Planck constant is Lorentz invariant?

[PeterDonis]

I don't think you are using the Einstein's Doppler formula in above analysis.

Sure I am; I'm just trying to give more details about what it actually means, physically. If my idealization of a single space dimension (so what is actually a spherical wave front being emitted by the moving point source is modeled as two plane waves emitted in opposite directions) disturbs you, I suppose we could work with the full mathematical apparatus of spherical waves in 3 space dimensions, but that seems like overkill.

 

[keji8341]

Yes, it is applicable.

Einstein’s Doppler formula is not applicable when a moving point light source is close enough to the observer; for example, it may break down or cannot specify a determinate value when the point source and the observer overlap. How to explain to it?

 

[keji8341]

By a moving point source, I assume you mean a spherical wave emanating from a moving point source? A spherical wave can't be described, mathematically, by a single 4-vector.



Why not? Bear in mind that I'm only considering a single spatial dimension; in the full 3 space dimensions you would have a spherical wave front being emitted, as I noted above. That's how point sources work.



My inclination would be to say that the observer wouldn't detect the photon at all in this case. Certainly that's what would happen in any real experiment.

Einstein proved that (k,w/c) for a plane wave is Lorentz covariant. Can you prove that (k,w/c) for a moving point light source is Lorentz covariant?

 

[Dale]

Einstein’s Doppler formula is not applicable when a moving point light source is close enough to the observer; for example, it may break down or cannot specify a determinate value when the point source and the observer overlap. How to explain to it?

So what? Lots of things break down in the limit as r->0. Coulomb's law, Schwarzschild spacetime, Newtonian gravity, angular momentum, etc. If that were a reason to reject something we would have to get rid of a lot of physics.

Einstein's Doppler formula is definitely applicable to a moving point source.

http://teachers.web.cern.ch/teachers/archiv/hst2002/bubblech/mbitu/wave_4.htm

 

[PeterDonis]

Einstein proved that (k,w/c) for a plane wave is Lorentz covariant. Can you prove that (k,w/c) for a moving point light source is Lorentz covariant?

No, because there is no such thing. Did you read the part where I said that a spherical wave, which is what a point source emits, can't be described by a single 4-vector?

What I *can* prove is that the spherical wave emitted by a point source is Lorentz invariant. That's simple: the spherical wave is just the future null cone (actually the term should probably be "null hypercone" since its spatial slices are 2-spheres, not circles) of the emission event. Null cones are always left invariant by Lorentz transformations. QED.

Edit: Perhaps I should expand on this more. If we are looking at the entire spherical wave emitted by the point source, there isn't really any "energy-momentum" object to compare it to in order to evaluate Planck's constant. So that case is really irrelevant to the question in the OP.

But we can decide to pick out a particular null ray from this spherical wave, by looking at a particular pair of events, the given emission event (the source of the entire spherical wave--this is some event on the source's worldline), and a particular reception event (some point further out on the future null cone, where the receiver's worldline intersects it). Then we can associate a particular null 4-vector (k, w/c) with the null ray from the emission event to the specified reception event.

The Doppler effect (more precisely, the "longitudinal" Doppler effect) is simply the observation that the actual value of k (or w/c, since the vector is null they are equal in magnitude) depends on the relative velocity beta of the source and the observer, by the Einstein Doppler formula. (The n in that formula is just the spatial direction of the null ray we specified, so n.beta is the angle between that ray and the moving source's spatial velocity.)

But we can also observe that, once we've chosen the reception event, for a given beta, the 4-vector (k, w/c) *is* Lorentz covariant. We could show this by modeling the chosen null ray as a plane wave. (If you want to say that the plane wave approximation breaks down when the events are too close together, I suppose that's true, but it has nothing to do with any "discontinuity" when the source passes the observer; it's simply due to the curvature of the actual spherical wavefront, which makes the plane wave approximation less accurate the closer the emission and reception events are in space.) However, we can show it even more easily by simply observing that, by definition, null rays and their associated null 4-vectors are always Lorentz covariant. (This is because Lorentz transformations always leave null cones invariant, so individual null rays can never be made non-null; they can only be conformally mapped into other null rays. Such a conformal mapping preserves inner products of null rays, which is the definition of "Lorentz covariant".)

