LOG#049. Ludicrous speed.

We are going to learn about the different notions of velocity that the special theory of relativity provides.

The special theory of relativity is a simple wonderful theory, but it comes with many misconceptions due to bad teaching/science divulgation. It is not easy to master the full theory of relativity without the proper mathematical background and physical insight. In the internet era where knowledge is shared, a fundamental issue is to understand things properly. There are many people who thinks they understand the theory of relativity when they don’t. Even at the academia.

Moreover, you can find many people in the blogsphere/websphere trying to sell false theories and wrong theories. It is the same like the so-called alternative medicine: they are not medicine at all. Bad science is not science, it is simply a lie and not science at all. It is religion. Science can be critized, but nobody can critize that Earth revolves around the Sun, it is common knowledge and truth. So, we can make critics to scientist, but not the scientific method and well established theories. We can try to understand better or in a novel way, but we can not deny facts and experiments. Gerard ‘t Hooft, Nobe Prize, explain it in his web page www.phys.uu.nl/~thooft/.

It is important to remark that Science revolutions come when we extend the theories we know they are correct, like special relativity and not with a full destruction of the current and well-tested theories. Newtonian relativity is a limit of General Relativity. Galilean relativity is a limit of Special Relativity. Quantum Mechanics is a limit of QFT and so on. The issue is not that. Said these words, I am quite sure that scientists and particularly physicists wish to overcome current theories with new ones. However, the process to create a new theory is not easy. Specially, if you don’t understand the traps and theories that have passed every known test till now.

What is velocity? Classically, the answer is short and very clear/neat: velocity is the rate of change of position with respect to time. It is a vector magnitude. Mathematically speaking is the quotient between the displacement vector and the time interval, or in the infinitesimal limit, the derivative of the position vector with respect to time.

$\boxed{\mathbf{v_m}=\dfrac{\Delta \mathbf{r}(t)}{\Delta t}\leftrightarrow \mbox{Average velocity}}$

$\boxed{\mathbf{v}=\dfrac{d\mathbf{r}(t)}{dt}\leftrightarrow \mbox{Instantaneous velocity}}$

In the special theory of relativity, due to the fact that time is not universal but relative we can build different notions of velocity. And it matters. There are some clear concepts from relativity you should master till now:

a) You can attach a clock to any yardstick you could physically use for measurements of space and time.

b) You must distinguish the notions of coordinate velocity (map coordinate is another commonly used notion/concept) and proper velocity. The latter is sometimes called hyperbolic (or imaginary) velocity. These two notions are caused by the presence of two “natural” elections of time: the proper time and the coordinate time.

c) Due to the previous two facts, you must also distinguish between proper acceleration and geometric acceleration. Proper-accelerations caused by the tug of external forces and geometric accelerations caused by choice of a reference frame that’s not geodesic i.e. a local reference coordinate-system that is not ”in free-fall”. Proper-accelerations are felt through their points of action e.g. through forces on the bottom of your feet. On the other hand geometric accelerations give rise to inertial forces that act on every ounce of an object’s being. They either vanish when seen from the vantage point of a local free-float frame, or give rise to non-local force effects on your mass distribution that cannot be made to disappear. Coordinate acceleration goes to zero whenever proper-acceleration is exactly canceled by that connection term, and thus when physical and inertial forces add to zero.

People who are not aware of the previous comments, don’t understand relativity and the physics behind it. They even don’t undertand what experiments and their data say.

Let me review the main magnitudes, 3-vectors and 4-vectors which the special theory of relativity studies in the next tables:

The two notions of 3-velocity we do have from the special theory of relativity, i.e., from the 4-velocity $\mathbb{U}=\dfrac{d\mathbb{X}}{d\tau}$,  are:

1) Coordinate velocity, $\mathbf{v}$:

$\mathbf{v}=\dfrac{d\mathbf{r}}{dt}$

It is the common notion of 3-velocity, measured from an inertial observer with respect to the coordinate time t. Note that the coordinate time is not a true invariant in SR!

2) Proper velocity (or the hyperbolic velocity/imaginary angle velocity related to it):

$\mathbf{w}\equiv \dfrac{d\mathbf{r}}{d\tau}=\gamma \mathbf{v}$

where $\tau$ is the proper time. This velocity can intuitively defined as the distance per unit traveler-time, retains many of the properties that ordinary velocity loses at high speed. In addition to these two definitions, we also have:

1)Proper-acceleration $\alpha$, is the acceleration experienced relative to a locally co-moving free-float-frame, and it helps when we are accelerating, speeding, and in curvy space-time.

