# Relativistic Doppler Effect: Intensity change

My physics prof. mentioned briefly, that in the framework of the relativistic Doppler effect, not only the frequency (alternatively, the wavelengh) changes when objects move with respect to each other (which I would understand), but also the intensity (which I do not understand). He said, it has to do with the fact, that the source strength, divided by the cube of the frequency, is a Lorentz invariant, but did not (want to) explain it any further. I found a publication by Johnson, M. H. and Teller, E. "Intensity changes in the Doppler effect", which is available here:

But it is the usual short version of an understandable explanation. I saw here, where it says: “A more-sophisticated method of deriving the beaming equations starts with the quantity $$\displaystyle\frac{S}{\nu^3}$$ This quantity is a Lorentz invariant, so the value is the same in different reference frames”. I think, that “sophisticated'' method is what I am looking for. I also have a hunch, that light gets bundled in high gravitational environments (edge of black hole) and therefore the intensity is higher, but I need a good explanation for dummies. I am Relativity level of about Schutz, Wald, Caroll and similar lecture texts.

• Somebody had answered my question, and I had commented on that, and it is all gone! Who does that? And why don't I get notified when somebody messes with my posting?! May 1, 2022 at 9:30

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$$\texttt{C O N T E N T S}$$

$$\bl\S\texttt{ A. Lorentz boost and transformation of velocity 3-vectors}$$

$$\bl\S\texttt{ B. Aberration of light}$$

$$\bl\S\texttt{ C. Relativistic Doppler Shift}$$

$$\bl\S\texttt{ D. Intensity changes in the Doppler effect}$$

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$$\texttt{R E F E R E N C E S}$$

$$\texttt{Reference-01:}$$ Intensity changes in the Doppler effect, M.H. Johnson and E.Teller.

$$\texttt{Reference-02:}$$ My answer in About de Broglie relations, what exactly is E? Its energy of what?.

$$\texttt{Reference-03:}$$ My answer in Deriving relativistic Doppler shift in terms of wavelength.

$$\texttt{Reference-04:}$$'Relativity-Special, General, and Cosmological' by W.Rindler, 2nd Ed.

$$\texttt{Reference-05:}$$'Modern Classical Physics. Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics' by Kip S.Thorne and Roger D. Blandford, 2017.

$$\texttt{Reference-06:}$$High Energy Astrophysics - Lecture 3, Frank Rieger.

$$\texttt{Reference-07:}$$Phase space volume and relativity.

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$$\bl\S$$ A. Lorentz boost and transformation of velocity 3-vectors

The $$\,1\p 1\m$$Lorentz boost transformation with velocity $$\:\bl\upsilon\e\upsilon\,\mb e_x \:$$ along the common $$x'\m,x\m$$axis of two inertial frames $$\:\mr S',\mr S\:$$ expressed in differential form is \begin{align} \mr dx &\e\gamma\plr{\mr dx'\p\upsilon\mr dt'} \qquad \plr{\m c\les\upsilon\les c} \tl{A-01a}\\ \mr dy &\e\mr dy' \tl{A-01b}\\ \mr dz &\e\mr dz' \tl{A-01c}\\ \mr dt &\e\gamma\plr{\mr dt'\p\dfrac{\upsilon\mr dx'}{c^2}} \tl{A-01d}\\ \gamma &\e\plr{1\boldsymbol{-}\dfrac{\upsilon^2}{c^2}}^{\m\frac12} \tl{A-01e} \\ \beta &\e\dfrac{\,\upsilon\,}{c} \qquad \plr{\m 1\les\beta\les 1} \tl{A-01f} \end{align} see Figure-01.

