[ShadySquid]

There is nothing in Relativity that suggests any object could not undergo constant acceleration forever. If you disagree, please show your work.

It seems to me like it doesn't matter what frame of reference the Earth is accelerating in.  Regardless of the frame of reference, if the Earth is accelerating, it is gaining speed, and eventually, that speed will reach the speed of light, ipso facto, the speed can no longer increase and the Earth will stop accelerating.  The speed of light is the hard limit for speed in the universe, and an object under constant acceleration is going to reach it eventually.

You can think of it this way:  You have a 10 liter water bottle.  Your task is to constantly pour water into the water bottle at a rate of one liter per hour, forever, without spilling any.  This is completely impossible, because the water bottle can only hold ten liters.  No matter what you do to the water bottle, no matter where you put it, it is physically impossible to add more than 10 liters to the water bottle.  The speed of light is like that water bottle; you can only fill it with so much velocity.  If you are constantly adding velocity, no matter where you put the speed-of-light bottle, no matter what angle you look at it from, it has a hard limit to how much velocity you can add; ipso facto, infinite constant acceleration is impossible.

I know it's not a perfect analogy, because water and speed are two very different substances, but it gets the idea across.

So, am I missing something here?  Why does the frame of reference even matter?

[BRrollin]

Furthermore, think back to your physics course. Did you compute a block sliding down an incline? I’m sure you did. How did you do this problem without introducing fictitious forces that arise from a noninertial frame?

How would you do this without the fictitious force of gravity?

You do not need a clock at the inner core, because on the timescales of the experiment (really it’s the spacetime interval) the clock on the core and another on the surface will remain synchronized (enough).

I agree. But this is equally true in FET as in RET, which is my whole point. (Except, of course, that there is no inner core in FET, so replace "the core" with "an inertial frame of reference beginning at rest with respect to the clock on the surface".)

Fictitious forces are identified by not having a corresponding potential. The coriolis and centrifugal forces are examples. Gravity has a potential, and so derives from potential theory. Hence, by definition, gravity is not a fictitious force.

Going back to the muon then. If the Earth was accelerating with UA, then we would experience time dilation. So the muon’s lifetime would be shorter - it would “age” faster. If the muon instead was relativistic - rather than the Earth, then IT would be time dilated. We would age faster relative to it, and from our perspective, the muon would live longer.

Since the muon lifetime is measured experimentally, we can compare that value to muon production from cosmic ray cascades onto the atmosphere.

The results show that the muon lives longer. Ergo, it is relativistic. Not us. Ergo, UA fails this experimental observation.

[xasop]

Fictitious forces are identified by not having a corresponding potential. The coriolis and centrifugal forces are examples. Gravity has a potential, and so derives from potential theory. Hence, by definition, gravity is not a fictitious force.

If you accept general relativity and its description of gravitation, then gravity is a fictitious force. Do you accept general relativity?

Going back to the muon then. If the Earth was accelerating with UA, then we would experience time dilation.

Ditto if the Earth is round with Einsteinian gravitation.

So the muon’s lifetime would be shorter - it would “age” faster.

No, that does not follow. The muon is moving at relativistic speeds relative to us, therefore we observe its decay to take longer. Unlike the equivalence principle, this isn't even getting into general relativity, this is basic special relativity.

If the muon instead was relativistic - rather than the Earth

The results show that the muon lives longer. Ergo, it is relativistic. Not us.

Being "relativistic" is a relative property. I don't feel like this should need to be pointed out, as it is in the name, but here we go anyway. If two observers A and B are moving at 0.9c with respect to one another, then A is relativistic from B's frame of reference and B is relativistic from A's frame of reference. This is, again, basic special relativity.

Thus, a muon moving at 0.9c with respect to the Earth is the same thing as the Earth moving at 0.9c with respect to a muon. This follows directly from the idea that there is no absolute frame of reference, which you earlier agreed with.

It is difficult to respond to your points when you make such contradictory statements, because you are now throwing points we have already established agreement upon out of the window. Do we need to go back to basics?