2) How some of the space-like effect of sideways ”felt” forces moves into the reference-frame’s time-domain at high speed, making the relatively unknown bound (from special relativity!)

$\dfrac{dp}{dt}\leq m\alpha$

With the above definitions, the relativistic momentum can be expressed in termns of coordinate velocity or proper velocity as follows:

$\mathbf{P}=m\mathbf{w}=M\mathbf{v}=m\gamma \mathbf{v}$

where

$\gamma=\dfrac{dt}{d\tau}=\dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}=\sqrt{1+\dfrac{\omega^2}{c^2}}$

is the Lorentz factor. The last equal sign in the previous equation can be easily derived from the relativistic relationship:

$\left(c\dfrac{dt}{d\tau}\right)^2-\left(\dfrac{d\mathbf{r}}{d\tau}\right)^2=c^2$

and the definition of $\gamma$ above.

Thanks to the metric-equation’s assignment of a frame-invariant traveler or proper-time $\tau$ to the displacement between events in context of a single map-frame of comoving yardsticks and synchronized clocks, proper velocity becomes one of three related derivatives in special relativity (coordinate velocity $\mathbf{v}$, proper-velocity $\mathbf{w}$, and Lorentz factor $\gamma$) that describe an object’s rate of travel. For unidirectional motion, in units of lightspeed c (i.e. c=1 if we want to) each of these is also simply related to  a traveling object’s hyperbolic velocity angle or rapidity $\eta$ by the next set of equations:

$\eta=\sinh^{-1}\left( \dfrac{w}{c}\right)=\tanh^{-1}\left(\dfrac{v}{c}\right)=\pm \cosh^{-1}\left(\gamma\right)$

The next table illustrates how the proper-velocity of $w_0 \equiv c$ or “one map-lightyear per traveler-year” is a natural benchmark for the transition from a sub-relativistic coordinate frame to a (fake) auxiliary super-relativistic motion (in imaginary units of $i=\sqrt{-1}$). Note that the velocity angle or pseudorapidity $\eta$ and the proper-velocity $w$ run from 0 to infinity and track the physical coordinate-velocity when $w<. On the other hand when $w>>c$, the (hyperbolic or imaginary) proper-velocity tracks Lorentz factor $\gamma$ while velocity angle $\eta$ is logarithmic and hence increases much more slowly:

LUDICROUS SPEED AND WARP SPEED

Hyperbolic velocities CAN exceed c! They can reach even the ludicrous speed of $\infty$ when the coordinate velocity approaches c! However, you must never forget the fact that the velocity-angle/hyperbolic velocity IS imaginary in value. It is quite clear from the above table. Indeed, being somehow “trekkie” or a Sci-Fi “romantic” person, you could “define” warp-speeds as “imaginary/hyperbolic” velocities, i.e., in terms of proper velocity. In that case, you could get the correspondence

$\mbox{WARP}0.25=\mbox{WARP}1/4=\dfrac{\sqrt{17}}{17}c\approx 0.24c$

$\mbox{WARP}0.5=\mbox{WARP}1/2=\dfrac{\sqrt{5}}{5}c\approx 0.45c$

$\mbox{WARP}1=\dfrac{\sqrt{2}}{2}c\approx 0.71c$

$\mbox{WARP}2=\dfrac{2\sqrt{5}}{5}c\approx 0.89c$

$\mbox{WARP}3=\dfrac{3\sqrt{10}}{10}c\approx 0.95c$

$\mbox{WARP}7=\dfrac{7\sqrt{2}}{10}c\approx 0.99c$

$\mbox{WARP}9=\dfrac{9\sqrt{82}}{82}c\approx 0.994c$

$\mbox{WARP}10=\dfrac{10\sqrt{101}}{101}c\approx 0.995c$

$\mbox{WARP}\infty\equiv c$

In general, we can define the WARP speed as $W=w/c$ and so, the proper velocity can be expressed in terms of the warp speed W in a very simple way $w=Wc$. Thus, the real or coordinate velocity would be connected with warp-speed through the relativistic equation:

$\boxed{v=c\tanh\sinh^{-1}(W)=c\tanh\sinh^{-1}\left(\dfrac{w}{c}\right)}$

Of course, the point is that, unlike the Sci-Fi franchise, the real velocity has never exceeded c, only the hyperbolic velocity and the proper velocity (note that in terms of SR, velocities approaching c imply very boosted frames, so despite we could travel to any point of the Universe in SR only approaching c very closely with respect to the traveler proper time-one human life-, but in terms of the “Earth” (or rest) reference frame millions of years would have passed away!).

