Einstein's Clock Synchronization Convention

[JesseM]

I don't see why it would--if the metric just tells you the proper time ticked by a clock moving along any given path through spacetime, this shouldn't change even if we discover a preferred frame (unless we discover a new type of 'clock' that does not exhibit the same type of time dilation as normal clocks, in which case I'm not sure what you'd have to do then--have two separate metrics for the different classes of clocks, maybe?)

If a Lorentz violation leads to a preferred frame and that leads to absolute simultaneity, then I suppose that a Lorentz violation leads to an absolute clock.

Sure. But do you disagree with my speculation that this would require us to have separate metrics for the two types of clocks, rather than modifying the original metric?

Even if you say it's not "just" a function, mathematically it would still be a function defining a notion of distance in spacetime, and I don't understand what it means to say this function would "represent something physical". Can you pin down what you mean by "represent"?

The components of the metric can be regarded as representing gravitational potentials.

Are you sure about that? I thought the concept of gravitational potential didn't make sense in GR. But either way, what does this have to do with aether? If you're saying the "something physical" the metric represents is gravitational potential, then doesn't it already represent that even if there is no aether?

That doesn't really answer my question of why you said "Acknowledged. Not only for speeds, but every possible empirical measurement", which seems to suggest the exact opposite (namely, that whatever is true about speeds is also true for every possible empirical measurement). But I suppose it's not really important, now that we're clear on the fact that you agree all measurements of speed are coordinate-dependent while other, more truly "physical" quantities are not.

I meant that speed is a dimensionful quantity, and no experiment can directly measure any dimensionful quantity.

That still doesn't help me make sense of your original comment that you were acknowledging something was true "not only for speeds, but for every possible empirical measurement". If you're now saying the thing you were acknowledging is that speed is a dimensionful quantity, does this sentence say that all empirical measurements are dimensionful quantities? That certainly isn't true--for instance, the ratio between the number of ticks on two clocks which take different paths through spacetime is an empirical quantity that's not dimensionful, and likewise the empirical determination of whether two events coincide at the same point in spacetime is not a dimensionful quantity (it's not a 'quantity' at all). edit: also, I just wanted to add that even dimensionful quantities need not be coordinate-dependent--if we ask how many seconds are ticked by a clock that takes a certain path through spacetime, different observers in different frames systems won't disagree on this, assuming they all define "1 second" in the same way (say, as 9,192,631,770 oscillations of a cesium atom). Do you understand the difference between saying a quantity is dependent on your choice of coordinate system and saying it is dimensionful? They are not the same thing at all.

Convinced of what? Of my points about observers in windowless boxes being able to construct devices whose measurements correspond to the coordinates of the Lorentz transform? The idea of constructing coordinate systems out of rulers and clocks synchronized using the Einstein synchronization procedure (which does not require any knowledge of the outside world) is the whole basis for the derivation of the Lorentz transformation, so it's pretty important that you understand it--if you're in doubt about this, please respond to my last comments in that post.

I haven't gone through everything carefully yet, so I'm not saying that it's unconvincing; I'm saying that I haven't completed a thorough review of it all yet.

Fair enough. But again, this isn't an argument original to me, it's the standard way that I've always seen the Lorentz transform derived.

but I am now viewing the Lorentz transform as an analogy to the homogeneity and isotropy of the Universe; e.g., that there is one and only one local frame in which the universe looks homogeneous and isotropic.

Are you suggesting that the "homogeneity and isotropy of the universe" defines a preferred frame?Doesn't a "local frame" in GR by definition only deal with an arbitrarily small patch of spacetime, so a statement about what the universe as a whole looks like could only be made in a global coordinate system? Also, why would the Lorentz transform be an "analogy" for the idea of a global preferred frame? An analogy in what sense? And what does this have to do with the question of whether the Lorentz transform is the most "natural" one for us to use given what we know about the laws of physics?

I'm saying that what I read in MTW about what is special about comoving coordinates are that the universe looks the same from every perspective because there is no "handle" rings a bell here.

When you say "rings a bell", does that mean it's just sort of a vague intuition that there's some analogy there but don't have a clear idea of what the analogy is? Because this explanation still doesn't give me any clear idea of what you meant when you said "I am now viewing the Lorentz transform as an analogy to the homogeneity and isotropy of the Universe".

Again, the issue of absolute vs. relative simultaneity is fundamentally just a matter of what coordinate systems you use, even with the known laws nothing is stopping you from using the Mansouri-Sexl coordinate systems to describe the world...experimental results only come into play in deciding which set of coordinate systems is more of a "natural" choice, and in that sense the results seen in particle accelerators are relevant because they keep showing that all the fundamental laws have the property of Lorentz-symmetry rather than "LET transform symmetry" (or whatever you want to call it).

If "the aether" is the same thing as "absolute simultaneity", then what you are describing is not "relativity without the aether" is it?

But the aether is not the same thing as absolute simultaneity, that's what I just said in the paragraph you were responding to! Simultaneity is purely a question of your choice of coordinate systems, and you are free to use any set of coordinate systems you want regardless of what the laws of physics are. Again, the laws of physics may make one choice of coordinate system more "natural" than another, but that doesn't mean you can't use unnatural coordinate systems. Even if there is no aether, you can still pick an arbitrary observer to be the "preferred" one and have all other observers synchronize their clocks so that their definition of simultaneity agrees with his; likewise, even if there is an aether you are still free to have each observer synchronize their clocks under the assumption that light moves at c in their rest frames, which will lead different observers to have different definitions of simultaneity.

 

[Aether]

Sure. But do you disagree with my speculation that this would require us to have separate metrics for the two types of clocks, rather than modifying the original metric?

No, I don't disagree, but I'm not settled on how to model every possible Lorentz violation yet. The Minkowski metric is never going to change no matter what new experiments might prove, so an asymmetrical metric would have to be a new metric. However, I like the Minkowski metric just fine as it is. So I may end up keeping the Minkowski metric but treating ds as a vector rather than as a scalar for example. I may also choose to leave both of those alone and just use different coordinates.

Are you sure about that? I thought the concept of gravitational potential didn't make sense in GR. But either way, what does this have to do with aether? If you're saying the "something physical" the metric represents is gravitational potential, then doesn't it already represent that even if there is no aether?

One of my GR textbooks says that "In a more general coordinate system, this Newtonian potential would be dispersed throughout the $$g_{\mu \nu}$$, so there is a sense in which all the components $$g_{\mu \nu}$$ can be regarded as gravitational potentials."

That still doesn't help me make sense of your original comment that you were acknowledging something was true "not only for speeds, but for every possible empirical measurement". If you're now saying the thing you were acknowledging is that speed is a dimensionful quantity, does this sentence say that all empirical measurements are dimensionful quantities? That certainly isn't true--for instance, the ratio between the number of ticks on two clocks which take different paths through spacetime is an empirical quantity that's not dimensionful, and likewise the empirical determination of whether two events coincide at the same point in spacetime is not a dimensionful quantity (it's not a 'quantity' at all).

Dimensionless ratios are measurable, and they are coordinate independent. Nothing else is directly measurable.

edit: also, I just wanted to add that even dimensionful quantities need not be coordinate-dependent--if we ask how many seconds are ticked by a clock that takes a certain path through spacetime, different observers in different frames systems won't disagree on this, assuming they all define "1 second" in the same way (say, as 9,192,631,770 oscillations of a cesium atom). Do you understand the difference between saying a quantity is dependent on your choice of coordinate system and saying it is dimensionful? They are not the same thing at all.

Defining "1 second" that way makes your measurement a dimensionless ratio of the number of oscillations of a cesium atom in your traveling clock per 9,192,631,770 oscillations of a reference cesium atom in one second. Isn't the frame of this reference cesium atom coordinate dependent?

When you say "rings a bell", does that mean it's just sort of a vague intuition that there's some analogy there but don't have a clear idea of what the analogy is? Because this explanation still doesn't give me any clear idea of what you meant when you said "I am now viewing the Lorentz transform as an analogy to the homogeneity and isotropy of the Universe".