The apparent "discontinuity" when the moving source passes the observer is due to switching null rays in mid-stream, so to speak, by switching the pair of events (emission, reception) that we are considering, which also means switching which particular null cone we are picking the events out of. This has to be the case, because at anyone particular emission event, the moving source cannot both be approaching and receding from the observer. So as soon as we pick a particular emission event, we have implicitly also picked a particular n.beta in the Einstein Doppler formula, and a particular 4-vector (k, w/c).

Only by looking at two *different* null rays, one with the source approaching and one with the source receding, and then inappropriately combining them into a single "measurement" of frequency or wavelength, can we see any discontinuity. But in doing that, we are combining two *different* 4-vectors (k, w/c) and (k', w'/c), that are associated with two different null rays between two different pairs of events on two different null cones. It's not surprising that such a combination is not well-behaved, and all this doesn't prove or disprove anything about Planck's constant.

In summary: for any case where there is actually a unique, valid 4-vector (k, w/c) for a photon, it is Lorentz covariant, and therefore is consistent with Planck's constant being a Lorentz scalar. For any case where there appears to be a photon "4-vector" that is not Lorentz covariant, it's because there is not one unique 4-vector involved; instead, information from multiple different 4-vectors is being inappropriately combined into a single "measurement".

 

[keji8341]

So what? Lots of things break down in the limit as r->0. Coulomb's law, Schwarzschild spacetime, Newtonian gravity, angular momentum, etc. If that were a reason to reject something we would have to get rid of a lot of physics.

Einstein's Doppler formula is definitely applicable to a moving point source.

http://teachers.web.cern.ch/teachers/archiv/hst2002/bubblech/mbitu/wave_4.htm

That I take the overlap-point as an example is just for clarity. Actually when the point source is close enough, Einstein's formula is not applicable. The closer, the bigger the error is. Theoretically, you don't have any grouds to say that Einstein's Doppler formula is applicable to the moving point light source, unless you can prove it, or you just set it as an artificial assumption.

In the classical electromagnetic theory, Coulomb's law is valid even in the limit as r->0, because it satisfies Maxwell equations.

 

[keji8341]

So what? Lots of things break down in the limit as r->0. Coulomb's law, Schwarzschild spacetime, Newtonian gravity, angular momentum, etc. If that were a reason to reject something we would have to get rid of a lot of physics.

Einstein's Doppler formula is definitely applicable to a moving point source.

http://teachers.web.cern.ch/teachers/archiv/hst2002/bubblech/mbitu/wave_4.htm

I think you must know Planck length physics.
In physics, the Planck length, denoted ℓP, is a unit of length, equal to 1.616252(81)×10**(−35) metres. The physical significance of the Planck length is a topic of research.
http://en.wikipedia.org/wiki/Planck_length

 

[keji8341]

No, because there is no such thing. Did you read the part where I said that a spherical wave, which is what a point source emits, can't be described by a single 4-vector?

What I *can* prove is that the spherical wave emitted by a point source is Lorentz invariant. That's simple: the spherical wave is just the future null cone (actually the term should probably be "null hypercone" since its spatial slices are 2-spheres, not circles) of the emission event. Null cones are always left invariant by Lorentz transformations. QED.

Sorry, I don't understand what the "null cone". Cerenkov cone?

 

[Dale]

That I take the overlap-point as an example is just for clarity. Actually when the point source is close enough, Einstein's formula is not applicable. The closer, the bigger the error is.

What are you talking about? Can you derive the error you think is there?

Theoretically, you don't have any grouds to say that Einstein's Doppler formula is applicable to the moving point light source, unless you can prove it, or you just set it as an artificial assumption.

Any EM field that has a definite phase obeys the Doppler formula. The phase is the Minkowski inner product between the position four-vector and the wave four-vector. Since the wave four-vector is a four-vector it transforms like any other four-vector.

In the classical electromagnetic theory, Coulomb's law is valid even in the limit as r->0, because it satisfies Maxwell equations.

And the Doppler formula is valid because it satisfies the Lorentz transform.

 

[keji8341]

What are you talking about? Can you derive the error you think is there?

