth's clock "jumps backward" in my example for the same reason that Earth's clock "jumps ahead" in the standard twins paradox resolutions. It's just an artifact of using the SR simultaneity convention to assign coordinates to distant events.

 

[yossell]

starthaus, I know you're the hard man of forum and that you run a tight ship, but won't you take a look at this diagram? Red dotted line a null beam, green arrow path of rocket. It starts stationary and then gently accelerates to a velocity where the slanty line is its line of simultaneity. As X is general, it's easy to find an X where it can gently accelerate to a v and a point where the line of simultaneity hits (-T 0 0 0)

 

[bcrowell]

In SR, for any two events A and B that are time like separated, if A is earlier than B in one frame, it is earlier than B in all frames.

This is correct for inertial reference frames, but not true in general for arbitrary reference frames. Consider a ship moving toward Earth at 0.8c that decelerates to come to rest with Earth at a distance of 10 light years. Using the standard SR simultaneity convention, wrt the ship observer, people who were dead on Earth simultaneous with the ship beginning deceleration are alive on Earth simultaneous with the ship stopping its deceleration.

Yep, dead people can rise from the grave in SR.

The reason we care about frame-independence of time ordering in SR is that it's necessary for causality. Your example refers to simultaneity between events that are spacelike separated, so it doesn't have any implications for causality. Depending on how you interpret yossell's statement of the principle, I'm not even sure that your example is a counterexample.

Here is a somewhat different statement of the principle that may be more clearcut. Let A and B be the end-points of a curve S in spacetime, such that the curve's orientation is always in the positive timelike direction as we traverse it from A to B. If the spacetime is flat, then this property of S is coordinate-independent.

 

[starthaus]

starthaus, I know you're the hard man of forum and that you run a tight ship,

Yes, I do :-)

but won't you take a look at this diagram? Red dotted line a null beam, green arrow path of rocket. It starts stationary and then gently accelerates to a velocity where the slanty line is its line of simultaneity. As X is general, it's easy to find an X where it can gently accelerate to a v and a point where the line of simultaneity hits (-T 0 0 0)

bcrowell beat me to the disproof but I can disprove your statement mathematically (Al68 offered nothing but some handwaving). Are you interested in the mathematical disprroof?

 

[yossell]

bcrowell beat me to it but I can disprove your statement mathematically (Al68 offered nothing but some handwaving). Are you interested in the mathematical disprroof?

I'd be interested, but I now think we must be talking at cross-purposes. I (and I think Al68), are considering examples where an object, by accelerating or decelerating, changes inertial frames in a way which affects the T-coordinate of distant events - bcrowell is talking about the property of particular CURVES in spacetime, and the question of whether they are timelike or not is not a coordinate dependent matter. This latter I think is true, and am not arguing against it.

 

[Rasalhague]

In #33, I made a calculation based on my interpretation of Al68's resurrection statement. Is this what you had in mind, Al68? Are other people here interpreting it differently? Did I get the calculation right?

 

[Al68]

The reason we care about frame-independence of time ordering in SR is that it's necessary for causality. Your example refers to simultaneity between events that are spacelike separated, so it doesn't have any implications for causality. Depending on how you interpret yossell's statement of the principle, I'm not even sure that your example is a counterexample.

You're right, it's not a counterexample (to causality), yossel clearly meant to refer strictly to inertial reference frames, not an arbitrary coordinate system. And you're right that there are no implications for causality. Assigning coordinates to distant events with the SR simultaneity convention doesn't cause anything.

 

[Al68]

In #33, I made a calculation based on my interpretation of Al68's resurrection statement. Is this what you had in mind, Al68? Are other people here interpreting it differently? Did I get the calculation right?

Yeah your interpretation is right, but I think you missed that the coordinate distance between the ship and Earth is only 6 ly in the ship's frame prior to decel. So it should be 8 yrs instead of 13.33 yrs.

 

[Al68]

...Al68 offered nothing but some handwaving...

LOL. You didn't ask for a proof. You asked for the reference I used. I provided it.

Edit: Rasalhague shows the math in the next post, but I assumed an experimental physicist like yourself wouldn't need it to be shown.

 

[Rasalhague]

Yeah your interpretation is right, but I think you missed that the coordinate distance between the ship and Earth is only 6 ly in the ship's frame prior to decel. So it should be 8 yrs instead of 13.33 yrs.

