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## Implications of the statement "Acceleration is not relative"

 Quote by GregAshmore I do not assume that the non-inertial observer in the rocket can legitimately claim that he is at rest throughout. That must be proven.
Do you assume that an *inertial* observer can legitimately claim that he is at rest? If so, what's the difference? What makes an inertial observer special?

 Quote by GregAshmore So what makes a claim to be at rest "legitimate"?
To me the answer is "mu": the question itself presupposes that "at rest" has some absolute meaning. It doesn't; "at rest" is relative. That means the only requirement is indeed this:

 Quote by GregAshmore If the only requirement is the development of one set of equations that can be used by all at-rest observers to correctly predict the behavior of physical systems, then I don't think I have any quarrel with relativity.
 Quote by GregAshmore if the requirement is that the observed universe be "real" to every at-rest observer, then I'm not so sure.
"Real" is too vague a term to be useful here, IMO. Perhaps the term "fictitious" has given a wrong impression:

 Quote by GregAshmore I'm fine with fictitious quantities if they allow me to correctly calculate what is going to happen. (I'm an engineer, not a theoretician.) But if the proceedings seen by the observer in the resting rocket are fictitious, I am not comfortable affirming the statement that all coordinate systems are equally valid.
Events are real to all observers. Does that help? For example, if two twins meet up and find that one's clock has less elapsed time than the other's, that's not a "fictitious proceeding". But if one twin says "well, your clock had more elapsed time because you were higher up than me in a gravitational field while my rocket was firing", the gravitational field could be termed "fictitious". But that's more a matter of terminology or interpretation than physics; the traveling twin wants to interpret everything in his "rest frame", but that frame is non-inertial, so physics doesn't look as simple. "Gravitational field" is just a label he puts on the lack of simplicity; but the lack of simplicity is there because of the coordinates he chose. Nothing forces him to use coordinates in which he is always at rest.

 Quote by GregAshmore It seems to me that a coordinate system that has the stars violating a law of physics--traveling faster than light speed--has a lesser validity than a coordinate system that has the stars behaving within the laws of physics.
The stars aren't violating any laws of physics. The actual law is not "things can't travel faster than light"; it is "things can't move outside the local light cones". All of the stars' worldlines are within their local light cones.

Once again, you appear to want to have it both ways; you want the "laws of physics" to look simple, but you want to be able to choose any coordinates you like. You can't have both of those things.

 Quote by GregAshmore I don't know how proper acceleration is calculated.
You calculate the path curvature of the rocket's worldline. That can be done in any coordinates, including ones in which the rocket is at rest.

 Quote by GregAshmore It is claimed that motions are not absolute. That must be proven.
How do you want it proven? It has already been shown that "motion" depends on the coordinates you choose. What more do you need?

 Quote by GregAshmore The premise is that every observer may legitimately consider himself to be at rest. For the premise to have any meaning at all, "at rest" must mean "absolutely at rest".
I disagree; you can't call anything "absolute" if it depends on the coordinates you adopt.

 Quote by GregAshmore Every observer develops the laws of physics on the assumption that he is at rest in absolute space.
Really? When you're driving your car, do you intuit its physics based on the assumption that you are at rest in absolute space and everything else is moving? If you do, you're pretty unusual; most people talk about "going somewhere" in their car (or walking or bicycling or any other way, for that matter), not "making the grocery store come to me using my car".

 Quote by GregAshmore If all such observers are able to agree on one set of physical laws ("of the same form"), then the premise that motion is not absolute is proven to be true.
Then it's proven; GR provides just such a set of laws. But in some coordinates, a bunch of the terms in the equations become zero, so the laws look simpler in those coordinates.

 Quote by GregAshmore Your definition of physical assumes the truth of the premise that is to be proven.
Then give an alternative definition that doesn't make any assumptions relevant to the argument.