Einstein's Doppler formula is the Doppler formula for a plane wave: w'=w*gamm*(1-n.beta), which can be seen in university physics textbooks. If it is applied to the moving point light source, when the observer and the point source overlap, n.beta is an inderterminate value, because the angle between n and beta can be arbitrary. From this we can deduce that it is not applicable when the point source is close enough to the observer.

 

[keji8341]

Any EM field that has a definite phase obeys the Doppler formula. The phase is the Minkowski inner product between the position four-vector and the wave four-vector. Since the wave four-vector is a four-vector it transforms like any other four-vector.

It is not necessarily. For example, the spherical wave has a phase function of phi=(wt -|k||x|) where k and x has a strong constraint and the Lorentz covariance of (k,w/c) is destroyed.

Note: For a plane wave, the phase function is given by phi=wt-k.x where there is no constraint between k and x, and from the invariance of phase, (k,w/c) must be Lorentz covariant as shown by Einstein in 1905.

 

[keji8341]

And the Doppler formula is valid because it satisfies the Lorentz transform.

Since (k,w/c) for a moving point light source is not Lorentz covariant as mentioned above, its Doppler formula cannot be obtained directly from the Lorentz transformation of (k,w/c).

 

[Dale]

It is not necessarily. For example, the spherical wave has a phase function of phi=(wt -|k||x|) where k and x has a strong constraint and the Lorentz covariance of (k,w/c) is destroyed.

All that means is that the wave four-vector is a function of position for anything other than a plane wave. In other words, it is a tensor field of rank 1. But there is ample experimental evidence that radiation from point sources Doppler shifts ala Einstein.

 

[Dale]

Since (k,w/c) for a moving point light source is not Lorentz covariant as mentioned above, its Doppler formula cannot be obtained directly from the Lorentz transformation of (k,w/c).

Sure it can. Look, the phase is a scalar and the four-position is the prototypical four-vector, so the object which is multiplied with a vector to get a scalar is a vector and transforms as a vector.

In other words given a scalar phi and a vector x how else can you get
$$\phi=f(x)$$
besides
$$\phi=x^{\mu}k_{\mu}$$

In this formula, given phi and x, k must clearly be a vector. That vector is called the wave four vector.

 

[PeterDonis]

Sorry, I don't understand what the "null cone". Cerenkov cone?

A null cone is just the set of all points in a spacetime that are at a null interval from a given point (the "source"). If we adopt coordinates such that the source is at (t, x, y, z) = (0, 0, 0, 0), then the null cone is just the set of points for which:

$$t^{2} - x^{2} - y^{2} - z^{2} = 0$$

The future null cone is the portion of this set for which t > 0.

 

[keji8341]

But there is ample experimental evidence that radiation from point sources Doppler shifts ala Einstein.

I don't understand; please give references.

 

[keji8341]

Sure it can. Look, the phase is a scalar and the four-position is the prototypical four-vector, so the object which is multiplied with a vector to get a scalar is a vector and transforms as a vector.

In other words given a scalar phi and a vector x how else can you get
$$\phi=f(x)$$
besides
$$\phi=x^{\mu}k_{\mu}$$

In this formula, given phi and x, k must clearly be a vector. That vector is called the wave four vector.

Please note: that is for plane waves. I think you just copy them from textbooks which are all talking about plane waves.

 

[keji8341]

A null cone is just the set of all points in a spacetime that are at a null interval from a given point (the "source"). If we adopt coordinates such that the source is at (t, x, y, z) = (0, 0, 0, 0), then the null cone is just the set of points for which:

$$t^{2} - x^{2} - y^{2} - z^{2} = 0$$

The future null cone is the portion of this set for which t > 0.

Thanks. That is a math description of the hypothesis of constancy of light speed, which is used in the derivation of Lorentz transformation.

 

[keji8341]

All that means is that the wave four-vector is a function of position for anything other than a plane wave. In other words, it is a tensor field of rank 1.

You are kidding; what do you mean for "wave four-vector is a function of position", give your specific expression for your wave four-vector, please, so that I can check how it depends on position and also it follows Lorentz transformation.

 

[keji8341]

Planck's constant is assumed to be a Lorentz scalar, and quantum theory can be built in an explicitly Poincare-covariant way with this assumption. The resulting theory (which is relativistic quantum field theory) is one of the most successful scientific results ever, and thus we can be pretty sure that our assumption of $\hbar$ being a scalar universal constant is good. That's the nature of any model building in the natural sciences: You make assumptions and look where they lead you in terms of observable predictions. Then you do an experiment to check, whether these predictions are correct and within which limits of physical circustances they are valid etc.