If a particle is moving with respect to the frame $$\:\mr S'\:$$ with velocity $$$$\begin{split} &\mb u'\e\plr{\mr u_x',\mr u_y',\mr u_z'}\e\mr u'\plr{\mr n_x',\mr n_y',\mr n_z'}\e\mr u'\mb n'\\ &\m c\leseq\mr u'\leseq c \qquad \Vlr{\mb n'}\e 1 \\ \end{split} \tl{A-02}$$$$ then to find its velocity with respect to the frame $$\:\mr S\:$$ $$$$\begin{split} &\mb u\e\plr{\mr u_x,\mr u_y,\mr u_z}\e\mr u\,\plr{\mr n_x,\mr n_y,\mr n_z}\e\mr u\,\mb n\\ &\m c\leseq\mr u\leseq c \qquad \Vlr{\mb n}\e 1 \\ \end{split} \tl{A-03}$$$$ we divide side-by-side each one of equations \eqref{A-01a},\eqref{A-01b},\eqref{A-01c} by equation \eqref{A-01d}, and setting $$$$\begin{split} \mb u'&\e\plr{\mr u_x',\mr u_y',\mr u_z'}\e\plr{\dfrac{\mr dx'}{\mr dt'}\:,\:\dfrac{\mr dy'}{\mr dt'}\:,\:\dfrac{\mr dz'}{\mr dt'}}\\ \mb u &\e\plr{\mr u_x,\mr u_y,\mr u_z}\e\plr{\dfrac{\mr dx}{\mr dt}\:,\:\dfrac{\mr dy}{\mr dt}\:,\:\dfrac{\mr dz}{\mr dt}}\\ \end{split} \tl{A-04}$$$$ we end up with the following Lorentz transformation of the velocity 3-vectors $$$$\mr u_x\e\dfrac{\mr u_x'\p\upsilon}{1\p\dfrac{\upsilon\mr u_x'}{c^2}}\:,\quad \mr u_y\e\dfrac{\mr u_y'}{\gamma\plr{1\p\dfrac{\upsilon\mr u_x'}{c^2}}}\:,\quad \mr u_z\e\dfrac{\mr u_z'}{\gamma\plr{1\p\dfrac{\upsilon\mr u_x'}{c^2}}} \tl{A-05}$$$$ essentially the relativistic addition of the velocities $$\:\mb u'\:$$ and $$\:\bl\upsilon$$.

$$\bl\S$$ B. Aberration of light

Consider that a light source at rest in frame $$\:\mr S'\:$$ is emitting a photon in the direction $$\:\mb n'\:$$ on the $$x'y'\m$$plane by an angle $$\:\theta'\:$$ with respect to $$\:\bl\upsilon$$. The velocity of the photon is $$$$\mb u'\e c\,\mb n'\e c\plr{\cos\theta',\sin\theta',0}\e\plr{\mr u_x',\mr u_y',\mr u_z'} \tl{B-01}$$$$ Inserting its components in equations \eqref{A-05} we find the velocity of the photon with respect to the frame $$\:\mr S$$ $$$$\mb u\e c\,\mb n\:\e c\plr{\cos\theta\:,\sin\theta\:,0}\e\plr{\mr u_x,\mr u_y,\mr u_z} \tl{B-02}$$$$ where $$$$\boxed{\:\:\cos\theta\e\dfrac{\cos\theta'\p\beta}{1\p\beta\cos\theta'}\:, \quad \sin\theta\e\dfrac{\sin\theta'}{\gamma\plr{1\p\beta\cos\theta'}}\Vp{\dfrac{\dfrac{a}{b}}{\dfrac{a}{b}}}\:\:}\quad \texttt{(Aberration 1)} \tl{B-03}$$$$ that is the equation for the aberration of light.