[BRrollin]

Fictitious forces are identified by not having a corresponding potential. The coriolis and centrifugal forces are examples. Gravity has a potential, and so derives from potential theory. Hence, by definition, gravity is not a fictitious force.

If you accept general relativity and its description of gravitation, then gravity is a fictitious force. Do you accept general relativity?

Going back to the muon then. If the Earth was accelerating with UA, then we would experience time dilation.

Ditto if the Earth is round with Einsteinian gravitation.

So the muon’s lifetime would be shorter - it would “age” faster.

No, that does not follow. The muon is moving at relativistic speeds relative to us, therefore we observe its decay to take longer. Unlike the equivalence principle, this isn't even getting into general relativity, this is basic special relativity.

If the muon instead was relativistic - rather than the Earth

The results show that the muon lives longer. Ergo, it is relativistic. Not us.

Being "relativistic" is a relative property. I don't feel like this should need to be pointed out, as it is in the name, but here we go anyway. If two observers A and B are moving at 0.9c with respect to one another, then A is relativistic from B's frame of reference and B is relativistic from A's frame of reference. This is, again, basic special relativity.

Thus, a muon moving at 0.9c with respect to the Earth is the same thing as the Earth moving at 0.9c with respect to a muon. This follows directly from the idea that there is no absolute frame of reference, which you earlier agreed with.

It is difficult to respond to your points when you make such contradictory statements, because you are now throwing points we have already established agreement upon out of the window. Do we need to go back to basics?

So we are not at the scale of General relativity, we are in the weak field limit which reduces to Newton. But even if you appeal to GR’s construction of gravity as geometry, that does not posit gravity as a fictitious force in classical mechanics. That is just not the correct definition. I understand your argument, but GR does not somehow make gravity a fictitious force in CM. In CM, you define fictitious forces as any force which lacks a potential function in the lagrangian formulation. Gravity has this. The definitions are just more rigorous than how you are using them.

“Ditto if the Earth is round with Einsteinian gravitation.”

What is Einsteinian gravitation? You mean GR? 

“Thus, a muon moving at 0.9c with respect to the Earth is the same thing as the Earth moving at 0.9c with respect to a muon. This follows directly from the idea that there is no absolute frame of reference, which you earlier agreed with.”

This is true. You are correct. But you’re missing the whole point. I am failing at getting you to see this. The fault is mine. Let me try one more time:

In UA, we are not an inertial frame. Yes? I think that is clear.  So you can’t USE special relativity.

This is what I’ve been trying to tell you this whole time. To use special relativity in the first place, you MUST have inertial frames. It’s in the postulates!

This is why the twin paradox is not really a paradox. This puzzle is used to teach students which situations can use SR and which cannot.

Hence, if you hold that UA is correct, then you automatically lose relativity as method for investigating consequences.

UA must either:

(A) invent a new theory to replace SR that is consistent with UA and predicts the muon lifetimes and dilations,

(B) bow out of the muon debate.

(C) some other option I haven’t thought of...

But, you absolutely cannot use SR principles and apply them to UA.

I hope this is helpful. I’m not trying to “beat” you in a debate, rather it is unfortunate to see anyone spend time on a dead-end endeavor.

[xasop]

So we are not at the scale of General relativity, we are in the weak field limit which reduces to Newton.

Please stop with this double standard. Either you can treat a 9.8 m s^-2 acceleration as negligible in both RET and UA, or neither. Given that your whole argument is based on treating UA as non-negligible, we must use general relativity in RET to compare apples to apples.

But even if you appeal to GR’s construction of gravity as geometry, that does not posit gravity as a fictitious force in classical mechanics.

Indeed, GR instead posits that classical mechanics is incorrect, albeit useful in many non-relativistic situations. For the purposes of this discussion, the distinction between classical mechanics being wrong and classical mechanics treating gravity as a fictitious force is academic.

What is Einsteinian gravitation? You mean GR?

Yes.

In UA, we are not an inertial frame. Yes? I think that is clear.  So you can’t USE special relativity.

Then we go back to my point that we are not in an inertial frame of reference according to RET, either. Your argument for why this doesn't matter was that the discrepancy between a non-inertial clock and an inertial one would be negligible in the time it takes for a muon to decay.