At this point it is more than vague intuition, but not quite a clear and convincing understanding. I understand that the homogeneity and isotropy of the universe is not something that is generally true for an observer; to make this true, he must take great pains to seek out a unique local frame. Once he is in that particular frame, then the universe looks homogeneous and isotropic. The Lorentz transform is analogous in that there is one and only one local frame in which the observer sees "something" in the same way as every other observer, and this happens to be his own rest frame (which happens to be much easier to locate than the comoving frame).

But the aether is not the same thing as absolute simultaneity, that's what I just said in the paragraph you were responding to! Simultaneity is purely a question of your choice of coordinate systems, and you are free to use any set of coordinate systems you want regardless of what the laws of physics are.

OK, well I'm using "aether" to mean "absolute simultaneity" as opposed to a luminiferous fluid. I suppose that we may discuss "objective absolute simultaneity" which is a physical consequence of any Lorentz violation, and "subjective absolute simultaneity" which is merely a coordinate system trick. So may we agree that "relativity without at least the 'subjective aether' is pseudoscience"?

Again, the laws of physics may make one choice of coordinate system more "natural" than another, but that doesn't mean you can't use unnatural coordinate systems. Even if there is no aether, you can still pick an arbitrary observer to be the "preferred" one and have all other observers synchronize their clocks so that their definition of simultaneity agrees with his; likewise, even if there is an aether you are still free to have each observer synchronize their clocks under the assumption that light moves at c in their rest frames, which will lead different observers to have different definitions of simultaneity.

This does not describe SR per se. It shows that SR is one carefully chosen slice of a more general science which also allows for LET.

 

[Aether]

From post #45: $$\frac {u_{LET}}{u_{SR}} = \frac { T_{SR1} - T_{SR0} }{ T_{SR1} - T_{SR0} + \frac{v \left( X1-X0 \right) }{c^2} }$$

which gives anisotropy of velocity for v=c and for v<c.

OK, but SR and LET are supposed to be empirically equivalent. So, I wonder if we aren't also supposed to synchronize the two SR clocks before using them with the LET transform and thereby annihilate any differences between the two.

DrGreg, after studying the Mansouri-Sexl papers, do you still think that this is exactly how they intended for their ether transform to be applied? If so, then I'll agree that velocities are anisotropic in LET.

 

[Hans de Vries]

Speeds are anisotropic in LET.

OK, but SR and LET are supposed to be empirically equivalent. So, I wonder if we aren't also supposed to synchronize the two SR clocks before using them with the LET transform and thereby annihilate any differences between the two.

DrGreg, after studying the Mansouri-Sexl papers, do you still think that this is exactly how they intended for their ether transform to be applied? If so, then I'll agree that velocities are anisotropic in LET.

Speeds are anisotropic in LET


Take an SR frame moving at v relative to the preferred frame. Define two
velocities w and -w in this frame moving in the same and in the opposite
direction of v. Via Einsteins velocity addition rule we can calculate what
these two speeds are in the preferred frame:

$$v_1\ = \ \frac{v+w}{1+\frac{vw}{c^2}}, \ \ \ \ \ \ \ v_2\ = \ \frac{v-w}{1-\frac{vw}{c^2}}$$

(see http://www.theory.caltech.edu/people/patricia/minkc.html or
http://www.phys.unsw.edu.au/einsteinlight/jw/module4_Lorentz_transforms.htm)

Now take the M&S formula's for the transformation to the moving LET frame:

$$t\ =\ \sqrt{1-v^2/c^2}\ \ T,\ \ \ \ \ \ \ \ x\ =\ \frac{1}{\sqrt{1-v^2/c^2}}\ (X-vT)$$

Which we may write in differential form:

$$dt\ =\ \sqrt{1-v^2/c^2}\ \ dT,\ \ \ \ \ \ \ \ dx\ =\ \frac{1}{\sqrt{1-v^2/c^2}}\ (dX-vdT)$$

From this we can derive the transformation of velocities from the preferred
frame to the moving LET frame:

$$\frac{dx}{dt}\ =\ \frac{1}{1-v^2/c^2}\ \left(\frac{dX}{dT}-v \right)$$

Now, substitute with $v1$ and $v2$:

$$w_1\ = \ \frac{1}{1-v^2/c^2}\ \left(\frac{v+w}{1+\frac{vw}{c^2}} -v \right), \ \ \ \ \ \ \ w_2\ = \ \frac{1}{1-v^2/c^2}\ \left(\frac{v-w}{1-\frac{vw}{c^2}}-v \right)$$

Which can be simplified further. Now, w and -w in the moving SR frame become:

$$w_1\ = \ \frac{w}{1+\frac{vw}{c^2}}, \ \ \ \ \ \ \ w_2\ = \ \frac{-w}{1-\frac{vw}{c^2}}$$

In the moving LET frame. (You better believe it )

------------------------------------


Regards, Hans

 

[JesseM]

No, I don't disagree, but I'm not settled on how to model every possible Lorentz violation yet. The Minkowski metric is never going to change no matter what new experiments might prove, so an asymmetrical metric would have to be a new metric. However, I like the Minkowski metric just fine as it is. So I may end up keeping the Minkowski metric but treating ds as a vector rather than as a scalar for example.

How can ds be a vector? And since the whole concept of a metric is to define a notion of distance between points on the manifold, you still need a scalar rather than a vector to use in the definition of the metric, don't you?

I may also choose to leave both of those alone and just use different coordinates.

But again, using different coordinates doesn't involve any different physical assumptions in itself.

Are you sure about that? I thought the concept of gravitational potential didn't make sense in GR. But either way, what does this have to do with aether? If you're saying the "something physical" the metric represents is gravitational potential, then doesn't it already represent that even if there is no aether?

One of my GR textbooks says that "In a more general coordinate system, this Newtonian potential would be dispersed throughout the $$g_{\mu \nu}$$, so there is a sense in which all the components $$g_{\mu \nu}$$ can be regarded as gravitational potentials."

Well, I'd like to know the details of what that "sense" is, whether it only works in some limiting case or whether it these components could rigourously be viewed as potentials in an arbitrary curved spacetime. In any case, you didn't answer the second part of my question, the one beginning with "But either way, what does this have to do with aether?"

That still doesn't help me make sense of your original comment that you were acknowledging something was true "not only for speeds, but for every possible empirical measurement". If you're now saying the thing you were acknowledging is that speed is a dimensionful quantity, does this sentence say that all empirical measurements are dimensionful quantities? That certainly isn't true--for instance, the ratio between the number of ticks on two clocks which take different paths through spacetime is an empirical quantity that's not dimensionful, and likewise the empirical determination of whether two events coincide at the same point in spacetime is not a dimensionful quantity (it's not a 'quantity' at all).

Dimensionless ratios are measurable, and they are coordinate independent. Nothing else is directly measurable.

I agree, but you pretty much ignored the point I was making in that paragraph, which is that plenty of "empirical measurements" are dimensionless ratios, like the two I mentioned above. So it still seems wrong for you to have said "not only for speeds, but for every possible empirical measurement" there.

edit: also, I just wanted to add that even dimensionful quantities need not be coordinate-dependent--if we ask how many seconds are ticked by a clock that takes a certain path through spacetime, different observers in different frames systems won't disagree on this, assuming they all define "1 second" in the same way (say, as 9,192,631,770 oscillations of a cesium atom). Do you understand the difference between saying a quantity is dependent on your choice of coordinate system and saying it is dimensionful? They are not the same thing at all.

Defining "1 second" that way makes your measurement a dimensionless ratio of the number of oscillations of a cesium atom in your traveling clock per 9,192,631,770 oscillations of a reference cesium atom in one second. Isn't the frame of this reference cesium atom coordinate dependent?

What measurement are we talking about here? I wasn't talking about measuring the ratio of ticks of two clocks moving at different velocities, I was just talking about measuring the amount of proper time ticked on a single clock which takes a certain path through spacetime. If everyone agrees to define "1 second" the same way--say, by 9,192,631,770 oscillations of a cesium atom moving alongside the clock--then don't you agree that although proper time is a dimensionful quantity, it is coordinate-independent in the sense that all frames will agree on the amount of proper time ticked by a clock that takes a certain path through spacetime?