Note that due to the trigonometric identity $$$$\tan\plr{\dfrac{\,\theta\,}{2}}\e\dfrac{\sin\theta}{1\p \cos\theta} \tl{B-04}$$$$ equations \eqref{B-03} yield $$$$\tan\plr{\dfrac{\,\theta\,}{2}}\e\dfrac{\sin\theta}{1\p \cos\theta}\e \dfrac{1}{\gamma\plr{1\p \beta}}\dfrac{\sin\theta'}{1\p \cos\theta'}\e\sqrt{\dfrac{1\m \beta}{1\p\beta}}\tan\plr{\dfrac{\,\theta'}{2}} \nonumber$$$$ that is the more simple equation $$$$\boxed{\:\:\tan\plr{\dfrac{\,\theta\,}{2}}\e \plr{\dfrac{c\m\upsilon}{c\p\upsilon}}^{\frac12}\tan\plr{\dfrac{\,\theta'}{2}}\Vp{\dfrac{\dfrac{a}{b}}{\dfrac{a}{b}}}\:\:}\quad\texttt{(Aberration 2)} \tl{B-05}$$$$ We meet above equation as (4.9) in $$\texttt{Reference-04}$$.

$$\bl\S$$ C. Relativistic Doppler Shift

According to Louis de Broglie to a massive particle moving with $$''$$subluminal$$''$$ velocity $$\:\mb u\e\mr u\,\mb n, \vert\mr u\vert\les c \:$$, there corresponds a $$''$$superluminal$$''$$ plane phase wave, see $$\texttt{Reference-02}$$.

A subluminal particle has time-like energy-momentum Lorentz 4-vector $$$$\mb P\e\plr{\gamma_{\mr u}\,m\,c, \gamma_{\mr u}\,m\,\mb u}\e\plr{\dfrac{E}{c},\mb p } \tl{C-01}$$$$ where $$$$\begin{split} \!\!\!\!\!\!\!\!E&\e\gamma_{\mr u}\,m\,c^2\e \texttt{energy of the particle}\\ \!\!\!\!\!\!\!\!\mb p&\e \gamma_{\mr u}m\mb u\e \gamma_{\mr u}m\mr u\mb n\e\texttt{linear momentum 3-vector of the particle}\\ &\Vlr{\mb n}\e 1\\ \end{split} \tl{C-02}$$$$ while a superluminal plane phase wave has time-like angular frequency Lorentz 4-vector $$$$\bl\Omega\e\plr{2\pi\nu, c\dfrac{2\pi}{\lambda}\,\mb m}\e\plr{\omega, c\,\mb k} \tl{C-03}$$$$ where $$$$\begin{split} \omega &\e2\pi\,\nu\e \texttt{angular frequency of the plane phase wave}\\ \mb k&\e \dfrac{2\pi}{\lambda}\,\mb m\e\texttt{wave number 3-vector of the plane phase wave}\\ \lambda &\e \texttt{wavelength of the plane phase wave}\\ &\Vlr{\mb m}\e 1\\ \end{split} \tl{C-04}$$$$

This superluminal plane phase wave is moving with velocity $$$$\mb w\e\dfrac{\omega}{\Vlr{\mb k}}\mb m\e\lambda\,\nu\,\mb m\e\dfrac{\omega}{\Vlr{\mb k}^2}\mb k\,,\qquad \Vlr{\mb w}\e \mr w\e\lambda\,\nu \tl{C-05}$$$$

The vector $$\:\bl\Omega\:$$ of equation \eqref{C-03} is a Lorentz 4-vector. This means that the vector $$\:\plr{\omega,c\,k_x,c\,k_y,c\,k_z}\:$$ is transformed as the infinitesimal displacement vector $$\:\plr{c\mr dt,\mr dx,\mr dy,\mr dz}\:$$ in equations \eqref{A-01a}-\eqref{A-01f}.

The de Broglie relation connects the energy-momentum Lorentz 4-vector of the particle $$\:\mb P$$, equation \eqref{C-01}, with the angular frequency of its accompanying plane phase wave $$\:\bl\Omega$$, equation \eqref{C-03} $$$$\boxed{\:\:c\,\mb P\e \hbar\,\bl\Omega\:\:\vp}\quad\texttt{(de Broglie)} \tl{C-06}$$$$

The directions of the particle motion $$\:\mb n\:$$ and of propagation of the plane phase wave $$\:\mb m\:$$ are identical, while the product of their speeds is a Lorentz scalar invariant $$$$\mb m\bl\equiv \mb n\,,\qquad \mr u\,\mr w\e \mr u'\,\mr w'\e c^2\e\texttt{ Lorentz invariant} \tl{C-07}$$$$