 Quote by GregAshmore The task is for every resting observer to explain what he sees, expressed as laws of physics.
I agree with this. But that doesn't imply this:

 Quote by GregAshmore The observer in the resting rocket sees the Earth moving by itself; that is an effect which must be explained.
No, the observer in the rocket sees the Earth moving *relative to him*. That's how the laws work. You don't get to declare by fiat that the laws *must* take a certain form, or *must* deal with "absolute rest" or "absolute motion". You have to find out whether they do by finding out what the laws are. It turns out that the actual laws--the laws of GR--do *not* talk about absolute rest or absolute motion; they talk only about relative rest and relative motion. If you want laws that talk about absolute rest and absolute motion, you're going to be disappointed, because there aren't any.

 Quote by GregAshmore The laws of physics must explain the motion of the Earth by itself, and the rocket by itself, if the principle of relativity is true.
No, the principle of relativity says that *relative* motion is what matters; it says that there is no such thing as absolute motion or absolute rest.

 Quote by GregAshmore Einstein says that "a gravitational field appears" when the rocket is fired. Then, "when the clock U1 has reached velocity v the gravitational field disappears."
Yes, but he's using the term "gravitational field" in different ways (possibly without realizing it). When he talks about the field being produced by "induction", there *has* to be a time delay involved; but that means there has to be *something* propagating even when the rocket is not firing. That something is the "gravitational field" that is produced by "induction", with the distant stars as the source. When he talks about the field appearing and disappearing, he's using "gravitational field" to mean the force that is felt only when the rocket is firing; but the "field" (the underlying whatever-it-is that produces the induction effect) is there whether or not the rocket is firing.

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 Quote by GregAshmore Does this mean that we can put an engine-less pod in space (without gravity due to mass, per the scenario) and then select the appropriate mapping, at will and as needed, to accelerate the Earth and stars until the star of our choice meets up with the pod?
The meeting of worldlines is a frame-invariant geometric fact which cannot be changed through a choice of coordinates.

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 Quote by GregAshmore Specifically, I do not assume that the non-inertial observer in the rocket can legitimately claim that he is at rest throughout. That must be proven.
What do you think of the following:
- The rocket cannot consider themselves inertial. This means that the simplest form of laws of physics cannot be used. A more complex way of expressing laws, that is also true (by natural vanishing of extra terms) for inertial motion, can be used. Thus, if you choose the more complex expression, laws are the same for all motion; however, this in no way changes that inertial and non-inertial motions are in-equivalent.
- The rocket is clearly at rest relative relative to itself. There is no escaping this, so it is clearly a legitimate thing to recognize.

 Quote by GregAshmore .... But if the requirement is that the observed universe be "real" to every at-rest observer, then I'm not so sure.
What can you possibly mean by this? How can the universe become less real because you are in a rocket firing thrust? I assume this isn't really what you mean, but I am at a loss for what you possibly could mean.
 Quote by GregAshmore As I read the posts in this thread regarding what the observer in the resting rocket observes, I saw the word "fictitious" many times. I'm fine with fictitious quantities if they allow me to correctly calculate what is going to happen. (I'm an engineer, not a theoretician.) But if the proceedings seen by the observer in the resting rocket are fictitious, I am not comfortable affirming the statement that all coordinate systems are equally valid. It seems to me that a coordinate system that has the stars violating a law of physics--traveling faster than light speed--has a lesser validity than a coordinate system that has the stars behaving within the laws of physics. The two coordinate systems may be equally useful, depending on circumstances, but they are not equally valid: one of them is telling a lie about the stars.
I can sympathize with this. Normally, you do not picture that distant mountains move rapidly when you turn your head. However, what would lead to a real problem is trying to apply the simplest form of laws to the 'turning head' frame. The simplest form of laws (only valid in inertial frames) says no matter can travel faster than the speed c. In the turning head frame, there are no limits on coordinate speed; but this law remains in a different way: no matter catches a pulse of light.
 Quote by GregAshmore I don't know how proper acceleration is calculated. If the calculation has the rocket moving, then the observer in the at-rest rocket will dispute the definition. His claim is that he is absolutely at rest, as explained below.
Proper acceleration calculation has been explained a few times in this thread. I will not repeat. However, I stress that proper acceleration can easily be non-zero for an observer at rest in non-inertial coordinates. As I tried to express it above, the rocket is indisputably at rest relative to itself. However, it is also indisputably non-inertial, which allows proper acceleration to be nonzero for an observer at coordinate rest.