At this point it is more than vague intuition, but not quite a clear and convincing understanding. I understand that the homogeneity and isotropy of the universe is not something that is generally true for an observer; to make this true, he must take great pains to seek out a unique local frame. Once he is in that particular frame, then the universe looks homogeneous and isotropic. The Lorentz transform is analogous in that there is one and only one local frame in which the observer sees "something" in the same way as every other observer, and this happens to be his own rest frame (which happens to be much easier to locate than the comoving frame).

Wait, what do you mean when you say there's only one local frame in which the observer sees "something"? What is this "something"? The whole point of Lorentz-symmetry is that the laws of physics will look the same in every frame, there's no frame that's "special" in any way. Also, even leaving aside the question of the laws of physics, the Lorentz transform is also symmetric in the sense that any observer who sees another observer moving at velocity v relative to him will use the same equations to transform into that observer's frame, unlike in the LET transform where you don't use the same equation to transform from the preferred frame to another frame as you would to transform from a non-preferred frame to the preferred frame, or to another non-preferred frame.

OK, well I'm using "aether" to mean "absolute simultaneity" as opposed to a luminiferous fluid. I suppose that we may discuss "objective absolute simultaneity" which is a physical consequence of any Lorentz violation, and "subjective absolute simultaneity" which is merely a coordinate system trick.

But all statements about simultaneity are just properties of your coordinate system, I have no idea what you mean by "objective absolute simultaneity". Again, I agree that Lorentz violations might make a coordinate transform which preserves absolute simultaneity to be more "natural" than the Lorentz transform, but it seems like you're talking about something more than just which type of coordinate transform is most "natural".

So may we agree that "relativity without at least the 'subjective aether' is pseudoscience"?

I don't understand what you mean by "relativity without at least the 'subjective aether'". Do you mean relativity that denies the possibility of using coordinate transforms other than the Lorentz transform? I would agree that it is wrong to deny that you're allowed to use any set of coordinate systems you like, if that's what you're saying.

Again, the laws of physics may make one choice of coordinate system more "natural" than another, but that doesn't mean you can't use unnatural coordinate systems. Even if there is no aether, you can still pick an arbitrary observer to be the "preferred" one and have all other observers synchronize their clocks so that their definition of simultaneity agrees with his; likewise, even if there is an aether you are still free to have each observer synchronize their clocks under the assumption that light moves at c in their rest frames, which will lead different observers to have different definitions of simultaneity.

This does not describe SR per se. It shows that SR is one carefully chosen slice of a more general science which also allows for LET.

I think we might disagree on the meaning of "SR". I would say that SR just refers to the hypothesis that the laws of physics are all Lorentz-symmetric (including any unobserved entities which play a fundamental role in the laws of physics, ruling out 'aether'), but does not specify that you must use the coordinate systems produced by the Lorentz transform. For example, I have seen physicists say that SR allows for accelerated coordinate systems, which are clearly not allowed under the Lorentz transform.

 

[Aether]

How can ds be a vector?And since the whole concept of a metric is to define a notion of distance between points on the manifold, you still need a scalar rather than a vector to use in the definition of the metric, don't you? But again, using different coordinates doesn't involve any different physical assumptions in itself.

$$ds^2=dA^2+dB^2+dC^2$$

Well, I'd like to know the details of what that "sense" is, whether it only works in some limiting case or whether it these components could rigourously be viewed as potentials in an arbitrary curved spacetime. In any case, you didn't answer the second part of my question, the one beginning with "But either way, what does this have to do with aether?"

That's only intended as an example of what it might mean for the metric to "represent" something physical.

I agree, but you pretty much ignored the point I was making in that paragraph, which is that plenty of "empirical measurements" are dimensionless ratios, like the two I mentioned above. So it still seems wrong for you to have said "not only for speeds, but for every possible empirical measurement" there.

My original statement may not have been self explanatory, but I don't see how there can still be any room left for doubt that I'm saying that all empirical measurements are dimensionless ratios.

What measurement are we talking about here? I wasn't talking about measuring the ratio of ticks of two clocks moving at different velocities, I was just talking about measuring the amount of proper time ticked on a single clock which takes a certain path through spacetime. If everyone agrees to define "1 second" the same way--say, by 9,192,631,770 oscillations of a cesium atom moving alongside the clock--then don't you agree that although proper time is a dimensionful quantity, it is coordinate-independent in the sense that all frames will agree on the amount of proper time ticked by a clock that takes a certain path through spacetime?

OK, so there is no reference cesium atom needed in this clock, you're counting oscillations and dividing by 9,192,631,770. Perhaps each oscillation of the cesium atom is what is actually measured as a dimensionless ratio against a voltage standard or something like that, and each count is its own dimensionless ratio.

Wait, what do you mean when you say there's only one local frame in which the observer sees "something"? What is this "something"? The whole point of Lorentz-symmetry is that the laws of physics will look the same in every frame, there's no frame that's "special" in any way. Also, even leaving aside the question of the laws of physics, the Lorentz transform is also symmetric in the sense that any observer who sees another observer moving at velocity v relative to him will use the same equations to transform into that observer's frame, unlike in the LET transform where you don't use the same equation to transform from the preferred frame to another frame as you would to transform from a non-preferred frame to the preferred frame, or to another non-preferred frame.

This is not a developed notion just a beginning of appreciation.

But all statements about simultaneity are just properties of your coordinate system, I have no idea what you mean by "objective absolute simultaneity".

"Objective" would apply if I could actually detect a locally preferred frame, but "subjective" applies while I still can't.

Again, I agree that Lorentz violations might make a coordinate transform which preserves absolute simultaneity to be more "natural" than the Lorentz transform, but it seems like you're talking about something more than just which type of coordinate transform is most "natural". I don't understand what you mean by "relativity without at least the 'subjective aether'". Do you mean relativity that denies the possibility of using coordinate transforms other than the Lorentz transform? I would agree that it is wrong to deny that you're allowed to use any set of coordinate systems you like, if that's what you're saying. I think we might disagree on the meaning of "SR". I would say that SR just refers to the hypothesis that the laws of physics are all Lorentz-symmetric (including any unobserved entities which play a fundamental role in the laws of physics, ruling out 'aether'), but does not specify that you must use the coordinate systems produced by the Lorentz transform. For example, I have seen physicists say that SR allows for accelerated coordinate systems, which are clearly not allowed under the Lorentz transform.

One postulate specifies that the laws of physics are all Lorentz symmetric, but the other specifies that the speed of light is a constant; that second postulate seems to specify that you must use the coordinate systems produced by the Lorentz transform. So, by "relativity without the aether", I'm referring to the apparent specification in SR that "you must use the coordinate systems produced by the Lorentz transform". That's OK as long as you only apply it to SR per se, but when that is understood as "experimental proof that LET is false" then that is at best unscientific, and possibly even pseudoscientific depending on the circumstances.

 

[DrGreg]

DrGreg, after studying the Mansouri-Sexl papers, do you still think that this is exactly how they intended for their ether transform to be applied? If so, then I'll agree that velocities are anisotropic in LET.

t_LET means time measured by a clock synchronized to the observer's proper-time clock via "absolute simultaneity".

t_SR means time measured by a clock synchronized to the observer's proper-time clock via the Einsteinian method.

The transformation formulas all take into account the method of synchronization and so no further adjustment is necessary.

I'm glad we now agree on something.

 

[DrGreg]

$$ds^2=dA^2+dB^2+dC^2$$

You seem to think this implies that ds is a vector.

Everything in this equation is a real number (i.e. one-dimensional) even if dA, dB, and dC are components of a vector.

This particular equation can be interpreted as the metric for 3D Euclidean geometry

$$ds^2=dx^2+dy^2+dz^2$$

which is essentially Pythagoras's Theorem in 3 dimensions.