Equating time and space components in equation \eqref{C-06} we have $$$$\begin{split} E&\e\hbar \omega \e h\,\nu\\ \mb p&\e\hbar\mb k\e\dfrac{\,h\,}{\lambda}\,\mb n\\ \end{split} \tl{C-08}$$$$

Now, all these relations are valid in the limiting case of a $$''$$luminal$$''$$ particle, that is a photon, and its accompanying $$''$$luminal$$''$$ phase wave, that is light or electromagnetic wave. In this case we have $$\: \mr u\e\mr w\e c\e\lambda\,\nu\:$$ and equations \eqref{C-08} give the energy-momentum 4-vector of a photon $$$$\mb P\e\plr{\dfrac{E}{c},\mb p}\e \dfrac{h\,\nu}{c}\plr{1,\mb n} \tl{C-09}$$$$ The vector $$\:c\mb P\:$$ of equation \eqref{C-09} is a Lorentz 4-vector. This means that the components of this vector $$\:\plr{h\,\nu,h\,\nu\,n_x,h\,\nu\,n_y,h\,\nu\,n_z}\:$$ are transformed as the components of the infinitesimal displacement vector $$\:\plr{c\mr dt,\mr dx,\mr dy,\mr dz}\:$$ in equations \eqref{A-01a}-\eqref{A-01f}.

So, let again the photon emitted by the light source in its rest frame $$\:\mr S'\:$$ as shown in Figure-01. For its energy-momentum 4-vector we have $$$$c\mb P'\e h\,\nu'\plr{1,\mr n_x',\mr n_y',\mr n_z'}\e\plr{h\,\nu',h\,\nu'\cos\theta',\,h\,\nu'\sin\theta',0} \tl{C-10}$$$$ Its energy-momentum 4-vector in the frame $$\:\mr S\:$$ is $$$$c\mb P\:\e h\,\nu\:\plr{1,\mr n_x,\mr n_y,\mr n_z}\e\plr{h\,\nu,h\,\nu\cos\theta,\,h\,\nu\sin\theta,0} \tl{C-11}$$$$ Inserting these vectors in equations \eqref{A-01a}-\eqref{A-01d} in place of $$\:\plr{c\,\mr dt',\mr d\mb r' }\:$$ and $$\:\plr{c\,\mr dt,\mr d\mb r}\:$$ respectively we have in details \begin{align} h\,\nu\cos\theta &\e\gamma\plr{h\,\nu'\cos\theta'\p\beta\,h\,\nu'} \tl{C-12a}\\ h\,\nu\sin\theta &\e h\,\nu'\sin\theta' \tl{C-12b}\\ 0 &\e 0 \tl{C-12c}\\ h\,\nu &\e\gamma\plr{h\,\nu'\p \beta\,h\,\nu'\cos\theta'} \tl{C-12d} \end{align} that is \begin{align} \dfrac{\nu}{\nu'} &\e\dfrac{\gamma\,\plr{\cos\theta'\p\beta}}{\cos\theta} \tl{C-13a}\\ \dfrac{\nu}{\nu'} &\e\dfrac{\sin\theta'}{\sin\theta} \tl{C-13b}\\ \dfrac{\nu}{\nu'} &\e\gamma\plr{1\p\beta\cos\theta'} \tl{C-13c} \end{align} Equating the right hand sides firstly of equations \eqref{C-13a}, \eqref{C-13c} and secondly of equations \eqref{C-13b}, \eqref{C-13c} we get the following equations respectively $$$$\cos\theta\e\dfrac{\cos\theta'\p\beta}{1\p\beta\cos\theta'}\:, \quad \sin\theta\e\dfrac{\sin\theta'}{\gamma\plr{1\p\beta\cos\theta'}} \tl{C-14}$$$$ identical to equations \eqref{B-03}. We meet again the aberration of light as discussed in $$\:\bl\S\texttt{B}$$.