Oh yeah, of course! 10*Sqrt[1-0.8^2]*0.8/Sqrt[1-0.8^2] = 10*0.8 = 8. Sorry deads! At least we get some back, and with a bigger boost as many as we want...

 

[Al68]

Oh yeah, of course! 10*Sqrt[1-0.8^2]*0.8/Sqrt[1-0.8^2] = 10*0.8 = 8. Sorry deads! At least we get some back, and with a bigger boost as many as we want...

Yep. Too bad the ship's observer can never actually observe them alive again. They're alive "now" according to the SR simultaneity convention, but no "they're alive" signal will ever reach the ship "later" regardless of its subsequent motion. In fact, the faster the ship tries to reach earth, the quicker they will die off again.

 

[starthaus]

LOL. You didn't ask for a proof. You asked for the reference I used. I provided it.

I asked you for a valid reference to your claim that acceleration can produce event order reversal. So, you provided no reference. Not only that, you got the wrong answer.

 

[JesseM]

Do you have a reference for that? My calculations show that your claim is false. The line of simultaneity keeps rotating wrt the x-axis in the Minkowski diagram as the speed varies but it never crosses over itself, even at the inflection points.

It depends on how fast the acceleration is, i.e. how long it takes the ship to go from 0.8c to rest in the Earth frame. For example, suppose the ship is traveling towards Earth at 0.8c until it is 10 light years from Earth at t=0 years in the Earth frame (so if the Earth is at x=0 light years, the ship could be at x=-10 light years), then it accelerates for a year until it is at rest at t=1 year the Earth frame. In that case, in the ship's inertial rest frame at the moment it begins to accelerate, the moment of its beginning to accelerate is simultaneous with an event that occurs on Earth at t=8 years in the Earth frame (since in the Earth's frame we have dx=10 and dt=8 between these events, meaning in the ship's frame dt'=gamma*(dt - v*dx/c^2)=0), but in the ship's inertial rest frame at the moment it stops accelerating, the moment it stops accelerating is simultaneous with an event that occurs on Earth at t=1 year in the Earth frame.

 

[starthaus]

It depends on how fast the acceleration is, i.e. how long it takes the ship to go from 0.8c to rest in the Earth frame. For example, suppose the ship is traveling towards Earth at 0.8c until it is 10 light years from Earth at t=0 years in the Earth frame (so if the Earth is at x=0 light years, the ship could be at x=-10 light years), then it accelerates for a year until it is at rest at t=1 year the Earth frame. In that case, in the ship's inertial rest frame at the moment it begins to accelerate, the moment of its beginning to accelerate is simultaneous with an event that occurs on Earth at t=8 years in the Earth frame (since in the Earth's frame we have dx=10 and dt=8 between these events, meaning in the ship's frame dt'=gamma*(dt - v*dx/c^2)=0), but in the ship's inertial rest frame at the moment it stops accelerating, the moment it stops accelerating is simultaneous with an event that occurs on Earth at t=1 year in the Earth frame.

I don't think that the above is correct. The issue in discussion was whether acceleration can invert the ordering of events. I will repeat the proof I gave for inertial frames and I will generalize to accelerated frames:

$$dt'=\gamma(\tau-vx/c^2)$$

$$dt'=\gamma (d\tau-vdx/c^2)$$

$$dt'=\gamma d\tau (1-v/c^2*\frac{dx}{d\tau})$$

Since $$v \frac{dx}{d\tau}<c^2$$ it follows that $$dt'$$ and $$d\tau$$ always have the same sign. If you are ok with the above, I can post the generalization to accelerated frames.