I am not going to address the rest of your post because I am curious to your reaction to the above, first.

 Quote by GregAshmore As I read the posts in this thread regarding what the observer in the resting rocket observes, I saw the word "fictitious" many times. I'm fine with fictitious quantities if they allow me to correctly calculate what is going to happen. (I'm an engineer, not a theoretician.) But if the proceedings seen by the observer in the resting rocket are fictitious, I am not comfortable affirming the statement that all coordinate systems are equally valid. It seems to me that a coordinate system that has the stars violating a law of physics--traveling faster than light speed--has a lesser validity than a coordinate system that has the stars behaving within the laws of physics. The two coordinate systems may be equally useful, depending on circumstances, but they are not equally valid: one of them is telling a lie about the stars.
The laws of physics are not violated by noninertial coordinates, it's just that they have a different form when they are expressed in noninertial coordinates.

For example, Newton's laws of motion, when described using inertial Cartesian coordinates x and y, look like this:

$\dfrac{d^2 x}{dt^2} = \dfrac{1}{m} F^x$
$\dfrac{d^2 y}{dt^2} = \dfrac{1}{m} F^y$

If we change to a new coordinate system
$r = \sqrt{x^2 + y^2}$
$\theta = arctan(\dfrac{y}{x})$

then the same equations of motion look like this:

$\dfrac{d^2 r}{dt^2} - r (\dfrac{d\theta}{dt})^2 = \dfrac{1}{m} F^{r}$
$\dfrac{d^2 \theta}{dt^2} + \dfrac{2}{r} \dfrac{dr}{dt} \dfrac{d \theta}{dt} = \dfrac{1}{m} F^{\theta}$

They're the same laws of motion, except written in different coordinates. The form of the laws change in different coordinates, but the physical content does not.

Similarly, the rule of light-speed is, in differential form: If an object travels a distance $\delta x$ in time $\delta t$, then

$(c \delta t)^2 - (\delta x)^2 \geq 0$

That's what the law looks like in Cartesian coordinates. In general coordinates, the same law looks like this:

$g_{\mu \nu}\ \delta x^\mu\ \delta x^\nu \geq 0$ (summed over all indices $\mu$ and $\nu$)

where $g_{\mu \nu}$ are the components of the metric tensor in the new coordinate system. In an inertial Cartesian coordinate system, the metric tensor has the simple form

$g_{tt} = c^2$
$g_{xx} = g_{yy} = g_{zz} = -1$
(with all other components zero).

The laws of physics look different in noninertial or curvilinear coordinates, but they have the same physical content.

 Quote by nitsuj lol that's funny. It's little tough for me to accept Dalespam's wording. But imo it's accurate. From which perspective is physics described? I would say it is from the perspective of modeling. [..]
Somewhat yes; but a physicist never looses sight of the things that he is modelling - in the context of the topic here Einstein also didn't. Never confound physical entities with their mathematical representation.

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 Quote by harrylin Somewhat yes; but a physicist never looses sight of the things that he is modelling - in the context of the topic here Einstein also didn't. Never confound physical entities with their mathematical representation.
Nobody here is doing that either.

 Quote by GregAshmore The two coordinate systems may be equally useful, depending on circumstances, but they are not equally valid: one of them is telling a lie about the stars.
You've got it backwards. Frames are preferred based on usefulness, not because some are more valid. For example, if you find it useful to use conservation of momentum, you prefer to use inertial frames, where conservation of momentum applies. But if you don't need conservation of momentum for your calculations, you might prefer some non-inertial frame. The fact that total momentum is not conserved in the non-inertial frame doesn’t make it less valid, just less practical in some cases. Nobody is lying here. Velocity is simply frame dependent.

 Quote by DaleSpam The meeting of worldlines is a frame-invariant geometric fact which cannot be changed through a choice of coordinates.
I'd like a yes or no answer.

After the resting rocket twin fires his engine and sees the Earth accelerate away, he eventually sees his target star approach. He can measure the distance to the star at intervals and verify that it is indeed getting closer. You have said repeatedly that the motion of the Earth is caused by the choice of coordinates, independent of the firing of the rocket. I give you the very same scenario, except without an engine in the rocket. Can you make the very same events happen? Can you cause the Earth and stars to move by selecting a certain coordinate system?