From the first of the aberration equations \eqref{C-14} we have $$$$\cos\theta'\e\dfrac{\cos\theta\m\beta}{1\m\beta\cos\theta} \tl{C-15}$$$$

Inserting this expression of $$\:\cos\theta'\:$$ firstly in the second of the aberration equations \eqref{C-14} we get $$$$\sin\theta'\e\dfrac{\sin\theta}{\gamma\plr{1\m\beta\cos\theta}} \tl{C-16}$$$$ and secondly by insertion in equation \eqref{C-13c} we have $$$$\dfrac{\nu}{\nu'}\e\dfrac{1}{\gamma\plr{1\m\beta\cos\theta}} \tl{C-17}$$$$ Finally differentiating any of the equations \eqref{C-14},\eqref{C-15} or \eqref{C-16} yields $$$$\dfrac{\mr d\theta'}{\mr d\theta}\e\dfrac{\nu}{\nu'} \tl{C-18}$$$$

Defining the Doppler factor $$$$\mr D\bl\equiv\dfrac{\nu}{\nu'} \tl{C-19}$$$$ all these relations are given in one stroke below $$$$\!\!\!\!\!\!\!\!\!\!\boxed{\:\mr D\e\dfrac{\nu\texttt{(shifted)}}{\nu'\texttt{(unshifted)}}\e\gamma\plr{1\p\beta\cos\theta'}\e\dfrac{1}{\gamma\plr{1\m\beta\cos\theta}}\e\dfrac{\sin\theta'}{\sin\theta}\e\dfrac{\mr d\theta'}{\mr d\theta}\Vp{\dfrac{\dfrac{a}{b}}{\dfrac{a}{b}}}\:} \tl{C-20}$$$$

The relation between the infinitesimal solid angles $$\:\mr d\Theta'\e\sin\theta'\mr d\theta'\mr d\phi'\:$$ and $$\:\mr d\Theta\e\sin\theta\mr d\theta\mr d\phi\:$$ could be derived from equations \eqref{C-20}, the Figure-02 and the fact that for the azimuth around $$\:\bl\upsilon\:$$ angle $$\:\phi\:$$ we have $$\:\mr d\phi\e\mr d\phi'\:$$ $$$$\dfrac{\mr d\Theta'}{\mr d\Theta}\e\dfrac{\sin\theta'\mr d\theta'\mr d\phi'}{\sin\theta\,\mr d\theta\,\mr d\phi}\e\underbrace{\plr{\dfrac{\sin\theta'}{\sin\theta}}}_{\mr D}\underbrace{\plr{\dfrac{\mr d\theta'}{\mr d\theta}}}_{\mr D}\underbrace{\plr{\dfrac{\mr d\phi'}{\mr d\phi}}}_{1}\e\mr D^2 \tl{C-21}$$$$ that is $$$$\boxed{\:\dfrac{\mr d\Theta'}{\mr d\Theta}\e \mr D^2\e\plr{\dfrac{\nu}{\nu'}}^2\Vp{\dfrac{\dfrac{a}{b}}{\dfrac{a}{b}}}\:} \tl{C-22}$$$$

$$\bl\S$$ D. Intensity changes in the Doppler effect

The proof of the Lorentz invariance of the scalar $$\:I_\nu/\nu^3\:$$ is based on the Lorentz invariance of the infinitesimal volume element in phase space. But at first we must define the Specific Intensity $$\:I_\nu$$. So consider a set of rays and construct the infinitesimal area $$\:\mr dA\:$$ normal to a given ray and look at all rays passing through area element within solid angle $$\:\mr d\Omega\:$$ of the given ray as shown in the Figure(1) below.