Of course, the momentum-four vector of a photon is Lorentz covariant. Otherwise it would not be a four vector to begin with! How do you come to the conclusion, it's not?

Of course, as an artificial assumption, there is nothing wrong as long as no contradiction shows up. Unfortunately, (k,w/c) for a moving point light source is not Lorentz covariant, which questions the Lorentz invariance of Planck constant.

 

[Dale]

I don't understand; please give references.

Certainly:
http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html#Tests_of_time_dilation

The Ives and Stilwell experiment is the prototypical example which used atoms as point sources, but there are others as well. Even the Mossbauer rotor experiments do not produce plane waves, although they are not point sources either.

 

[Dale]

You are kidding; what do you mean for "wave four-vector is a function of position", give your specific expression for your wave four-vector, please, so that I can check how it depends on position and also it follows Lorentz transformation.

I did. For any wave which has a definite phase the definition of the wave four-vector is:
$$\phi=x^{\mu}k_{\mu}=g_{\mu\nu}x^{\mu}k^{\nu}$$
For a plane wave in an inertial frame k is constant and equal to the usual expression $k=(\omega,\mathbf k)$, but the above expression is more general and always works for any wave with a definite phase.

Are you not aware that the tensors in such expressions are functions of position and time in general? If not, please see page 11 after equation 1.36 in

http://arxiv.org/abs/gr-qc/9712019

"Of course in spacetime we will be interested not in a single vector space, but in fields of vectors and dual vectors."

 

[keji8341]

Certainly:
http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html#Tests_of_time_dilation

The Ives and Stilwell experiment is the prototypical example which used atoms as point sources, but there are others as well. Even the Mossbauer rotor experiments do not produce plane waves, although they are not point sources either.

That is because the observer is far away from the source, wavelength/distance is too small, Einstein formula is a good approximation.

 

[Dale]

Can you demonstrate that there are errors which become large at smaller distances? I certainly see no indication of that.

 

[keji8341]

I did. For any wave which has a definite phase the definition of the wave four-vector is:
$$\phi=x^{\mu}k_{\mu}=g_{\mu\nu}x^{\mu}k^{\nu}$$
For a plane wave in an inertial frame k is constant and equal to the usual expression $k=(\omega,\mathbf k)$, but the above expression is more general and always works for any wave with a definite phase.

Are you not aware that the tensors in such expressions are functions of position and time in general? If not, please see page 11 after equation 1.36 in

http://arxiv.org/abs/gr-qc/9712019

"Of course in spacetime we will be interested not in a single vector space, but in fields of vectors and dual vectors."

Please note: (1) For a plane wave, the phase phi=wt-k.x=(k,w/c).(x,ct); (k,w/c) and (x,ct) are independent, and both are Lorentz covariant. (2) But for a spherical wave, phi=wt-|k||x|, where (x,ct) must be Lorentz covariant while (k,w/c) can't be Lorentz covariant because k and x must be parallel, required by wave equation; this is an additional constraint. It is the additional constraint that destroyed the covariance of (k,w/c) for a moving point light source.

 

[Dale]

k and x must be parallel

This is not true in general. It is only true for a spherical wave centered at the origin. In general k and x can have any arbitrary relationship. Perhaps that is the source of your misunderstanding?

 

[keji8341]

Can you demonstrate that there are errors which become large at smaller distances? I certainly see no indication of that.

The relative error is proportional to (atom's radiation wavelength)/(distance between the source and the observer). Usually, radiation wavelength is < microns, and the distance >cm ----> the error is smaller than 10**(-4), all past experiments cannot identify the error. As you know the experimentally obtained Planck constant also has errors of ~10**(-4).

 

[keji8341]

This is not true in general. It is only true for a spherical wave centered at the origin. In general k and x can have any arbitrary relationship. Perhaps that is the source of your misunderstanding?

As you kow, I am talking about the Doppler effect from a moving point light source. For a moving point light source, the phase factor is exp[i(wt-|k||x|)] required by wave equation, the wave vector k is always parallel to the radial vector (the point source fixed at x'=0; at t=t'=0, x=x'=0).