 

[JesseM]

I don't think that the above is correct. The issue in discussion was whether acceleration can invert the ordering of events. I will repeat the proof I gave for inertial frames and I will generalize to accelerated frames:

$$dt'=\gamma(\tau-vx/c^2)$$

$$dt'=\gamma (d\tau-vdx/c^2)$$

$$dt'=\gamma d\tau (1-v/c^2*\frac{dx}{d\tau})$$

How would you define the non-inertial frame for a non-inertial observer? There's no single "correct" way, but I think the most common type of non-inertial frame used by physicists in SR would be one that has the following properties:

1. For point on the non-inertial observer's worldline, the coordinate time is just equal to the observer's proper time at that point

2. At any point on the observer's worldline, a line of simultaneity in the non-inertial coordinate system which goes through that point would be identical to a line of simultaneity in the inertial frame where the observer has an instantaneous velocity of zero at that point

3. For two events which lie on a single line of simultaneity in the non-inertial coordinate system, the coordinate distance between them should be the same as the coordinate distance in the inertial frame which also has a line of simultaneity going through both events

4. The non-inertial observer has a coordinate position that doesn't change with coordinate time in the non-inertial system

If you define the non-inertial coordinate system in this way, then (3) implies the system's judgments about simultaneity always agree with those of the observer's instantaneous inertial rest frame, so this was the basis for my comments above (although I think it would actually be more common to just say the coordinate system "ends" at the point where lines of simultaneity would cross over one another, as with Rindler coordinates which don't extend past the Rindler horizon). I think Al68 was also assuming this sort of coordinate system when he commented about the dead rising from the grave in post #30. If you're assuming a different type of non-inertial coordinate system, you'll have to explain how it's defined for an observer with a non-inertial worldline that has a known parametrization x(t) relative to some inertial frame (for an observer experiencing constant proper acceleration a, their x(t) in an inertial frame where they started at rest would be x(t) = (c^2/a) (sqrt[1 + (at/c)^2] - 1) according to the http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html ).

 

[starthaus]

How would you define the non-inertial frame for a non-inertial observer?

I'll show you next. First off, do you agree that inertial motion cannot reverse the order of events between frames? Yes or no?

 

[JesseM]

I'll show you next. First off, do you agree that inertial motion cannot reverse the order of events between frames? Yes or no?

Can you clarify what you mean by that? It is certainly true that for events with a space-like separation (|dx| > |c*dt| in any given inertial frame), you can find a pair of inertial frames which disagree on their order, are you denying that? If you don't deny it, why doesn't this qualify as an example of "inertial motion reversing the order of events between frames" as you define this phrase?

 

[yossell]

$$dt'=\gamma(\tau-vx/c^2)$$

$$dt'=\gamma (d\tau-vdx/c^2)$$

$$dt'=\gamma d\tau (1-v/c^2*\frac{dx}{d\tau})$$

Since $$v \frac{dx}{d\tau}<c^2$$ it follows that $$dt'$$ and $$d\tau$$ always have the same sign. If you are ok with the above, I can post the generalization to accelerated frames.

Your question is whether inertial motion can invert the order of events.

I do not think you have shown this claim in its full generality, but I think you have shown a weaker one, that inertial motion cannot invert the order of timelike events.

First, an aside: what's going on in step 1 to 2 - why the replacement of x and \tau with dx and d\tau? Just notational?

Secondly, I take the strategy to be this: dt' represents the difference in time between two events A and B in one inertial frame; d\tau and dx the difference in time and the difference in x-coordinate between these two events in another inertial frame. You then try to show that d\tau and dt' have the same sign, and therefore that any two frames must agree on the temporal order of the two events.

The trouble is that dx/d\tau is not necessarily a velocity of anything - it is *just* a ratio. The events may be spacelike separated. In which case, the ratio is not less than c. In such cases, your proof can be used to show that the temporal order can be inverted.

However, when attention is restricted to events which are timelike separated, then your proof seems to go through.

 

[starthaus]

Your question is whether inertial motion can invert the order of events.

More precisely, if order of timelike events (see the OP) is maintained in SR. We aren't talking about spacelike events.

I do not think you have shown this claim in its full generality, but I think you have shown a weaker one, that inertial motion cannot invert the order of timelike events.

...which is precisely what I have been talking about at posts 12 and 13. This answer is for both you and JesseM. In mathematical terms, as already shown twice, the proof is for $$\frac{dx}{d\tau}<1$$ (no "faster than light signalling"). I am addressing the OP, not the case of spacelike events.

First, an aside: what's going on in step 1 to 2 - why the replacement of x and \tau with dx and d\tau? Just notational?

No, this is standard differentiation.

Secondly, I take the strategy to be this: dt' represents the difference in time between two events A and B in one inertial frame; d\tau and dx the difference in time and the difference in x-coordinate between these two events in another inertial frame. You then try to show that d\tau and dt' have the same sign, and therefore that any two frames must agree on the temporal order of the two events.

yes

The trouble is that dx/d\tau is not necessarily a velocity of anything - it is *just* a ratio. The events may be spacelike separated.