 

[Aether]

You seem to think this implies that ds is a vector.

Everything in this equation is a real number (i.e. one-dimensional) even if dA, dB, and dC are components of a vector.

This particular equation can be interpreted as the metric for 3D Euclidean geometry

$$ds^2=dx^2+dy^2+dz^2$$

which is essentially Pythagoras's Theorem in 3 dimensions.

In post #14 I gave a quote from one of Paul Dirac's books which indicates that the instantaneous velocity of any particle is c, but that the direction of this vector oscillates very rapidly ($$>10^{20}$$ cycles per second) about a mean value of dv=dx+dy+dz. When dv=0, then ds=cdt, and at any given instant this is evidently a vector ds=dA+dB+dC=cdt (where dA is parallel to dx, dB is parallel to dy, and dC is parallel to dz). It only appears to be a scalar when you smear it over a relatively long time period. That looks like a violation of Lorentz symmetry to me, but it is changing so fast that the effect can easily be overlooked in experiments where speeds are averaged over any suitably long time period. Wouldn't it be interesting if two particles 1km apart had ds vectors that pointed instantaneously in the same direction at all times? I don't know that they actually do that, but would like to have metrics and transforms which are general enough to allow me to examine all such possibilities. Lorentz transforms and LET transforms each preserve the magnitude of the line interval ds, but I prefer to have a transform handy that preserves both the magnitude and direction of ds.

 

[DrGreg]

In post #14 I gave a quote from one of Paul Dirac's books...

I can't comment on this quote because

• I don't have a copy of this book and I can't see the context of the quote
• I don't know a great deal about quantum theory anyway

But you are misunderstanding the notation.

ds is always a scalar, never a vector. In the context of a (t, x, y, z) co-ordinate system, it is the length of the vector (dt, dx, dy, dz).

The equations in your last post make no sense to me and I can't understand what you are trying to say.


Note that in a printed book, scalars and other 1-dimensional quantities are written in italics, while vectors are often written in bold font. (Though not all authors follow this convention, especially in pure maths. In GR, vectors may be indicated instead by Greek superscripts such as $$X^\mu$$. (But, just to confuse you further, a numerical superscript like $$X^0$$ indicates a single component of a vector.))

 

[JesseM]

$$ds^2=dA^2+dB^2+dC^2$$

What do dA, dB and dC represent? In any case, this equation would still indicate that ds is a scalar rather than a vector, since it's the norm of the vector (dA, dB, dC).

That's only intended as an example of what it might mean for the metric to "represent" something physical.

But your original comment was not just that it represents "something" physical, but that it represents the aether--you said "I think that the metric represents a physical thing aka 'the aether'". So I still don't understand how a function defining distance in spacetime would "represent" the aether.

My original statement may not have been self explanatory, but I don't see how there can still be any room left for doubt that I'm saying that all empirical measurements are dimensionless ratios.

I agree that all truly empirical measurements are dimensionless, or can be stated in units which are defined in a dimensionless way...I wouldn't say all empirical measurements are dimensionless ratios though (I don't see how counting the number of oscillations of a cesium atom moving along a particular path is a ratio, for example). In any case, this still doesn't help me understand what it was you thought you were acknowledging in your original comment, the thing you were saying was true "not only for speeds, but for every possible empirical measurement"--surely you weren't acknowledging that both empirical measurements and speeds are all dimensionless ratios, because speeds clearly aren't, and the comment of mine you were responding to was one about how speeds are coordinate-dependent. Perhaps we can just agree that your original comment here was confused and move on?

But all statements about simultaneity are just properties of your coordinate system, I have no idea what you mean by "objective absolute simultaneity".

"Objective" would apply if I could actually detect a locally preferred frame, but "subjective" applies while I still can't.

But as I keep saying over and over, even if we did detect a locally preferred frame there would be nothing stopping us from using a coordinate transform that led different reference frames to define simultaneity differently. A coordinate transform where all frames agreed on simultaneity might be more "natural" in this case, but other coordinate transforms would still be permissable. Do you agree or disagree?

I think we might disagree on the meaning of "SR". I would say that SR just refers to the hypothesis that the laws of physics are all Lorentz-symmetric (including any unobserved entities which play a fundamental role in the laws of physics, ruling out 'aether'), but does not specify that you must use the coordinate systems produced by the Lorentz transform. For example, I have seen physicists say that SR allows for accelerated coordinate systems, which are clearly not allowed under the Lorentz transform.

One postulate specifies that the laws of physics are all Lorentz symmetric, but the other specifies that the speed of light is a constant; that second postulate seems to specify that you must use the coordinate systems produced by the Lorentz transform.

I think it's implicit in both of the postulates that they only refer to what would be true in inertial coordinate systems where the clocks are synchronized according to the Einstein synchronization procedure. After all, in Einstein's original paper he spends all of section 1 defining how to construct such coordinate systems out of physical measuring rods and clocks, and then only in section 2 does he lay out the two fundamental postulates. So the postulates don't deny the possibility of creating other types of coordinate systems where the speed of light is no longer constant, as I understand them. And like I said, I have seen physicists say that SR can handle accelerating coordinate systems, and the speed of light would not in general be a constant in such coordinate systems.

So, by "relativity without the aether", I'm referring to the apparent specification in SR that "you must use the coordinate systems produced by the Lorentz transform".

I don't think there is any such specification in "SR" as I understand it.

 

[Aether]

I can't comment on this quote because

• I don't have a copy of this book and I can't see the context of the quote
• I don't know a great deal about quantum theory anyway

The relevant section from the book is section "69. The motion of a free electron" which is only a little over two pages long. If you would like to download a scanned-in copy, let me know and I'll tell you where you can go to download it.

But you are misunderstanding the notation.

ds is always a scalar, never a vector. In the context of a (t, x, y, z) co-ordinate system, it is the length of the vector (dt, dx, dy, dz).

The equations in your last post make no sense to me and I can't understand what you are trying to say.

If I understand correctly what Paul Dirac is saying (and I think I do because I have several of his subsequently published papers and articles in Nature where he says explicitly that "an ether is rather forced upon us" or words to that effect) then I don't see why we can't represent $$\bold{c_0} dt$$ as a vector, and $$d \bold{s}=\bold{c_0} dt-dx-dy-dz$$ as a vector. I understand that for many practical purposes this would be a meaningless complication, but for my purposes I would like to figure out how to do it correctly. On second thought, it may be the vector $$\bold{c_0} dt$$ that should be held invariant in the transformation that I'm trying to develop.

Mansouri-Sexl add three arbitrary synchronization parameters to the LET transformation equation for the t coordinate; one for each direction in space. Making $$\bold{c_0} dt$$ a vector may imply three time coordinates; one for each direction in space.

This relates to my own personal theory, and you are free to ignore it. I'm only mentioning it in response to the questions that you and JesseM have asked.

 

[Aether]

What do dA, dB and dC represent? In any case, this equation would still indicate that ds is a scalar rather than a vector, since it's the norm of the vector (dA, dB, dC).

They are the components of this vector: $$d \bold{s}=\bold{c_0} dt-dx-dy-dz$$, where $$\bold{c_0}$$ is the instananeous velocity of an electron as described by Paul Dirac.

But your original comment was not just that it represents "something" physical, but that it represents the aether--you said "I think that the metric represents a physical thing aka 'the aether'". So I still don't understand how a function defining distance in spacetime would "represent" the aether.

The discussion of $$d \bold{s}$$ as a vector speaks directly and with some precision to the point of what I think that the metric may physically represent.

I agree that all truly empirical measurements are dimensionless, or can be stated in units which are defined in a dimensionless way...I wouldn't say all empirical measurements are dimensionless ratios though (I don't see how counting the number of oscillations of a cesium atom moving along a particular path is a ratio, for example). In any case, this still doesn't help me understand what it was you thought you were acknowledging in your original comment, the thing you were saying was true "not only for speeds, but for every possible empirical measurement"--surely you weren't acknowledging that both empirical measurements and speeds are all dimensionless ratios, because speeds clearly aren't, and the comment of mine you were responding to was one about how speeds are coordinate-dependent. Perhaps we can just agree that your original comment here was confused and move on?