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 Quote by GregAshmore I'd like a yes or no answer.
But you've asked two different questions. This actually illustrates well the difference between "motion" and relative motion.

 Quote by GregAshmore I give you the very same scenario, except without an engine in the rocket. Can you make the very same events happen?
This is question 1. The answer is no: you can't make the very same events happen because you can't change the relative motion of the Earth and the rocket (or the target star and the rocket) without a rocket engine. (We're assuming no other possibilities, i.e., no gravity, no aliens with tractor beams, etc. )

 Quote by GregAshmore Can you cause the Earth and stars to move by selecting a certain coordinate system?
This is question 2, and is *not* the same as question 1. The answer is yes, of course; just take the coordinate system in which the Earth, the rocket, and the target star are all at rest (since there's no rocket engine, they will always be at rest in this coordinate system), and boost it in some random direction. You now have a coordinate system in which all three are moving, at the same velocity.

What you can't do is get them moving at *different* velocities just by changing coordinate systems. But nobody was claiming that you could; that is, nobody was claiming that you could change their *relative* velocities just by changing coordinates. That's what takes the rocket engine; which is why relative motion is "physical" in a way that "motion" by itself is not.

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 Can you cause the Earth and stars to move by selecting a certain coordinate system?
Yes. Any observer moving relative to the earth and stars will see them moving relative to himself. Velocity is relative.

 Quote by PeterDonis Do you assume that an *inertial* observer can legitimately claim that he is at rest? If so, what's the difference? What makes an inertial observer special?
I don't assume anything with regard to which kind of observer can consider himself at rest. The concept was never of more than passing interest until I read Einstein's book Relativity. I didn't have any pre-conceived ideas going in.

 Quote by PeterDonis To me the answer is "mu": the question itself presupposes that "at rest" has some absolute meaning. It doesn't; "at rest" is relative. That means the only requirement is indeed this:
I don't think that is true, as a matter of logic. "At rest" must have an absolute meaning for the observer who claims it. Einstein's stated goal was to show that acceleration does not have any absolute quality. Absolute acceleration (as Einstein used the term acceleration in the book referenced above) implies absolute motion. The charge is, "You are accelerating; therefore I am certain that you are in absolute motion." Einstein counters the charge with, "No. I am permanently at rest." For that statement to have any effectiveness against the charge of absolute motion, the rest spoken of must be absolute. If the rest is not absolute, then the observer must admit that he is moving, or at least might be moving. The observer must believe that he is absolutely at rest; he must evaluate all the evidence on the presumption that he is absolutely at rest. You will note that I have not said that any observer is absolutely at rest in actual fact, only that he evaluates what he observes on that basis. So for the resting rocket twin, the Earth moves by itself; the Earth moves absolutely.

I'm going to skip replying to some of your responses. I need more time, and probably more study, to give a good answer.

 Quote by PeterDonis Events are real to all observers. Does that help? For example, if two twins meet up and find that one's clock has less elapsed time than the other's, that's not a "fictitious proceeding". But if one twin says "well, your clock had more elapsed time because you were higher up than me in a gravitational field while my rocket was firing", the gravitational field could be termed "fictitious". But that's more a matter of terminology or interpretation than physics; the traveling twin wants to interpret everything in his "rest frame", but that frame is non-inertial, so physics doesn't look as simple. "Gravitational field" is just a label he puts on the lack of simplicity; but the lack of simplicity is there because of the coordinates he chose. Nothing forces him to use coordinates in which he is always at rest.
The difference in elapsed clock times is not the issue--not any longer. The issue is whether the rocket twin is moving in some absolute sense, or can legitimately claim to be at rest.

As to why the rocket twin insists on claiming that he is at rest, I guess you'ld have to ask Einstein. I certainly never thought to make an issue of it until I read his book.