No, see above.

However, when attention is restricted to events which are timelike separated, then your proof seems to go through.

It does not "seem", it does. It is just a repeat of the one at post 12. Same conditions, same outcome.

 

[yossell]

I still don't understand what you're calling differentiation or what you're doing from 1 - 2.

From line 2 onwards we're agreed that it shows that, for any timelike related (and I emphasise this because you've not been explicitly putting in this rider) events, all inertial observers agree about the events' temporal order ---- as I in fact wrote way back in post 5.

Great.

Now, I'm interested in the generalisation you've mentioned, to deal with Al68's point, that a rocket ship can, beginning at space time point A and in an inertial frame where events A and space time point B are simultaneous, travel to a spacetime point C and an inertial frame where C is simultaneous with an event that lies in B's past.

 

[starthaus]

I still don't understand what you're calling differentiation or what you're doing from 1 - 2.

Because this is a well known operation in calculus. It is important that you understand it for the next step, accelerated motion. How familiar are you with differential calculus?

 

[JesseM]

...which is precisely what I have been talking about at posts 12 and 13. This answer is for both you and JesseM. In mathematical terms, as already shown twice, the proof is for $$\frac{dx}{d\tau}<1$$ (no "faster than light signalling")

OK, so when you said "First off, do you agree that inertial motion cannot reverse the order of events between frames?" were you only talking about pairs of events for which |dx/dt| <= c? If so, I agree that different inertial frames won't disagree on their order.

 

[starthaus]

OK, so when you said "First off, do you agree that inertial motion cannot reverse the order of events between frames?" were you only talking about pairs of events for which |dx/dt| <= c? If so, I agree that different inertial frames won't disagree on their order.

Excellent, let's wait for yossell, see if he's comfortable with some heavy duty differential calculus required for the case of accelerated frames.

Do you think that the above holds for accelerated frames?

 

[JesseM]

Do you think that the above holds for accelerated frames?

It depends what you allow to qualify as an accelerated frame. You can write down a coordinate transformation which relates a non-inertial coordinate system to an inertial one, with the equations of the transformation implying that lines of simultaneity in the accelerated frame will actually cross when plotted in the inertial frame, but some authors would say this is not a well-behaved frame since it doesn't assign unique coordinates to each event, so they'd require that the accelerated frame only be defined up to the point where lines of simultaneity would cross rather than throughout all of spacetime (this is what's done with Rindler coordinates, which only cover a region of spacetime to one side of the Rindler horizon)

Also, it depends on whether you impose the rule that the space coordinate must actually be spacelike at all times (i.e. every surface of constant time is a spacelike surface, meaning path in that surface is spacelike) and the time coordinate must be timelike everywhere (every line of constant position coordinate is a timelike worldline). If you don't impose that rule, then non-inertial frames can disagree on the order of events with a timelike separation even without any lines of simultaneity crossing in any of the non-inertial frames.

 

[Al68]

I asked you for a valid reference to your claim that acceleration can produce event order reversal. So, you provided no reference. Not only that, you got the wrong answer.

LOL. That's not the claim I made and you know it. I said that events could be in reverse order in an accelerated reference frame. Slight difference there. And I provided the only reference I used: the lorentz transformations.

And I got the right answer. All you need to do to see how obvious it is is to imagine a clock (clock C) at rest with Earth local to the ship's (almost instantaneous) deceleration, that is synched with Earth's clock in Earth's frame. Prior to deceleration, Earth's clock is ahead of clock C by 8 years in the ship's frame. After deceleration, Earth's clock matches clock C in the ship's frame. Clock C is (almost) local to the ship before and after decel and has (almost) the same reading before and after decel. Simple lorentz transformations. No kawcoolus required.

Why would an experimental physicist have so much trouble understanding such a simple scenario? What do you think the right answer is?

 

[starthaus]

LOL. That's not the claim I made and you know it. I said that events could be in reverse order in an accelerated reference frame.

This is false, the events will not be in reverse order, accelerated frames preserve event ordering, just as inertial frames do. This is the same erroneous claim you made earlier.

