# LOG#045. Fake superluminality.

Before becoming apparent superluminal readers, we are going to remember and review some elementary notation and concepts from the relativistic Doppler effect and the starlight aberration we have already studied in this blog.

Let us consider and imagine the next gedankenexperiment/thought experiment. Some moving object emits pulses of light during some time interval, denoted by $\Delta \tau_e$ in its own frame. Its distance from us is very large, say

$D>>c\Delta \tau_e$

Question: Does it (light) arrive at time $t=D/c$? Suppose the object moves forming certain angle $\theta$ according to the following picture

Time dilation means that a second pulse would be experiment a time delay $\Delta t_e=\gamma \Delta \tau_e$, later of course from the previous pulse, and at that time the object would have travelled a distance $\Delta x=v\Delta t_e\cos\theta$ away from the source, so it would take it an additional time $\Delta x/c$ to arrive at its destination. The reception time between pulses would be:

$\Delta t_r=\Delta t_e+\beta \Delta t_e\cos\theta=\gamma (1+\beta \cos\theta)\Delta \tau_e$

i.e.

$\boxed{\Delta t_r=(1+\beta\cos\theta)\gamma \Delta \tau_e}$

In the range $0<\theta<\pi$, the time interval separation measured from both pulses in the rest frame on Earth will be longer than in the rest frame of the moving object. This analysis remains valid even if the 2 events are not light beams/pulses but succesive packets or “maxima” of electromagnetic waves ( electromagnetic radiation).

Astronomers define the dimensionless redshift

$\boxed{(1+z)\equiv \dfrac{\Delta t_r}{\Delta \tau_e}=\gamma (1+\beta \cos\theta)}$

where, as it is common in special relativity, $\beta=v/c$, $\gamma^2=\dfrac{1}{1-\beta^2}$

The 3 interesting limits of the above expression are:

1st. Receding emitter case. The moving object moves away from the receiver. Then, we have $\theta=0$ supposing a completely radial motion in the line of sight, and then a literal “redshift” ( lower frequencies than the proper frequencies)

$(1+z)=\sqrt{\dfrac{1+\beta}{1-\beta}}$

2nd. Approaching emitter case. The moving object approaches and goes closer to the observer. Then, we get $\theta=\pi$, or motion inward the radial direction, and then a “blueshift” ( higher frequencies than those of the proper frequencies)

$(1+z)=\sqrt{\dfrac{1-\beta}{1+\beta}}$

3rd. Tangential or transversal motion of the source. This is also called second-order redshift. It has been observed in extremely precise velocity measurements of pulsars in our Galaxy.

$(1+z)=\gamma$

Furthermore, these redshifts have all been observed in different astrophysical observations and, in addition, they have to be taken into account for tracking the position via GPS, geolocating satellites and/or following their relative positions with respect to time or calculating their revolution periods around our planet.

Remark: Quantum Mechanics and Special Relativity would be mutually inconsistent IF we did not find the same formual for the ratios between energy and frequencies at different reference frames.

EXAMPLE: The emission line of the oxygen (II) [O(II)] is, in its rest frame, $\lambda_0=3727\AA$. It is observed in a distant galaxy to be at $\lambda=9500\AA$. What is the redshift z and the recession velocity of this galaxy?

Solution.  From the definition of wavelength in electromagnetism $cT=\lambda$, adn $c\tau=\lambda_0$. Then,

$(1+z)=\dfrac{T}{\tau}=\dfrac{\lambda}{\lambda_0}=\dfrac{9500}{3727}=2.55$, and thus $z=1.55$

From the radial velocity hypothesis, we get

$(1+z)=\sqrt{\dfrac{1+\beta}{1-\beta}}$ or

$\beta=\dfrac{(1+z)^2-1}{(1+z)^2+1}=0.73$

and thus $\beta=0.73$ or $v=0.73c$
Note that this result follows from the hypothesis of the expansion of the Universe, and it holds in the relativistic theory of gravity, General Relativity, and it should also holds in extensions of it, even in Quantum Gravity somehow!

Remember: Stellar aberration causes taht the positions on the sky of the celestial objects are changing as the Earth moves around the Sun. As the Earth’s velocity is about $v_E\approx 30km/s$, and then $\beta_E\approx 10^{-4}$, it implies an angular separation about $\Delta \theta\approx 10^{-4}rad$. Anyway, it is worth mentioning that the astronomer Bradley observed this starlight aberration in 1729! A moving observer observes that light from stars are at different positions with respect to a rest observer, and that the new position does not depend on the distance to the star. Thus, as the relative velocity increases, stars are “displaced” further and further towards the direction of observation.

Now, we are going to the main subject of the post. I decided to review this two important effects because it is useful to remember then and to understand that they are measured and they are real effects. They are not mere artifacts of the special theory of relativity masking some unknown reality. They are the reality in the sense they are measured. Alternative theories trying to understand these effects exist but they are more complicated and they remember me those people trying to defend the geocentric model of the Universe with those weird metaphenomenon known as epicycles in order to defend what can not be defended from the experimental viewpoint.