The photon's Specific Intensity $$\:I_\nu\:$$ is defined to be the total energy $$$$\mr dE\e h\nu\mr dN \tl{D-01}$$$$ (where $$\:\mr dN\:$$ is the number of photons) that crosses this area, per unit area $$\:\mr dA$$, per unit time $$\:\mr dt$$, per unit frequency $$\:\mr d\nu$$, and per unit solid angle $$\:\mr d\Omega$$ $$$$I_\nu\bl\equiv\dfrac{\mr dE}{\mr dA\,\mr dt\,\mr d\nu\,\mr d\Omega} \tl{D-02}$$$$ (i.e., per unit everything)

As respect to the invariance of the infinitesimal volume element in phase space we note in summary the following (see $$\texttt{Reference-05}$$, Chapter 3): As tools for the study of a collection of a very large number of identical particles (all with the same rest mass $$\:m$$) consider a tiny 3-dimensional volume $$\:\mr d\mc V_x\:$$ centered on some location $$\:\mb x \:$$ in physical space and a tiny 3-dimensional volume $$\:\mr d\mc V_p\:$$ centered on location $$\:\mb p\:$$ in momentum space. Together these make up a tiny 6-dimensional volume (in Newtonian theory) $$$$\mr d^2\mc V\bl\equiv\mr d\mc V_x\mr d\mc V_p \tl{D-03}$$$$ In any Cartesian coordinate system, we can think of $$\:\mr d\mc V_x\:$$ as being a tiny cube located at $$\:\plr{x,y,z}\:$$ and having edge lengths $$\:\mr dx,\mr dy,\mr dz$$, and similarly for $$\:\mr d\mc V_p$$. Then, as computed in this coordinate system, these tiny volumes are $$$$\mr d\mc V_x\e\mr dx\,\mr dy\,\mr dz\,,\quad \mr d\mc V_p\e\mr dp_x\,\mr dp_y\,\mr dp_z\,,\quad \mr d^2\mc V\e\mr dx\,\mr dy\,\mr dz\,\mr dp_x\,\mr dp_y\,\mr dp_z \tl{D-04}$$$$ Denote by $$\:\mr dN\:$$ the number of particles (all with rest mass $$\:m$$) that reside inside $$\:\mr dN\:$$ in phase space (at some moment of time $$\:t$$). Stated more fully: $$\mr dN\:$$ is the number of particles that, at time $$\:t$$, are located in the 3-volume $$\:\mr d\mc V_x\:$$ centered on the location $$\:\mb x\:$$ in physical space and that also have momentum vectors whose tips at time $$\:t\:$$ lie in the 3-volume $$\:\mr d\mc V_p\:$$ centered on location $$\:\mb p\:$$ in momentum space. Denote by $$$$\boxed{\:\:\mc N\plr{\mb x,\mb p,t}\bl\equiv\dfrac{\mr dN}{\mr d^2\mc V}\e\dfrac{\mr dN}{\mr d\mc V_x\mr d\mc V_p}\Vp{\dfrac{\dfrac{a}{b}}{\dfrac{a}{b}}}\:\:} \tl{D-05}$$$$ the number density of particles at location $$\:\plr{\mb x,\mb p}\:$$ in phase space at time $$\:t$$. This is also called the *distribution function. In Newtonian theory, the volumes $$\:\mr d\mc V_x\:$$ and $$\:\mr d\mc V_p\:$$ occupied by our collection of $$\:\mr dN\:$$ particles are independent of the reference frame that we use to view them. Not so in relativity theory: $$\mr d\mc V_x\:$$ undergoes a Lorentz contraction when one views it from a moving frame, $$\:\mr d\mc V_p\:$$ also changes; but their product $$\:\mr d^2\mc V\e\mr d\mc V_x\mr d\mc V_p\:$$ is the same in all frames. More precisely in relativity theory it has been proved on one hand that $$$$\boxed{\:\:E\,\mr d\mc V_x\e p_0\,\mr d\mc V_x\e\texttt{Lorentz invariant} \Vp{\dfrac{\dfrac{a}{b}}{\dfrac{a}{b}}}\:\:} \tl{D-06}$$$$ and on the other hand that $$$$\boxed{\:\:\dfrac{\mr d\mc V_p}{E}\e \dfrac{\mr d\mc V_p}{p_0}\e\texttt{Lorentz invariant} \Vp{\dfrac{\dfrac{a}{b}}{\dfrac{a}{b}}}\:\:} \tl{D-07}$$$$ Therefore, in relativity theory for the distribution function we have also $$$$\boxed{\:\:\mc N\bl\equiv\dfrac{\mr dN}{\mr d^2\mc V}\e\dfrac{\mr dN}{\mr d\mc V_x\mr d\mc V_p}\e\texttt{Lorentz invariant}\Vp{\dfrac{\dfrac{a}{b}}{\dfrac{a}{b}}}\:\:} \tl{D-08}$$$$ Although in derivations and proofs it is assumed nonzero rest mass ($$m\bl\ne 0$$), the conclusions \eqref{D-06} and \eqref{D-07} continue to hold if we take the limit as $$m\rightarrow 0$$ and the 4-momenta become null. Correspondingly, \eqref{D-06} to \eqref{D-08} are valid for particles with zero mass also, like photons.