You said "So I think you should stop harping on the speed of light and acknowledge that you are making a general point that all speeds are coordinate-dependent." and I said "Acknowledged. Not only for speeds, but every possible empirical measurement." This is an acknowledgment that I am making a general point which applies "not only for speeds, but for every possible empirical measurement" namely that no experiment can directly measure a dimensionful quantity.

But as I keep saying over and over, even if we did detect a locally preferred frame there would be nothing stopping us from using a coordinate transform that led different reference frames to define simultaneity differently. A coordinate transform where all frames agreed on simultaneity might be more "natural" in this case, but other coordinate transforms would still be permissable. Do you agree or disagree?

Agree.

I think it's implicit in both of the postulates that they only refer to what would be true in inertial coordinate systems where the clocks are synchronized according to the Einstein synchronization procedure. After all, in Einstein's original paper he spends all of section 1 defining how to construct such coordinate systems out of physical measuring rods and clocks, and then only in section 2 does he lay out the two fundamental postulates. So the postulates don't deny the possibility of creating other types of coordinate systems where the speed of light is no longer constant, as I understand them. And like I said, I have seen physicists say that SR can handle accelerating coordinate systems, and the speed of light would not in general be a constant in such coordinate systems.

My textbook prefaces the postulates by "The fundamental postulates of the theory concern inertial reference systems or intertial frames. Such a reference system is a coordinate system...such that when particle motion is formulated in terms of this reference system Newton's first law holds." When the scope of the postulates is limited in this way, then I welcome them with open arms. I suppose that every formal description of SR would be limited in this way, so my concerns would only ever arise when someone attempts to apply SR outside of this particular coordinate system.

I don't think there is any such specification in "SR" as I understand it.

OK. It's not an issue with SR per se, and only becomes an issue when the postulates aren't kept within the context of their unique coordinate system.

 

[JesseM]

They are the components of this vector: $$d \bold{s}=\bold{c_0} dt-dx-dy-dz$$, where $$\bold{c_0}$$ is the instananeous velocity of an electron as described by Paul Dirac.

But that vector has 4 components, while (dA, dB, dC) has only three. Why not just spell out what the vector is in terms of the quantities dt,dx,dy,dz instead of inventing a new set of symbols with no established meaning? And if you're just talking about the vector (dt,dx,dy,dz), that vector is already used in relativity, I don't understand what new idea you're trying to introduce. Like I said, the norm ds of this vector is still a scalar.

As for Dirac's comments, remember that in quantum physics particles don't have a well-defined velocity at all moments, and our measurements of particles affects their behavior. As I understood him, Dirac wasn't saying that all particles "naturally" move at c at all times, just that any attempt to measure "instantaneous velocity" by taking two position measurements arbitrarily close together will yield a velocity arbitrarily close to c, by some application of the uncertainty principle that I don't really understand (probably because I haven't studied relativistic quantum mechanics).

Anyway, I don't understand what connection you're trying to draw between Dirac's point about the instantaneous velocity of particles and relativity. If you're suggesting this might somehow imply a preferred frame, remember that his comments are based on theoretical calculations from relativistic QM, a Lorentz-symmetric theory.

The discussion of $$d \bold{s}$$ as a vector

But it isn't a vector, it's a scalar which is the norm of a vector.

You said "So I think you should stop harping on the speed of light and acknowledge that you are making a general point that all speeds are coordinate-dependent." and I said "Acknowledged. Not only for speeds, but every possible empirical measurement." This is an acknowledgment that I am making a general point which applies "not only for speeds, but for every possible empirical measurement" namely that no experiment can directly measure a dimensionful quantity.

This is still totally convoluted...when you said "acknowledged", you mean you weren't acknowledging the thing I said I thought you should acknowledge ('that you are making a general point that all speeds are coordinate-dependent'), but rather that you were acknowledging a different point of your own ('that no experiment can directly measure a dimensionful quantity') which you didn't spell out? Usually if one person says "you should acknowledge X" and the other person says "acknowleged", it's assumed that the second person is still referring to X.

And like I said, the issue of dimensionful vs. dimensionless quantities is entirely separate from the issue of coordinate-dependent vs. coordinate-independent quantities, as there are plenty of dimensionful quantities that are not coordinate-dependent, like the proper time of a particular path through spacetime. Likewise, it's possible to come up with dimensionless quantities that are coordinate-dependent, like the ratio of times between one tick of my clock and one tick of your clock (if we use the coordinate systems specified by the Lorentz transform, each of us will say it's the other guy's clock ticks which are longer).

But as I keep saying over and over, even if we did detect a locally preferred frame there would be nothing stopping us from using a coordinate transform that led different reference frames to define simultaneity differently. A coordinate transform where all frames agreed on simultaneity might be more "natural" in this case, but other coordinate transforms would still be permissable. Do you agree or disagree?

Agree.

OK, glad we agree on this.

My textbook prefaces the postulates by "The fundamental postulates of the theory concern inertial reference systems or intertial frames. Such a reference system is a coordinate system...such that when particle motion is formulated in terms of this reference system Newton's first law holds." When the scope of the postulates is limited in this way, then I welcome them with open arms. I suppose that every formal description of SR would be limited in this way, so my concerns would only ever arise when someone attempts to apply SR outside of this particular coordinate system.

OK, so as long as SR is described in a way that makes it clear the two postulates only apply in coordinate systems built to Einstein's specifications, then maybe you don't have a problem with SR at all. I think physicists all have this implicit understanding of what is meant by SR even if they don't always state it with complete clarity.

 

[Aether]

But that vector has 4 components, while (dA, dB, dC) has only three. Why not just spell out what the vector is in terms of the quantities dt,dx,dy,dz instead of inventing a new set of symbols with no established meaning? And if you're just talking about the vector (dt,dx,dy,dz), that vector is already used in relativity, I don't understand what new idea you're trying to introduce. Like I said, the norm ds of this vector is still a scalar.

That vector doesn't really have four components. I'll rewrite it this way: $$d \textbf{s} = \textbf{c} _0 dt- \textbf{v} dt$$.

As for Dirac's comments, remember that in quantum physics particles don't have a well-defined velocity at all moments, and our measurements of particles affects their behavior. As I understood him, Dirac wasn't saying that all particles "naturally" move at c at all times, just that any attempt to measure "instantaneous velocity" by taking two position measurements arbitrarily close together will yield a velocity arbitrarily close to c, by some application of the uncertainty principle that I don't really understand (probably because I haven't studied relativistic quantum mechanics).

Anyway, I don't understand what connection you're trying to draw between Dirac's point about the instantaneous velocity of particles and relativity.

I'm not trying to draw any connection between Dirac's point and relativity per se. Relativity is a coping strategy for life in a universe without a locally preferred frame. The connection that I'm drawing is between Dirac's point and Lorentz symmetry. If you take an instantaneous measurement of an electron's velocity and you get c in the direction of the constellation Leo, then that's pretty interesting to me if I understand it correctly.

If you're suggesting this might somehow imply a preferred frame, remember that his comments are based on theoretical calculations from relativistic QM, a Lorentz-symmetric theory.

OK. It only caught my eye because I was already looking for this particular phenomena based on a prediction from somewhere else. A successive approximation process of "propose new symmetry", "formulate new QM", "make new theoretical calculations", and "repeat" is to be expected. I'm not yet able to complete this particular cycle on my own though.

This is still totally convoluted...when you said "acknowledged", you mean you weren't acknowledging the thing I said I thought you should acknowledge ('that you are making a general point that all speeds are coordinate-dependent'), but rather that you were acknowledging a different point of your own ('that no experiment can directly measure a dimensionful quantity') which you didn't spell out? Usually if one person says "you should acknowledge X" and the other person says "acknowleged", it's assumed that the second person is still referring to X.