 Quote by PeterDonis Once again, you appear to want to have it both ways; you want the "laws of physics" to look simple, but you want to be able to choose any coordinates you like. You can't have both of those things.
Again, the issue for me is not the simplicity or complexity of the equations. The issue is whether the claim to be at rest makes physical sense. See the end of this post.

 Quote by PeterDonis You calculate the path curvature of the rocket's worldline. That can be done in any coordinates, including ones in which the rocket is at rest.
Ok. I don't recall seeing this definition before. If it was in this thread, I missed it.

 Quote by PeterDonis Really? When you're driving your car, do you intuit its physics based on the assumption that you are at rest in absolute space and everything else is moving? If you do, you're pretty unusual; most people talk about "going somewhere" in their car (or walking or bicycling or any other way, for that matter), not "making the grocery store come to me using my car".
In the context, I thought it was clear that I meant every resting observer.

 Quote by PeterDonis Yes, but he's using the term "gravitational field" in different ways (possibly without realizing it). When he talks about the field being produced by "induction", there *has* to be a time delay involved; but that means there has to be *something* propagating even when the rocket is not firing. That something is the "gravitational field" that is produced by "induction", with the distant stars as the source. When he talks about the field appearing and disappearing, he's using "gravitational field" to mean the force that is felt only when the rocket is firing; but the "field" (the underlying whatever-it-is that produces the induction effect) is there whether or not the rocket is firing.
Prompted by the above, I read this again:
 But all the stars that are in the universe, can be conceived as taking part in bringing forth the gravitational field; because during the accelerated phases of the coordinate system K' they are accelerated relative to the latter and thereby can induce a gravitational field, similar to how electric charges in accelerated motion can induce an electric field.
This is consistent with the idea that the field is present all the time. But there seems to be a causal conundrum here. The effect of the field is not felt until the Earth accelerates. But the Earth only accelerates due to the effect of the field.

If no plausible explanation for the workings of the gravitational field can be given, the field must be considered a pure fiction; ad-hoc hand-waving. In that case, the absoluteness of acceleration is not removed, at least with respect to SR.

Personally, I have no stake in the argument. I don't care if there is an absoluteness to acceleration. But the issue was important to Einstein, and having come this far, I'd like to be able to evaluate whether he succeeded in eliminating the problem.

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 Quote by GregAshmore If no plausible explanation for the workings of the gravitational field can be given, the field must be considered a pure fiction; ad-hoc hand-waving. In that case, the absoluteness of acceleration is not removed, at least with respect to SR.
Gravity produces coordinate acceleration but rockets produce proper acceleration. The field that Einstein conjures up produces the first kind - which is relative. But, as you have been told, proper acceleration is not coordinate dependent and cannot be made to disappear or appear by a change of coordinates. Proper acceleration is absolute in this sense.

 Personally, I have no stake in the argument. I don't care if there is an absoluteness to acceleration. But the issue was important to Einstein, and having come this far, I'd like to be able to evaluate whether he succeeded in eliminating the problem.
Again, it's been said before, Einstein does not make it clear which kind he refers to so he has not succeeded, in my opinion.

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 Quote by GregAshmore I don't assume anything with regard to which kind of observer can consider himself at rest.
Then we're missing a big piece of the puzzle, because we haven't established, have we, that the principle of relativity applies to *inertial* observers? If we haven't even done that, how can we do it for non-inertial observers?

 Quote by GregAshmore I don't think that is true, as a matter of logic. "At rest" must have an absolute meaning for the observer who claims it.
How can it be absolute if it's only true for the observer who claims it? Doesn't that make it relative? Isn't that the whole *point* of making the distinction between "relative" and "absolute"?

 Quote by GregAshmore Einstein's stated goal was to show that acceleration does not have any absolute quality.
Just like "motion" and "rest". That was his whole point. He wasn't trying to argue that motion or rest were absolute. If you think he was arguing that, you have seriously misunderstood his point.

 Quote by GregAshmore The charge is, "You are accelerating; therefore I am certain that you are in absolute motion." Einstein counters the charge with, "No. I am permanently at rest."
No, you're misunderstanding what he said. A better statement of his counter is: "No. There is no such thing as absolute motion. Acceleration, like motion and rest, is relative." Then he investigates what it takes to consistently maintain such a position, and finds out that to do so, we must also accept that a "gravitational field" (in the sense of "acceleration due to gravity"--mathematically, as DaleSpam pointed out, this corresponds to Christoffel symbols, first derivatives of the metric) is relative; it can be present in some frames and absent in others.