In order to make our discussion visual and phenomenological, I am going to consider a practical example. Certain radio-galaxy, denoted by 3C 273 moves with a velocity

$\omega=0.8 miliarc sec/yr=4\cdot 10^{-9}\dfrac{rad}{yr}$

Note that $1 miliarc sec=\left(\dfrac{10^{-3}}{3600}\right)^{\textdegree}$

Knowing the rate expansion of the universe and the redshift of the radiogalaxy, its distance is calculated to be about $2.6\cdot 10^9 lyr$. To obtain the relative tangential velocity, we simply multiply the angular velocity by the distance, i.e. $v_{r\perp}=\omega D$.

From the above data, we get that the apparent tangential radial velocity of our radiogalaxy would be about $v_{r\perp}\approx 10c$. Indeed, this observation is not isolated. There are even jets of matter flowin from some stars at apparent superluminal velocities. Of course this is an apparent issue for SR. How can we explain it? How is it possible in the SR framework to obtain a superluminal velocity? It shows that there is no contradiction with SR. The (fake and apparent) superluminal effect CAN BE EXPLAINED naturally in the SR framework in a very elegant way. Look at the following picture:

It shows:

-A moving object with velocity $v=\vert \mathbf{v}\vert$ with respect to Earth, approaching to Earth.

-There is some angle $\theta$ in the direction of observation. And as it moves towards Earth, with our conventions, $lates \theta\approx\pi=180\textdegree$

-The moving object emits flashes of light at two different points, A and B, separated by some time interval $\Delta t_e$ in the Earth reference frame.

-The distance between those two points A and B, is very small compared with the distance object-Earth, i.e., $d(A,B)<< D$.

Question: What is the time separation $\Delta t_r$ between the receptions of the pulses at the Earth surface?

The solution is very cool and intelligent. We get

A: time interval $\Delta t_e=t_A=\dfrac{D}{c}$

B: time interval $t_B=t_A+\dfrac{v\Delta t_e\cos\theta}{c}$

Note that $\cos\theta<0$!

From this equations, we get a combined equation for the time separation of pulses on Earth

$\boxed{\Delta t_r=\Delta t_e (1+\beta \cos\theta)}$

The tangential separation is defined to be

$\Delta Y=Y_B-Y_A=v\Delta t_e\sin\theta$

so, the apparent velocity of the source, seen from the Earth frame, is showed to be:

$\boxed{v_a=\dfrac{\Delta Y}{\Delta t_r}=\dfrac{\beta\sin\theta}{1+\beta\cos\theta}c}$

Remark (I): $v_a>>c$ IFF $\beta\approx 1$ AND $\cos\theta\approx -1$!

Remark (II): There are some other sources of fake superluminality in special relativity or general relativity (the relativist theory of gravity). One example is that the phase velocity and the group velocity can indeed exceed the speed of light, since from the equation $v_{ph}v_{g}=c^2$, it is obvious that whenever that one of those two velocities (group or phase velocity) are lower than the speed of light at vacuum, the another has to be exceeding the speed of light. That is not observable but it has an important rôle in the de Broglie wave-particle portrait of the atom. Other important example of apparent and fake superluminal motion is caused by gravitational (micro)lensing in General Relativity. Due to the effect of intense gravitational fields ( i.e., big concentrations of mass-energy), light beams from slow-movinh objects can be magnified to make them, apparently, superluminal. In this sense, gravity acts in an analogue way of a lens, i.e., as it there were a refraction index and modifying the propagation of the light emitted by the sources.

Remark (III): In spite of the appearance, I am not opposed to the idea of superluminal entities, if they don’t break established knowledge that we do know it works. Tachyons have problems not completely solved and many physicists think (by good reasons) they are “unphysical”.  However, my own experience working with theories beyond special/general relativity and allowing superluminal stuff (again, we should be careful with what we mean with superluminality and with “velocity” in general) has showed me that if superluminal objects do exist, they have observable consequences. And as it has been showed here, not every apparent superluminal motion is superluminal!Indeed, it can be handled in the SR framework. So, be aware of crackpots claiming that there are superluminal jets of matter out there, that neutrinos are effectively superluminal entities ( again, an observation refuted by OPERA, MINOS and ICARUS and in complete disagreement with the theory of neutrino oscillations and the real mass that neutrino do have!) or even when they say there are superluminal protons and particles in the LHC or passing through the atmosphere without any effect that should be vissible with current technology. It is simply not true, as every good astronomer, astrophysicist or theoretical physicist do know! Superluminality, if it exists, it is a very subtle thing and it has observable consequences that we have not observed until now. As far as I know, there is no (accepted) observation of any superluminal particle, as every physicist do know. I have discussed the issue of neutrino time of flight here before:

https://thespectrumofriemannium.wordpress.com/2012/06/08/

Final challenge: With the date given above, what would the minimal value of $\beta$ be in order to account for the observed motion and apparent (fake) superluminal velocity of the radiogalaxy 3C 273?