I acknowledged what you said because I recognized that all velocities were dimensionful quantities and therefore that they can't possibly be directly measurable, and "coordinate-dependent" seemed consistent with that concept although I didn't understand the implications of that then as well as I do now. Then I added to it. Take this in context; I'm still sorting out exactly what "coordinate-dependent" really means.

And like I said, the issue of dimensionful vs. dimensionless quantities is entirely separate from the issue of coordinate-dependent vs. coordinate-independent quantities, as there are plenty of dimensionful quantities that are not coordinate-dependent, like the proper time of a particular path through spacetime. Likewise, it's possible to come up with dimensionless quantities that are coordinate-dependent, like the ratio of times between one tick of my clock and one tick of your clock (if we use the coordinate systems specified by the Lorentz transform, each of us will say it's the other guy's clock ticks which are longer).

OK.

OK, so as long as SR is described in a way that makes it clear the two postulates only apply in coordinate systems built to Einstein's specifications, then maybe you don't have a problem with SR at all. I think physicists all have this implicit understanding of what is meant by SR even if they don't always state it with complete clarity.

I would never have noticed how "SR is described" in the first place if it wasn't being misrepresented.

 

[JesseM]

That vector doesn't really have four components. I'll rewrite it this way: $$d \bold{s}=\bold {c_0}dt-d \bold{v}$$.

That equation doesn't make sense to me. First off it isn't even correct according to the standard definition of ds, which would be $$ds = \sqrt{c^2 dt^2 - dx^2 - dy^2 - dz^2}$$. Also, $$\bold {c_0}dt-d \bold{v}$$ seems to be subtracting a vector from a scalar, it is not the same as $$cdt - dx - dy - dz$$, because it's not correct to write $$d \bold{v} = dx + dy + dz$$. Or if you mean to create a new meaning for $$\bold {c_0}dt$$ where it is a vector rather than a scalar, it would still need to have the same number of components as the velocity vector in order to subtract the velocity vector from it, so what are its three components?

I'm not trying to draw any connection between Dirac's point and relativity per se. Relativity is a coping strategy for life in a universe without a locally preferred frame. The connection that I'm drawing is between Dirac's point and Lorentz symmetry. If you take an instantaneous measurement of an electron's velocity and you get c in the direction of the constellation Leo, then that's pretty interesting to me if I understand it correctly.

Since Dirac says that the velocity of c comes about because uncertainty in momentum approaches infinity, presumably the direction would be entirely random.

I acknowledged what you said because I recognized that all velocities were dimensionful quantities and therefore that they can't possibly be directly measurable, and "coordinate-dependent" seemed consistent with that concept although I didn't understand the implications of that then as well as I do now. Then I added to it. Take this in context; I'm still sorting out exactly what "coordinate-dependent" really means.

OK, if you were thinking at the time that coordinate-dependent and dimensionful were the same thing, presumably that means you thought you were acknowledging the same point I was making even though the point you had in mind was not the one I had in mind.

I would never have noticed how "SR is described" in the first place if it wasn't being misrepresented.

By who, in what instances? Like I said, I think when people say things like "relativity says light always moves at c" there's an implicit assumption that we're using the most physically natural system of coordinates. And again, this is true of any statement about velocities, including ones like "that car is moving at 55 mph relative to me".

 

[Aether]

That equation doesn't make sense to me. First off it isn't even correct according to the standard definition of ds, which would be $$ds = \sqrt{c^2 dt^2 - dx^2 - dy^2 - dz^2}$$. Also, $$\bold {c_0}dt-d \bold{v}$$ seems to be subtracting a vector from a scalar, it is not the same as $$cdt - dx - dy - dz$$, because it's not correct to write $$d \bold{v} = dx + dy + dz$$. Or if you mean to create a new meaning for $$\bold {c_0}dt$$ where it is a vector rather than a scalar, it would still need to have the same number of components as the velocity vector in order to subtract the velocity vector from it, so what are its three components?

The $$\bold{c_0}$$ is in bold, has three components, and they represent the instantaneous velocity of an electron which has magnitude $$c_0$$ and whatever direction. Sorry, I meant: $$\textbf{v} dt = dx + dy + dz$$.

Since Dirac says that the velocity of c comes about because uncertainty in momentum approaches infinity, presumably the direction would be entirely random.

I don't think that the momentum approaches infinity, and have a precise finite prediction for what it approaches. Like I said, a successive approximation process seems more reasonable than infinite just-about-anything to me.

OK, if you were thinking at the time that coordinate-dependent and dimensionful were the same thing, presumably that means you thought you were acknowledging the same point I was making even though the point you had in mind was not the one I had in mind.

Seems that way, though my statement didn't even capture completely what I did mean, so apologies for that.

By who, in what instances? Like I said, I think when people say things like "relativity says light always moves at c" there's an implicit assumption that we're using the most physically natural system of coordinates. And again, this is true of any statement about velocities, including ones like "that car is moving at 55 mph relative to me".

OK. I'm not really that interested in comparing coordinate systems per se anyway. I'm looking for actual symmetry violations, and need to be able to distinguish empirical facts from coordinate system induced hallucinations.

 

[JesseM]

The $$\bold{c_0}$$ is in bold, has three components, and they represent the instantaneous velocity of an electron which has magnitude $$c_0$$ and whatever direction.

What electron? Normally ds just refers to the distance between two points (perhaps infinitesimally close points) in spacetime. Can you show how your new notation would work in practice? Like, if I want to know the interval ds between two points along a straight path through flat spacetime, and the first point is t=5,x=8,y=10,z=1 and the second is t=21,x=11,y=7,z=-5, then dt would just be 21-5, dx would be 11-8, and so on. Does your notation only apply to paths taken by actual particles like electrons rather than arbitrary paths through spacetime? If so it can't be used to make a full metric, it seems. And does it depend on assuming particles have a well-defined position and velocity at every instant?

Why isn't it correct to write $$d \bold{v} = dx + dy + dz$$, because if there is something wrong with that notation then that needs to be cleaned up first.

Because one is a vector and the other is just the sum of the vector's components, a scalar quantity. You can write $$d \bold{v} = (dx,dy,dz)$$ though.

I don't think that the momentum approaches infinity, and have a precise finite prediction for what it approaches. Like I said, a successive approximation process seems more reasonable than infinite just-about-anything to me.

I'm just going by what Dirac says--in the paper you posted, he writes:

It may easily be verified that a measurement of a component of the velocity must lead to the result ±c in relativity theory, simply from an elementary application of the principle of uncertainty of (24). To measure the velocity we must measure the position at two slightly different times and then divide the change of position by the time interval. (It will not do to measure the momentum and apply a formula, as the ordinary connexion between velocity and momentum is not valid.) In order that our measured velocity may approximate to the instantaneous velocity, the time interval between the two measurements must be very accurate. The great accuracy with which the position of the electron is known during the time-interval must give rise, according to the principle of uncertainty, to an almost complete indeterminacy in its momentum. This means that almost all values of the momentum are equally probable, so that the momentum is almost certain to be infinite. An infinite value for a component of momentum corresponds to the value of ±c for the corresponding component of velocity.

So it seems Dirac is saying that this is an unescapable consequence of the uncertainty principle, that as the accuracy of your position measurement approaches 100%, the uncertainty in momentum increases without bound, and I take it that in relativistic quantum theory this means that the likelihood that the velocity will be found to be as close to c as your measurement can resolve will approach 100%. If you don't think the uncertainty in momentum becomes arbitrarily large and so the expectation value of the momentum approaches infinity in this case, then I think your ideas must violate the uncertainty principle.

Seems that way, though my statement didn't even capture what I did mean, so apologies for that.

No problem, glad we got it cleared up.

 

[Aether]

What electron? Normally ds just refers to the distance between two points (perhaps infinitesimally close points) in spacetime. Can you show how your new notation would work in practice? Like, if I want to know the interval ds between two points along a straight path through flat spacetime, and the first point is t=5,x=8,y=10,z=1 and the second is t=21,x=11,y=7,z=-5, then dt would just be 21-5, dx would be 11-8, and so on. Does your notation only apply to paths taken by actual particles like electrons rather than arbitrary paths through spacetime? If so it can't be used to make a full metric, it seems. And does it depend on assuming particles have a well-defined position and velocity at every instant?