Either that argument holds, and special relativity applies with UA, or it doesn't hold and special relativity does not apply with RET. Once again, I would like to ask you to make up your mind.

Bottom line, I would like you to answer this one question. Is a 9.8 m s^-2 proper acceleration in the observer's frame of reference negligible for the purposes of the muon experiment?

[BRrollin]

So we are not at the scale of General relativity, we are in the weak field limit which reduces to Newton.

Please stop with this double standard. Either you can treat a 9.8 m s^-2 acceleration as negligible in both RET and UA, or neither. Given that your whole argument is based on treating UA as non-negligible, we must use general relativity in RET to compare apples to apples.

But even if you appeal to GR’s construction of gravity as geometry, that does not posit gravity as a fictitious force in classical mechanics.

Indeed, GR instead posits that classical mechanics is incorrect, albeit useful in many non-relativistic situations. For the purposes of this discussion, the distinction between classical mechanics being wrong and classical mechanics treating gravity as a fictitious force is academic.

What is Einsteinian gravitation? You mean GR?

Yes.

In UA, we are not an inertial frame. Yes? I think that is clear.  So you can’t USE special relativity.

Then we go back to my point that we are not in an inertial frame of reference according to RET, either. Your argument for why this doesn't matter was that the discrepancy between a non-inertial clock and an inertial one would be negligible in the time it takes for a muon to decay.

Either that argument holds, and special relativity applies with UA, or it doesn't hold and special relativity does not apply with RET. Once again, I would like to ask you to make up your mind.

Bottom line, I would like you to answer this one question. Is a 9.8 m s^-2 proper acceleration in the observer's frame of reference negligible for the purposes of the muon experiment?

Lol, there is no double standard here. You keep participating as though one of us must “win.” Why this insistence for an adversarial exchange? 

The existence of a free-fall acceleration does not guarantee the applicability of GR.

It looks like you are now trying to pivot to GR as a comparative theory rather than SR. That’s probably a more lucrative strategy, and I am happy to discuss GR with you, but be warned, it is incredibly difficult.

Yes, I know the proper definition of fictitious forces is academic. I’ll level with ya: most of the epicyclic subtopics we have been discussing have been academic, but with the intention of being helpful, I have been patient.

“Either that argument holds, and special relativity applies with UA, or it doesn't hold and special relativity does not apply with RET. Once again, I would like to ask you to make up your mind.”

This is a false dichotomy. Special relativity does not hold in UA, but does apply in RET.

“Bottom line, I would like you to answer this one question. Is a 9.8 m s-2 proper acceleration in the observer's frame of reference negligible for the purposes of the muon experiment?”

NO! It is a dealbreaker! In UA,
that is indeed your proper acceleration, and SR is inapplicable. In RET, your proper acceleration standing on the earth’s surface is ZERO - because the Earth’s surface is in the way! If you’re in free-fall, then yes, now you’re fucked.

In UA, just by standing on the Earth, you are in a noninertial frame. In RET, standing on the Earth, you are in an inertial frame.

[xasop]

“Bottom line, I would like you to answer this one question. Is a 9.8 m s-2 proper acceleration in the observer's frame of reference negligible for the purposes of the muon experiment?”

NO! It is a dealbreaker! In UA,
that is indeed your proper acceleration, and SR is inapplicable. In RET, your proper acceleration standing on the earth’s surface is ZERO - because the Earth’s surface is in the way! If you’re in free-fall, then yes, now you’re fucked.

In UA, just by standing on the Earth, you are in a noninertial frame. In RET, standing on the Earth, you are in an inertial frame.

Nope. GR says that free-fall is an inertial frame of reference because you are following a geodesic in space-time. Being in contact with the surface of the Earth causes an upward proper acceleration of 9.8 m s^-2 to stop you from falling, which is why gravity is a fictitious force. This is the equivalence principle.

So, given that your answer is "NO! It is a dealbreaker!", the clocks in the muon experiment cannot be considered inertial in either UA or RET with GR. As you have already indicated that you accept GR, I don't see how your case remains at all defensible.