 

[JesseM]

This is false, the events will not be in reverse order, accelerated frames preserve event ordering, just as inertial frames do. This is the same erroneous claim you made earlier.

What about the accelerated frame I defined in post #53 with properties 1-4? Do you disagree that lines of simultaneity can cross in such a frame, causing the frame to label timelike-separated events with a reversed order from the order in inertial frames? Or do you think my definition wasn't clear?

 

[Al68]

This is false, the events will not be in reverse order, accelerated frames preserve event ordering, just as inertial frames do. This is the same erroneous claim you made earlier.

LOL. I can only assume my claim to be perfectly correct given that an experimental physicist such as yourself has failed to show it to be erroneous in any way. Thanks for the confirmation! :!)

BTW, my claim was that (in an accelerated frame) events could be assigned time coordinates in reverse order by the SR simultaneity convention, not that they actually occur or are observed in reverse order. Unless someone was so pedantic as to take my "dead rising from the grave" statement seriously.

 

[yossell]

What about the accelerated frame I defined in post #53 with properties 1-4? Do you disagree that lines of simultaneity can cross in such a frame, causing the frame to label timelike-separated events with a reversed order from the order in inertial frames? Or do you think my definition wasn't clear?

I have a question about non-inertial frames: In the kinds of frames that you're talking about, one and the same space-time point will be assigned two different coordinates - e.g. the point of intersection of the lines of simultaneity. But does the concept of a frame allow for this to happen? The time of an event will be multivalued in such a frame. Do people know if this is ok?

Let me say that I don't see anything deeply physical going on here - I recognise that non-inertial frames are a somewhat artificial concept. it's just an interesting (to me) question about the definition of a frame.

 

[Ken Natton]

Wow. I can’t believe the pace that this thread has gone at. Just one day away and I’ve had four pages of challenging stuff to read and try to understand. I hate to act as a weight and drag the pace down, but my hope is that if anyone does answer me, that doesn’t necessarily have to impinge on the conversation between the rest of you.

I’m still on the difference between special and general relativity.

Special relativity mathematically takes place against a background of Minkowski space. General relativity takes place against a background of 4 dimensional Lorentzian manifolds of which Minkwoski space is one (special) example. The motivation for this is that choosing Lorentzian manifolds other than Minkowski space allows gravity to be modeled relativistically.

Okay, but like Cartesian co-ordinate systems and Euclidian geometry, these metrics are just intellectual constructs, tools of analysis, rather aspects of physical reality. Yes, the reality of the curvature of spacetime is part of general relativity theory, but that is always the true physical reality, a body isn’t actually transformed from being in curved spacetine to being in flat spacetime when it ceases to accelerate and starts to travel at a constant speed. An inertial reference frame is really just an idealised state to make the concept easier to understand, as is the case for Newton’s first law, when we are asked to imagine a body in a situation without gravity and without friction. The truth, is it not, is that all real bodies, from subatomic particles to galaxies, exist in permanent non-inertial reference frames?

There are those that argue that special relativity is superseded and made redundant by general relativity. The only defence against that is that special relativity is the route into an understanding. Special relativity is more graspable for those coming to relativity for the first time, and once you have got that, it becomes easier to extend the principle to cover all reference frames.

Yes I know Einstein was specifically interested in the equivalence of gravitational freefall and acceleration under some other force, and that this led him to the idea that mass actually warps spacetime and that this is the actual explanation for gravity. But the concept of equivalence is extendable to the equivalence of all accelerating reference frames, is it not? It seems unlikely to me that special relativity describes a physical reality that only exists in idealised conditions. Surely, the physical reality is always the same. Special relativity just covers a special case of it, general relativity generalises that principle.

So I suppose, to bring it back to the original subject of the thread, what you are telling me is that within the idealised constraints of an inertial reference frame, changing the sequence of events is not possible. Physical reality does not actually impose those constraints, and thus relativity of sequence is always possible. Once again, we have the undermining of the notion of cause and effect that worried me.

 

[Rasalhague]

This is false, the events will not be in reverse order, accelerated frames preserve event ordering, just as inertial frames do. This is the same erroneous claim you made earlier.

Is the issue here: (1) "Are there pairs of non-identical events with no defined coordinate time order or with multiple coordinate time orders in one single given accelerated coordinate system?" Al68: "Yes." starthaus: "No."