An electron is needed to define the instantaneous direction of $$\bold{c_0}$$ using Dirac's analysis, though I suppose that many other types of particles would do just as well. To know whether this notation would apply to paths taken by particles as well as individual particles one would need to know if two particles some distance apart both always prefer the same direction simultaneously (or any other predictable correlation). If it is a random orientation then we probably couldn't interpolate between particles. The particle doesn't have any instantaneous position and velocity in terms of (dx,dy,dz) at all as these are only averages of (dA,dB,dC) over relatively long periods of time. Assuming that flat spacetime also means that all particles point in the same direction at the same time, then you can recover the familiar concept of a spacetime interval by taking the magnitude of the time vectors. The actual vector intervals...need to know if there is a predictable function for Dirac's rapid oscillation of the electron to answer that.

Because one is a vector and the other is just the sum of the vector's components, a scalar quantity. You can write $$d \bold{v} = (dx,dy,dz)$$ though.

I changed that notation after I posted it: $$\textbf{v} dt = dx+dy+dz$$.

I'm just going by what Dirac says--in the paper you posted, he writes: So it seems Dirac is saying that this is an unescapable consequence of the uncertainty principle, that as the accuracy of your position measurement approaches 100%, the uncertainty in momentum increases without bound, and I take it that in relativistic quantum theory this means that the likelihood that the velocity will be found to be as close to c as your measurement can resolve will approach 100%. If you don't think the uncertainty in momentum becomes arbitrarily large and so the expectation value of the momentum approaches infinity in this case, then I think your ideas must violate the uncertainty principle. No problem, glad we got it cleared up.

Imagine Dirac doing an immitation of Don Adams ("Get Smart"): I said "almost certain to be infinite". My number is so large that you would double over laughing, but it isn't infinite. Which do you like better: infinite momentum, or a deterministic explanation for the uncertainty principle?

 

[JesseM]

An electron is needed to define the instantaneous direction of $$\bold{c_0}$$ using Dirac's analysis, though I suppose that many other types of particles would do just as well. To know whether this notation would apply to paths taken by particles as well as individual particles

What would it mean for it to apply to "individual particles" rather than their paths? What would dx, dy and dz represent if not the incremental distance traveled by the particle in the incremental time dt?

one would need to know if two particles some distance apart both always prefer the same direction simultaneously (or any other predictable correlation). If it is a random orientation then we probably couldn't interpolate between particles.

So if two particles take the same path you wouldn't have a single value for the integral of ds along that path, right? And like I said, you have no way to integrate along paths that don't happen to be taken by any actual particles. So whatever you're doing here, you aren't defining a new type of metric, since a metric is supposed to give a single distance for an arbitrary path in your spacetime, and the ds you use appears to have no relation to the ds used in relativity (for example, note that the integral of ds along a path in relativity is just c times the proper time along that path--is anything like that true of your ds?) So what is the purpose of the ds you're calculating? What do you do with it? How do you use it to make predictions about measurable things?

The particle doesn't have any instantaneous position and velocity in terms of (dx,dy,dz) at all as these are only averages of (dA,dB,dC) over relatively long periods of time.

Again, what are dA, dB, and dC? You seem to have introduced these symbols without defining them.

Assuming that flat spacetime also means that all particles point in the same direction at the same time

What possible reason is there to make this assumption, since the theory that predicts the instantaneous velocity of a particle is c in the first place would definitely not lead to this conclusion? I don't even know if the prediction about the instantaneous velocities has been experimentally tested to any great degree of precision. It seems like you're just picking and choosing predictions of quantum theory you think are neat and throwing out ones that don't fit with your personal intuitions, with very little understanding of the underlying theory and why it makes these predictions, and why they go together. Same with your picking symbols used in relativity and redefining them in ways that seem to have no relation to their original use...a lot of what you are doing looks like a kind of http://www.physics.brocku.ca/etc/cargo_cult_science.html to me.

then you can recover the familiar concept of a spacetime interval by taking the magnitude of the time vectors. The actual vector intervals...need to know if there is a predictable function for Dirac's rapid oscillation of the electron to answer that.

Since the prediction is based on total uncertainty in the momentum due to extremely precise position measurements, it's a safe bet that the direction of the momentum would be randomized along with its magnitude.

I changed that notation after I posted it. $$\bold{v} = dx+dy+dz$$.

Why did you get rid of the "d"? That makes even less sense, how do infinitesimal displacements along the x,y, and z lead to a non-infinitesimal velocity vector? And you didn't address my point about one side of your equation being a vector and the other being a scalar.

He says "almost certain to be infinite".

I'd guess that what he means is that the expectation value of the momentum approaches infinity as the uncertainty in position approaches zero.

My number is so large that you would double over laughing, but it isn't infinite. Which do you like better: infinite momentum, or a deterministic explanation for the uncertainty principle?

If it's an "explanation" for the uncertainty principle, then it should make the same empirical predictions as the uncertainty principle--if you're saying there's an upper limit to the momentum no matter how much you reduce the uncertainty in the position, that would seem to be a violation of the uncertainty principle.

Shouldn't this go in the Independent Research forum, anyway?

 

[Aether]

Again, what are dA, dB, and dC? You seem to have introduced these symbols without defining them.

I defined them in posts #79 & #83. However, in #79 I used $$dv$$ where I meant $$\textbf{v} dt$$, and didn't make the vectors bold. So, my apologies for causing confusion in that way.

Why did you get rid of the "d"? That makes even less sense, how do infinitesimal displacements along the x,y, and z lead to a non-infinitesimal velocity vector? And you didn't address my point about one side of your equation being a vector and the other being a scalar.

You're right about the notation. I changed it to: $$\textbf{v} dt=dx+dy+dt$$. Both sides of the equation are vectors.

Shouldn't this go in the Independent Research forum, anyway?

This forum looks like a great idea, but I wasn't looking to get into an open discussion of my own personal theories at this time. I was merely attempting to answer all of your questions as I have been doing all along. Your questions and comments have been generally helpful, and I appreciate that.

 

[DrGreg]

Technical note on vectors in TEX

I noticed in some posts in this thread that on this site's TEX system \bold text is sometimes hard to distinguish from normal italic TEX.

I'd suggest using \textbf instead.

$$\textbf r = (x, y)$$

Or, if you prefer, \vec

$$\vec r = (x, y)$$

 

[DrGreg]

If I understand correctly what Paul Dirac is saying (and I think I do because I have several of his subsequently published papers and articles in Nature where he says explicitly that "an ether is rather forced upon us" or words to that effect) then I don't see why we can't represent $$\textbf c_0 dt$$ as a vector, and $$d \textbf s =\textbf c_0 dt-dx-dy-dz$$ as a vector. I understand that for many practical purposes this would be a meaningless complication, but for my purposes I would like to figure out how to do it correctly. On second thought, it may be the vector $$\textbf c_0 dt$$ that should be held invariant in the transformation that I'm trying to develop.

I preface my remarks by saying I don't know a lot about quantum theory. This website has a separate forum devoted to the subject.

In quantum theory particles do not have an exact position or momentum. There is always some error in any measurment you make. The more accurately you measure one, the less accurately you measure the other.

So if you try to measure a quantum particle's velocity by measuring $$\delta x / \delta t$$, for very, very, very, small $$\delta x$$ and $$\delta t$$you are doomed to failure because the errors will overwhelm the tiny difference you are trying to measure. So "instantaneous velocity" calculated this way is pretty meaningless. Your best bet is to measure a whole sequence of distances and times, plot them on a graph and perform a straight line curve fit. This averages out all the measurement errors.

In this forum I assume that all particles are "classical" particles postulated to have a precise position and momentum.

Mansouri-Sexl add three arbitrary synchronization parameters to the LET transformation equation for the t coordinate; one for each direction in space. Making $$\textbf c_0 dt$$ a vector may imply three time coordinates; one for each direction in space.