Consider that photons hit the surface area $$\:\mr dA\:$$ in time interval $$\:\mr dt\:$$ as shown in the Figure(2) below.

Since the photons move at the speed of light $$\:c$$, the product of that surface area with $$\:c\:$$ times the time $$\:\mr dt\:$$ is equal to the volume they occupy at a specific moment of time: $$$$\mr d\mc V_x\e\mr dA\,\mr dt \tl{D-09}$$$$ Focus attention on a set $$\:S\:$$ of photons in this volume that all have nearly the same frequency $$\:\nu\:$$ and propagation direction $$\:\mb n$$. Their energies $$\:E\:$$ and momenta $$\:\mb p\:$$ are related to $$\:\nu\:$$ and $$\:\mb n\:$$ by equation \eqref{C-09}, that is $$$$E\e h\nu\,,\quad \mb p\e\plr{h\nu/c}\mb n \tl{D-10}$$$$ Their frequencies lie in a range $$\:\mr d\nu\:$$ centered on $$\:\nu\:$$, and they come from a small solid angle $$\:\mr d\Omega\:$$ centered on $$\m\mb n$$; the volume they occupy in momentum space is related to these quantities by $$$$\mr d\mc V_p\e \Vlr{\mb p}^2\mr d\Omega\,\mr d \Vlr{\mb p}\e\plr{h\nu/c}^2\mr d\Omega\,\plr{h\mr d\nu/c}\e\plr{h/c}^3\nu^2\mr d\Omega\,\mr d\nu \tl{D-11}$$$$ From the definition of specific intensity, equation \eqref{D-02}, using equations \eqref{D-01}, \eqref{D-09} and \eqref{D-11} we have $$$$\boxed{\:\:\mc N\bl\equiv\dfrac{\mr dN}{\mr d^2\mc V}\e\dfrac{\mr dN}{\mr d\mc V_x\mr d\mc V_p}\e\dfrac{c^2}{h^4}\dfrac{I_\nu}{\nu^3}\Vp{\dfrac{\dfrac{a}{b}}{\dfrac{a}{b}}}\:\:} \tl{D-12}$$$$ that is the scalar $$\:I_\nu/\nu^3\:$$ is except a constant identical to the distribution function $$\:\mc N\:$$ so by equation \eqref{D-08} a Lorentz invariant scalar.

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(1) Figure extracted from $$\texttt{Reference-06}$$

(2) Figure extracted from $$\texttt{Reference-05}$$

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• Thanks, but I don't see in your derivation, that the intensity canges with the cube of the frequency. But Planck's law of blackbody radiation (of a star) might do the job: If the spectral intensity is proportional to $\nu^3/\left(e^{h\nu/kT}-1\right)$, and the frequency $\nu$ changes because of the Doppler effect, then of course also the spectral intensity changes with $\nu^3$. Is that acceptable, or am I on the wrong track? Apr 30, 2022 at 10:32