 Quote by GregAshmore In the context, I thought it was clear that I meant every resting observer.
Huh? The position you have been taking is that *every* observer can consider himself to be at rest, so "resting observer" just means any observer whatsoever. *I* can draw distinctions like the one I drew in what you quoted--nobody in real life considers himself to be "at rest" and the grocery store to be moving--because I'm arguing that relative motion is what's important anyway. But how can *you* draw such distinctions without undermining your whole position?

 Quote by GregAshmore The effect of the field is not felt until the Earth accelerates. But the Earth only accelerates due to the effect of the field.
No, the effect of the field is always there. When the rocket engine is off, the rocket and the person inside are free-falling in the field along with the Earth. When the rocket engine fires, it holds the rocket and the person inside at rest in the field so they can't free-fall with the Earth. So the Earth appears to accelerate "downward". But the field is always there; firing the rocket engine just prevents the rocket from free-falling in it.

 Quote by PAllen What do you think of the following:
My reply will have to wait until tomorrow night.

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 Quote by GregAshmore I'd like a yes or no answer.
OK

 Quote by GregAshmore Does this mean that we can put an engine-less pod in space (without gravity due to mass, per the scenario) and then select the appropriate mapping, at will and as needed, to accelerate the Earth and stars until the star of our choice meets up with the pod?
No.

 Quote by GregAshmore After the resting rocket twin fires his engine and sees the Earth accelerate away, he eventually sees his target star approach. He can measure the distance to the star at intervals and verify that it is indeed getting closer. You have said repeatedly that the motion of the Earth is caused by the choice of coordinates, independent of the firing of the rocket. I give you the very same scenario, except without an engine in the rocket. Can you make the very same events happen?
No.

 Quote by GregAshmore Can you cause the Earth and stars to move by selecting a certain coordinate system?
Yes.

 Quote by GregAshmore If no plausible explanation for the workings of the gravitational field can be given, the field must be considered a pure fiction
All of physics is that kind of "fiction". But it is "fiction" that describes nature quantitatively and allows predictions. Newton didn't explain the workings of his gravitational field either. He just quantified it, and this "fiction" still works great for most purposes.

 Quote by stevendaryl The laws of physics are not violated by noninertial coordinates, it's just that they have a different form when they are expressed in noninertial coordinates. For example, Newton's laws of motion, when described using inertial Cartesian coordinates x and y, look like this: $\dfrac{d^2 x}{dt^2} = \dfrac{1}{m} F^x$ $\dfrac{d^2 y}{dt^2} = \dfrac{1}{m} F^y$ If we change to a new coordinate system $r = \sqrt{x^2 + y^2}$ $\theta = arctan(\dfrac{y}{x})$ then the same equations of motion look like this: $\dfrac{d^2 r}{dt^2} - r (\dfrac{d\theta}{dt})^2 = \dfrac{1}{m} F^{r}$ $\dfrac{d^2 \theta}{dt^2} + \dfrac{2}{r} \dfrac{dr}{dt} \dfrac{d \theta}{dt} = \dfrac{1}{m} F^{\theta}$ They're the same laws of motion, except written in different coordinates. The form of the laws change in different coordinates, but the physical content does not.
True, the physical content does not change in this example.

 Quote by stevendaryl The laws of physics look different in noninertial or curvilinear coordinates, but they have the same physical content.
This is not true in the case of the resting rocket. In that coordinate system, no force acts on the Earth, yet it accelerates. And, an unbalanced force acts on the rocket, yet it does not accelerate. Both of these phenomena violate Newton's law as quoted above. Additional physical content is proposed as an explanation: a gravitational field. Additional term(s) are needed in the equation to express the behavior of the additional physical content. I believe (but can't say for sure) that the equation presented by DaleSpam in #278 corresponds to the law of Newton quoted above, with the addition of Christoffel symbols to account for the gravitational field.
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