No they are not saying anything of the sort. The three parameters are something you, the observer, decide when you choose how to synchronize your clocks, they are not extra dimensions of anything.

 

[Aether]

In this forum I assume that all particles are "classical" particles postulated to have a precise position and momentum.

OK. What is the right term (Minkowski geometry? Topology?) for that science which emcompasses all possible coordinate systems (e.g., from which new coordinate systems may be contructed), all types of particles, which directly addresses all questions of Lorentz symmetry and violations thereof, and of which relativity is an infinitesimally thin slice (e.g., that it explicitly applies only to inertial frames and classical particles)?

No they are not saying anything of the sort. The three parameters are something you, the observer, decide when you choose how to synchronize your clocks, they are not extra dimensions of anything.

That's right. Only the first sentence describes what Mansouri-Sexl are saying. The second sentence follows it so that the two may be compared to show that there is a precedent for parameterizing the time coordinate with three arbitrary components, one for each direction in space.

 

[Jimmy Snyder]

Just a friendly reminder, the first postulate of the special theory of relativity, namely that the speed of light c is the same in all inertial frames, only holds true in view of Einstein's clock synchronization convention.

Do I have this right? These guys are trying to prove that a postulate is true? By the way, it's the second postulate. Who are these guys?

 

[Aether]

Do I have this right? These guys are trying to prove that a postulate is true?

Mansouri-Sexl do not try to prove, by experiment, that the postulate is true. On page 499 of their first paper they say "When clocks are synchronized according to the Einstein procedure the equality of the velocity of light in two opposite directions is trivial and cannot be the subject of an experiment."

By the way, it's the second postulate.

One of my GR textbooks, A Short Course in General Relativity, by Foster & Nightingale, lists the speed of light postulate first.

Who are these guys?

They developed a popular test theory of special relativity and published it in 1977. Since then, most published experiments testing for violations of local Lorentz invariance have referenced their work.

After lengthy discussion, I see more clearly now that the key limitation of the constancy of the speed of light postulate is that it is only true "in all inertial frames". Inertial frames are coordinate systems in which the speed of light is defined to be constant in all directions, and that is the end of it. There can be no such thing as an experiment to verify that this is true. If someone claims that there is, then they are not observing this explicit limitation that is built into the postulate. That would be like saying that "experiments have proven that the inches on an english ruler really are inches, so there".

 

[Aether]

The primary reason to synchronize clocks is to be able to measure velocities. When we demand that an object of mass m and velocity v moving north have an equal and opposite momentum to an object of mass m and velocity v moving south, we require Einsteinan clock synchronization.

Empirically, this means that we require an two objects of equal masses moving at the same speed in opposite directions to stop when they collide inelastically.

It is indeed *possible* to use non-Einsteinain clock synchronizations, and under some circumstances it is more-or-less forced on us. In such circumstances, one must not remember that momentum is not isotropic.

Note that Newton's laws assume that momentum is isotropic (an isotropic function of velocity). Therfore Newton's laws (with the definition of momentum as p=mv) cannot be used unless Einstein's clock synchronization is used. Some other definition of momentum other than p=mv must be used if it is to remain a conserved quantity when non-standard clock synchronizations are used.

The ability to use Newton's laws at low velocities was what motivated Einstein to define his method of clock synchronization.

pervect, I agree with you now. I did not fully appreciate that it is the very definition of what an inertial reference frame represents which causes momentum to be conserved as p=mv rather than a law of nature per se. That is as interesting to me as is the fact that the constancy of the speed of light is an artifact of the definition of an inertial frame.

I interpreted "Empirically, this means that we require an two objects of equal masses moving at the same speed in opposite directions to stop when they collide inelastically" to imply that SR and LET were not empirically equivalent, but I see now that by this you were correctly defining how "at the same speed in opposite directions" is used to define an intertial frame.

 

[Jimmy Snyder]

One of my GR textbooks, A Short Course in General Relativity, by Foster & Nightingale, lists the speed of light postulate first.

One of Einstein's papers, 'On the Electrodynamics of Moving Bodies', lists it second.

There can be no such thing as an experiment to verify that this [postulate] is true.

Do you know of a postulate that has been proven true?

 

[Aether]

One of Einstein's papers, 'On the Electrodynamics of Moving Bodies', lists it second.

OK.

Do you know of a postulate that has been proven true?

If you have a point to make, then please make it.

 

[DrGreg]

In this forum I assume that all particles are "classical" particles postulated to have a precise position and momentum.

Aether, I said this simply to suggest that if you want to discuss the subtleties of quantum theory, it would be a good idea to do that in the Quantum Theory forum of this website.

 

[Aether]

Aether, I said this simply to suggest that if you want to discuss the subtleties of quantum theory, it would be a good idea to do that in the Quantum Theory forum of this website.

I know, and agree. However, if QM is built on relativity (or at least an assumption of Lorentz symmetry) then it's not exactly what I am looking for any more than relativity per se. That is not intended as a criticism of either theory. I simply want to keep each theory within its proper context so as not to be tripped up by inadvertent applications of these theories outside of their proper domain, and am asking what is the appropriate global perspective from which to build new theories?

 

[Jimmy Snyder]

If you have a point to make, then please make it.

My point is that postulates are never proven true. I asked you if you knew of a counterexample because I knew that there aren't any. I hoped that by my asking this question, you would come to realize what my point was without my telling you. It's a method of argumentation first practiced by Socrates, and described by Plato. It tends to fall to pieces when challenged in this way.

 

[Aether]

My point is that postulates are never proven true. I asked you if you knew of a counterexample because I knew that there aren't any. I hoped that by my asking this question, you would come to realize what my point was without my telling you. It's a method of argumentation first practiced by Socrates, and described by Plato. It tends to fall to pieces when challenged in this way.

OK, here is a counterexample of Copernicus' 7 Postulates. I would judge that at least some of them have been proven to be true.

(from the Commentariolus)

There is no one centre of all the celestial circles or spheres.

The centre of the Earth is not the centre of the Universe, but only of gravity and the lunar sphere.

All the spheres rotate about the Sun as their midpoint, and so the centre of the Universe is near the Sun.

The Earth's distance from the Sun is...imperceptible when compared with the loftiness of the firmament [of fixed stars].

An apparent motion of the firmament is the result, not of the firmament itself moving, but of the Earth's motion. The Earth... goes through a complete rotation on its axis each day, while the firmament and highest heaven remain unaltered.

What appear to us as [annual] motions of the Sun result, not from its moving itself, but from the [linear] motion of the Earth and its sphere, with which we travel around the Sun just like any other planet. The Earth has, accordingly, more than one motion.

The apparent retrogradations and [returns to] direct motions of the planets are the result, not of their own motion, but of the Earth's. The motion of the Earth alone, therefore, is enough to explain many apparent anomalies in the heavens

 

[Jimmy Snyder]

Here is the Britannica article on the word 'axiom'. Postulates are unprovable by definition. Counterexamples cannot exist.

In mathematics or logic, an unprovable rule or first principle accepted as true because it is self-evident or particularly useful (e.g., “Nothing can both be and not be at the same time and in the same respect”).

The term is often used interchangeably with postulate, though the latter term is sometimes reserved for mathematical applications (such as the postulates of Euclidean geometry). It should be contrasted with a theorem, which requires a rigorous proof.

http://www.britannica.com/ebc/article-9356242?query=postulate&ct=

 

[Aether]

Here is the Britannica article on the word 'axiom'. Postulates are unprovable by definition.

OK, but why are you showing me an article on the word 'axiom' to convince me of something about the word 'postulate'?

Counterexamples cannot exist.

I just showed you a counterexample. I wouldn't be surprised if there was some formal context in which what you are saying is correct, but you haven't put your statement within such a context. This is similar to where this thread is ending up; the speed of light postulate is true within the context of inertial reference frames, but not outside of that context.

We recently had a discussion of this as it relates specifically to special relativity in the 'Consistency of the speed of light thread', have you looked at that? Someone may have said something there that you can use to make your point.