At first I thought the issue was: (2) "Does Al68's scenario correctly exemplify the fact that spacelike separated events don't have a frame-independent natural time order. (First simultaneity judgement in the instantaneous comoving inertial rest frame of the ship, 10 light years from the earth, traveling towards the Earth at 0.8c relative to the earth. Second simultaneity judgement in the instantaneous comoving inertial rest frame of the ship after it decelerates till it's at rest with respect to the earth, i.e. in the rest frame of the earth.)"

I thought starthaus imagined Al68 was talking about a pair of timelike separated events. I thought you were both just talking at cross purposes, particularly as starthaus's proof consisted of using the Lorentz boost formulas to show that timelike separated events can't change order under a boost.

But if the issue is actually (1), then I would have thought the answer was no because of the definition of "coordinate system", "reference frame", "chart", specifically the requirement that the coordinate functions be, as their name suggests, functions, i.e. single-valued. Events in spacetime for which a coordinate system doesn't behave this way, I'm thinking, wouldn't belong to the domain of that particular coordinate system. In this case, if we have an accelerated coordinate system something like Rindler coordinates (as JesseM says in the final paragraph of #53), events on Earth just aren't covered by these coordinates. Or, if they were, then, by definition of a chart, there'd have to be one single time order specified, even though it needen't necessarily agree with the time order of another coordinate system, and--at least for some pairs of events--not every possible coordinate system will agree.

Is that anywhere near the mark?

 

[yossell]

Okay, but like Cartesian co-ordinate systems and Euclidian geometry, these metrics are just intellectual constructs, tools of analysis, rather aspects of physical reality.

Since the rest of this paragraph looks right to me, there's probably nothing wrong with your understanding, but I would be careful about this way of putting it. The metrics of spacetime themselves aren't normally thought of as mere intellectual constructs. The metric is quite unlike a choice of coordinate system, and in fact conveys coordinate independent information about space-time itself, such as whether space is curved, and the coordinate-independent Minkowski `length' of a space-time curve.

There have been those (I think Poincare was one) who thought that the choice of geometry itself was as much a convention as choice of coordinate system. But I wouldn't say that this was a mainstream idea today.

The truth, is it not, is that all real bodies, from subatomic particles to galaxies, exist in permanent non-inertial reference frames?

Again, this is probable pit-nicking, but accelerated bodies can be analysed from the point of view of an inertial frame. What's (I think) true is that, the existence of gravity, inertial frames as understood in SR no longer exist globally. Rather, one can apply SR at a local level - working on a small scale, not extending your t and x coordinates too far, objects still behave approximately as SR says they do.

But the concept of equivalence is extendable to the equivalence of all accelerating reference frames, is it not?

I tentatively believe it's only to all *freely* falling frames.

It seems unlikely to me that special relativity describes a physical reality that only exists in idealised conditions. Surely, the physical reality is always the same. Special relativity just covers a special case of it, general relativity generalises that principle.

Since gravity is pervasive, all space time is warped a little - so I think in that sense SR does only describe a physical reality that exists in idealised conditions. At least, it's not clear to me that there's anywhere where the idealised conditions obtain. Do you see this as a problem? The way in which SR approximates GR is mathematically well defined and well understood.

Physical reality does not actually impose those constraints, and thus relativity of sequence is always possible. Once again, we have the undermining of the notion of cause and effect that worried me.

I don't think this was quite the lesson - though as you can see, there was controversy and...maybe something more...

I would summarise the main points as: (a) the temporal order of two space-like events is dependent on inertial frame, but, in standard interpretations of SR*, this has no causal implications as such events lie outside the light cone and thus are causally independent; (b) very crazy/artificial non-inertial frames may be constructed on which a later event (my death) has a smaller coordinate time than an earlier event (my birth); but such frames are so artificial - really little more than a choice of labelling or giving coordinates to distant events - that nobody should try and read off the causal story or physical story off the numbers of the resulting chart; (c) some very strange solutions of GR are possible, which allow a kind of circular causation, but these solutions seem removed from reality and, at least locally, there's an event by event causal story - it's just that it loops around on itself; Even in this model, though, this circular causal chain is not a frame dependent matter.

tl;dr
I don't think you should worry.

*e.g. no tachyons.