Indeed, since you have asserted that a consequence of such a proper acceleration is that our observations would be inconsistent with reality, you must now surely conclude that neither UA nor GR can be valid, but I'll let you grapple with that one on your own.

[BRrollin]

“Bottom line, I would like you to answer this one question. Is a 9.8 m s-2 proper acceleration in the observer's frame of reference negligible for the purposes of the muon experiment?”

NO! It is a dealbreaker! In UA,
that is indeed your proper acceleration, and SR is inapplicable. In RET, your proper acceleration standing on the earth’s surface is ZERO - because the Earth’s surface is in the way! If you’re in free-fall, then yes, now you’re fucked.

In UA, just by standing on the Earth, you are in a noninertial frame. In RET, standing on the Earth, you are in an inertial frame.

Nope. GR says that free-fall is an inertial frame of reference because you are following a geodesic in space-time. Being in contact with the surface of the Earth causes an upward proper acceleration of 9.8 m s^-2 to stop you from falling, which is why gravity is a fictitious force. This is the equivalence principle.

So, given that your answer is "NO! It is a dealbreaker!", the clocks in the muon experiment cannot be considered inertial in either UA or RET with GR. As you have already indicated that you accept GR, I don't see how your case remains at all defensible.

Indeed, since you have asserted that a consequence of such a proper acceleration is that our observations would be inconsistent with reality, you must now surely conclude that neither UA nor GR can be valid, but I'll let you grapple with that one on your own.

I think you may benefit from some additional study of inertial frames. This also serves as published evidence for my statements:

https://arxiv.org/pdf/1310.4465.pdf

The read is technical and long, but you WERE warned.

Since you seem adamant about not believing the current dictates of physics, I will let the article speak for itself. We can discuss the points within it, but I won’t further discuss issues beyond it - unless you likewise provide published evidence as context.

The discussion has devolved into:

Me: this is what physics says.

You: no, THIS is what physics says.

So we turn to published physics, and discuss what it says

[xasop]

We can discuss the points within it, but I won’t further discuss issues beyond it - unless you likewise provide published evidence as context.

That's fine by me. As far as I'm concerned, your objections to UA have been adequately refuted, and we can now all get on with more productive discussions in peace.

[BRrollin]

We can discuss the points within it, but I won’t further discuss issues beyond it - unless you likewise provide published evidence as context.

That's fine by me. As far as I'm concerned, your objections to UA have been adequately refuted, and we can now all get on with more productive discussions in peace.

Well, no, they haven’t, lol. The article will help you learn why you are mistaken in your understanding of inertial frames in GR. Why in UA it is a noninertial frame, and why in RET it is an inertial frame.

I have provided published evidence to support my position - that’s the whole point of linking it.

It is bizarre that you think the matter has somehow now been closed! I have evidence in support, but as far as I can tell, you have only the recalcitrant insistence that your claims are de facto correct.

It is your prerogative of course, but I am more concerned with my beliefs being true

[xasop]

It is your prerogative of course, but I am more concerned with my beliefs being true

And I'm fine with letting you believe what you want. Just please stop wasting people's time with poorly concocted arguments for why they shouldn't maintain the existing UA model.

[Groit]

Being "relativistic" is a relative property. I don't feel like this should need to be pointed out, as it is in the name, but here we go anyway. If two observers A and B are moving at 0.9c with respect to one another, then A is relativistic from B's frame of reference and B is relativistic from A's frame of reference. This is, again, basic special relativity.

Thus, a muon moving at 0.9c with respect to the Earth is the same thing as the Earth moving at 0.9c with respect to a muon. This follows directly from the idea that there is no absolute frame of reference, which you earlier agreed with.

Alright then, so we can say that the muons are travelling at some non-relativistic speed, and the Earth is accelerating towards them. When scientists perform the experiment, they find that their relative velocity is 0.9 c, that's fine, UA works and it seems reasonable...

However, one week later the scientist carry out the same experiment again, and bear in mind that for the past week the Earth has been accelerating so its velocity has increased by
They now find that more muons are reaching the surface of the Earth and thus their relative velocity and time dilation has increased. A week later and they do another test, again different results... and so on.