 

[Dale]

As you kow, I am talking about the Doppler effect from a moving point light source. For a moving point light source, the phase factor is exp[i(wt-|k||x|)] required by wave equation, the wave vector k is always parallel to the radial vector (the point source fixed at x'=0; at t=t'=0, x=x'=0).

A point source cannot be both fixed at the origin and moving. In any frame where the source is moving k is not parallel to x in general.

 

[keji8341]

A point source cannot be both fixed at the origin and moving. In any frame where the source is moving k is not parallel to x in general.

In the source-rest frame X'Y'Z', phi'=w't'-|k'||x'|, where x' denotes the radial vector from the source point to the observation point. Under Lorentz transformation, phi=phi'=wt-|k||x|, because the wavefront fired at t=t'=0 and x=x'=0 is always a spherical surface observed in both frames at any of the same times.

 

[Dale]

In the source-rest frame X'Y'Z', phi'=w't'-|k'||x'|. Under Lorentz transformation, phi=phi'=wt-|k||x|, because the wavefront fired at t=t'=0 and x=x'=0 is always a spherical surface observed in both frames at any of the same times.

In the frame where the source is moving the wavefront fired at t=t'=0 and x=x'=0 w varies over the wavefront. I will work out the problem in detail for you later, probably tomorrow.

 

[keji8341]

In the frame where the source is moving the wavefront fired at t=t'=0 and x=x'=0 w varies over the wavefront. I will work out the problem in detail for you later, probably tomorrow.

I think that, Einstein’s Doppler formula is not applicable when a moving point light source is close enough to the observer; for example, it may break down or cannot specify a determinate value when the point source and the observer overlap. Do you agree? This is easy to judge.

 

[PeterDonis]

Thanks. That is a math description of the hypothesis of constancy of light speed, which is used in the derivation of Lorentz transformation.

You're not reading very carefully. The Lorentz transformation is defined as keeping the interval constant; the interval is defined as the quantity $t^{2} - x^{2} - y^{2} - z^{2}$. (Strictly speaking, that's the interval of a given point (t, x, y, z) from the origin (0, 0, 0, 0).)

The equation I wrote down says more than that; it says that this quantity, the interval, is not just constant, but *equal to zero*. In other words, it defines a set of points in spacetime that are separated from the origin (0, 0, 0, 0) by a zero interval. This set of points is called a null cone. The "future" portion of the null cone is the subset of these points for which t > 0; in other words, it's the portion of the null cone that lies to the future of the origin (0, 0, 0, 0).

Since a Lorentz transformation keeps the interval constant, it must map the null cone into itself; in other words, it maps null rays into other null rays. The mapping is conformal, so it preserves the inner product; thus, the null cone is Lorentz covariant.

 

[keji8341]

You're not reading very carefully. The Lorentz transformation is defined as keeping the interval constant; the interval is defined as the quantity $t^{2} - x^{2} - y^{2} - z^{2}$. (Strictly speaking, that's the interval of a given point (t, x, y, z) from the origin (0, 0, 0, 0).)

The equation I wrote down says more than that; it says that this quantity, the interval, is not just constant, but *equal to zero*. In other words, it defines a set of points in spacetime that are separated from the origin (0, 0, 0, 0) by a zero interval. This set of points is called a null cone. The "future" portion of the null cone is the subset of these points for which t > 0; in other words, it's the portion of the null cone that lies to the future of the origin (0, 0, 0, 0).

Since a Lorentz transformation keeps the interval constant, it must map the null cone into itself; in other words, it maps null rays into other null rays. The mapping is conformal, so it preserves the inner product; thus, the null cone is Lorentz covariant.

Thanks. I know Lorentz transformation is more than the constancy of light speed: (1) isotropy of time and space, (2) homogeneous, (3) constancy of light speed, (4) physical laws are covariant.

 

[keji8341]

In the frame where the source is moving the wavefront fired at t=t'=0 and x=x'=0 w varies over the wavefront. I will work out the problem in detail for you later, probably tomorrow.

x and x' should be taken the radial vectors from the source point to the observation point. The point source is fixed in the X'Y'Z' frame, and the source point is constant observed in X'Y'Z' frame, but it is moving observed in the XYZ frame.