UA model does not fit into the way we measure muons at the surface of the Earth. This experiment has been done time and time again over many years and the results are always the same, this is because the Earth is moving at a non-relativistic speed i.e. in its orbit around the sun etc... and the muons are travelling towards the Earth at approx 0.98 c. There is no acceleration for the Earth or muons.

[xasop]

However, one week later the scientist carry out the same experiment again, and bear in mind that for the past week the Earth has been accelerating so its velocity has increased by
They now find that more muons are reaching the surface of the Earth and thus their relative velocity and time dilation has increased. A week later and they do another test, again different results... and so on.

We don't observe this. What are you basing this claim on?

[BRrollin]

However, one week later the scientist carry out the same experiment again, and bear in mind that for the past week the Earth has been accelerating so its velocity has increased by
They now find that more muons are reaching the surface of the Earth and thus their relative velocity and time dilation has increased. A week later and they do another test, again different results... and so on.

We don't observe this. What are you basing this claim on?

Precisely, this change in the muon spectrum is NOT observed - but should be in UA.

Here’s an article that provides details:

https://cds.cern.ch/record/427778/files/0002052.pdf

Since an increasing muon energy would impact other observables, such as neutron production, so we should see an increase in those energies too. But we dont:

https://scholar.google.com/scholar?q=cosmic+ray+muons&hl=en&as_sdt=0&as_vis=1&oi=scholart#d=gs_qabs&u=%23p%3D7KRZ0fQ4F94J

The Particle Data Group has a nice summary article of muon observations. Note the lack of increasing muon energies with respect to time:

http://pdg.lbl.gov/2011/reviews/rpp2011-rev-cosmic-rays.pdf

I’ll finally note that these observations have been been conducted for decades. So there is a literal anthology of observations that UA doesn’t match.

[xasop]

Precisely, this change in the muon spectrum is NOT observed - but should be in UA.

You didn't answer my question. Why do you claim that it should be in UA?

[BRrollin]

Precisely, this change in the muon spectrum is NOT observed - but should be in UA.

You didn't answer my question. Why do you claim that it should be in UA?

We’ve...already discussed this. Are you messing with me? I don’t want to just be repeating myself....

Let’s go point by point. Maybe you can indicate where we are missing each other.

What follows is with respect to a muon detection event on the Earth’s surface using probably a scintillator and amplifier. The muon origin is a cosmic ray that creates a cascade upon interaction with the atmosphere.

Sorry, not trying to be pedantic, just getting super precise in order to find our mismatch. If you disagree with the situation above, please indicate. The points then follow:

1. In UA, we are in a noninertial frame. Agree?

2. In RET, we are in an inertial frame. Agree?

3. If agree to 1 above then SR is invalid to use. Agree?

4. If agree to 2 above then SR is valid to use. Agree?

We may end up stopped by this point to discuss. But in case not:

5. GR on the earth’s surface is a weak field limit. The spacetime curvature is small - that is, the Riemann tensor is perturbative in Minkowski space. Upon completing the expansion, Einstein’s field equations reduce to Newtonian mechanics.

Therefore, to appeal to GR in UA means to accept newton - it is hence self-contradictory, because UA assumes no Newton.

You have constructed a proof by contradiction against UA in using GR.

The only way to avoid this is to refuse the weak field limit. This is incorrect GR, but presumably you wish to modify it in some fashion. I assume so, otherwise why are we even discussing it?

The most direct way to modify it is to absorb the weak field expansion terms into the stress energy tensor. This preserves the equality. I assume this is what you intend...

That means objects entering that spacetime will encounter a shift in their spectrum. But what reference frame do you COMPARE this shift?

I hope you got something out of that article I sent you on GR frames.

Anyway, you simply measure two inbound muons - with some time delay between them. Since you are in a noninertial frame, the stress energy tensor must absorb more terms as time proceeds. So you compare the spectrum to what you got a week ago, or whatever.

But this is not observed.

I’m trying to help you construct a different modification to GR that mighty be better - hence the articles I sent last time.

I look forward to your reply

[xasop]

1. In UA, we are in a noninertial frame. Agree?

Agree.

2. In RET, we are in an inertial frame. Agree?

Disagree, as I have stated previously. But even if we cannot come to an agreement on this point...

3. If agree to 1 above then SR is invalid to use. Agree?

Disagree, because for the purposes of the muon experiment, the difference in observation from an inertial frame of reference is negligible.

Let's try a thought experiment. Suppose we hollow out the Burj Khalifa into a vertical hyperloop. We can now send objects through it that can take measurements as inertial observers, unaffected by the Earth's acceleration.

Now, launch a clock upwards into the Burj Khalifa, which will rise to the top and then fall back down. As it is not being accelerated by UA, it is an inertial observer which can be analysed using SR.

My assertion is that the difference between the muon decay time measured by a clock on the surface of the Earth, and that measured by our Burj hyperloop clock, will be so tiny as to be immeasurable, and in any case insignificant for the purposes of the muon experiment. Agree?

[BRrollin]

1. In UA, we are in a noninertial frame. Agree?

Agree.

2. In RET, we are in an inertial frame. Agree?

Disagree, as I have stated previously. But even if we cannot come to an agreement on this point...

3. If agree to 1 above then SR is invalid to use. Agree?

Disagree, because for the purposes of the muon experiment, the difference in observation from an inertial frame of reference is negligible.

Let's try a thought experiment. Suppose we hollow out the Burj Khalifa into a vertical hyperloop. We can now send objects through it that can take measurements as inertial observers, unaffected by the Earth's acceleration.

Now, launch a clock upwards into the Burj Khalifa, which will rise to the top and then fall back down. As it is not being accelerated by UA, it is an inertial observer which can be analysed using SR.

My assertion is that the difference between the muon decay time measured by a clock on the surface of the Earth, and that measured by our Burj hyperloop clock, will be so tiny as to be immeasurable, and in any case insignificant for the purposes of the muon experiment. Agree?

Good, this is useful for us.

For disagreement on 2. Yes, the statement is false. We are not in inertial frame in RET, because the Earth is spinning and in orbit. But on the timescales and distance scales of muon experiment, those accelerations are negligible. This is why we can perform table top experiments on earth and not get fictitious terms.

For the Burj example.

Absolutely disagree. The launched clock is not in an inertial frame, because it must accelerate to change from going up to going down.

We will probably be at an impasse until we thoroughly examine this disagreement.

You might want to look up the twin paradox - it’s resolution explains why your example cannot be accurate.

As for your assertion on synchronous clocks in the muon experiment. If you attempt to apply SR to this situation, you will not have unique world lines for your events. Moreover, if you assume synchronous clocks, you will identify contradictory space time events that will not be light-like.

Hence, it cannot be a correct approximation to treat UA as inertial. 

Not to be bossy, but it would probably be illuminating if you went through the computation. That way, you would “see it for yourself.” Which always seems to be more impactful than if someone else shows it.

[xasop]

Absolutely disagree. The launched clock is not in an inertial frame, because it must accelerate to change from going up to going down.

This just shows that you don't even understand the basic principle of UA, let alone being able to discredit it. Under UA, the Earth only accelerates up. Free-falling projectiles do not accelerate down. A projectile that goes up and then down again to an observer on Earth is not accelerating, it is the observer on Earth who accelerates.

[BRrollin]

Absolutely disagree. The launched clock is not in an inertial frame, because it must accelerate to change from going up to going down.

This just shows that you don't even understand the basic principle of UA, let alone being able to discredit it. Under UA, the Earth only accelerates up. Free-falling projectiles do not accelerate down. A projectile that goes up and then down again to an observer on Earth is not accelerating, it is the observer on Earth who accelerates.

Not according to the definition of acceleration. An object that changes its direction of motion accelerates. An object starting on the Earth in UA is already accelerating. As it moves up, it must accelerate even more in order to do so. It then must decelerate to stop, and the Earth to catch up.

Just draw a free body diagram for this. Or try to demonstrate your claim using kinematics equations. You will see this.