When the coordinate-speeds approach c, the respective coordinate velocities deviate from this simple addition rule in that rapidities (hyperbolic velocity angle boosts) add instead of velocities, i.e. $\eta_{12}=\eta_1+\eta_2$. Coordinate velocities add non-linearly. And it is a well-tested consequence of the Special Theory of relativity.  For highly relativistic objects (i.e. those with momentum per unit mass much larger than lightspeed) the result of the coordinate-velocity expression  familiar from most textbooks is rather uninteresting since the coordinate-velocities all peak out at c, i.e., as everybody knows, in special relativity $1c\boxplus 1c=1c$, because applying the relativistic addition of velocities rule, we get

$c\boxplus c=\dfrac{ (c + c)}{(1 + 1)}=c$

And it is a fact from both theory and experiment! It will remain as long as SR remains a valid theory. SR holds yet with an astonishing degree of precision and accuracy. So, you can not deny every data and experiment that confirms SR. That is completely nonsense but there are some people and pseudo-scientists out there building their own theories AGAINST the achievements and explanations that SR provides to every experiment we have done until the current time. I am sorry for all of them. They are totally wrong. Science is not what they say it is. Any theory going beyond SR HAS to explain every experiment and data that SR does explain, and it is not easy to build such a theory or to say, e.g., why we have not observed (apparently) superluminal objects. I will discuss more superluminal in a forthcoming post/log entry, some posts after the special 50th post/log that is coming after this one! Stay tuned!

Coming back to our discussion…Why is all this stuff important? High Energy Physics is the natural domain of SR! And there, SR has not provided ANY wrong result till, in spite that some researches going beyond the Standard Model include modified dispersion relationships that reduce to SR in the low energy regime, we have not seen yet ANY deviation from SR until now.

For unidirectional motion, at low speeds the coordinate velocity $v_{13}$ of object 1 from the point of view of oncoming object 3 might be described as the sum of the velocity $v_{12}$ of object 1 with respect to lab frame 2 plus the velocity $v_{23}$ of the lab frame 2 with respect to object 3, that is:

$v_{13}=v_{12}+v_{23}$

Compare this expression to the previously obtained expression for rapidities! Rapidities always add, coordinate velocities add (linearly) only at low velocities. In conclusion, you must be careful by what you mean by velocity is a boosted system!

By the other hand, for relative proper-velocity, the result is:

$w_{13}=\gamma_{13}v_{13}=\gamma_{12}\gamma_{23}(v_{12}+v_{23})$

This expression shows how the momentum per unit mass as well as the map-distance traveled per unit traveler time of object 1, as seen in the frame of oncoming particle 3, goes as the sum of the coordinate-velocities times the product of the gamma (energy) factors. The proper velocity equation is especially important in high energy physics, because colliders enable one to explore proper-speed and energy ranges much higher than accessible with fixed-target collisions. For instance each of two electrons (traveling with frames 1 and 3) in a head-on collision traveling in the lab frame (2) at

$\gamma_{12}mc^2=45\mbox{GeV}$

or equivalenty $w_{12}=w_{23}=\gamma v\approx 88000$ lightseconds per traveler second  would see the other coming toward them at coordinate velocity $v_{13}\approx c$ and $w_{13}=88000^2(1+1) \approx 1.55\cdot 10^{10}$ lightseconds per traveler second or $\gamma_{13}mc^2\approx 7.9 \mbox{PeV}$. From the target’s view, that is an incredible increase in both energy and momentum per unit of mass.

Other magnitudes and their frame dependence in SR can be read from the following table:

CAUTION: These results don’t mean that the “real” energy is that. Energy is relative and it depends on the frame! The fact that in colliders, seen from the target reference frame, the energy can be greater than the center of mass energy is not an accident. It is a consequence of the formalism of special relativity. A similar observation can be done for velocities. Coordinate velocities, IN THE FRAMEWORK OF SPECIAL RELATIVITY, can never exceed the speed of light. As long as SR holds, there is no particle whose COORDINATE velocity can overcome the speed of light. However, we have seen that PROPER velocities are other monsters. They serve as a tool to handle rotations along the temporal axis, i.e., to handle boosts mixing space and time coordinates. Proper (or hyperbolic) velocities CAN be greater than speed of light. But, it does not contradict the special theory of relativity at all since hyperbolic velocities ARE NOT REAL since they are imaginary quantities and they are not physical. We can only measure momentum and real quantities!  Moreover, remember that, in fact, group or phase velocities we have found before can ALSO be greater than c. So, you must be careful by what do you mean by velocity in SR or in any theory. Furthermore, you must distinguish the notion of particle velocity with those of the relative velocity between two inertial frames, since the particle velocities ( coordinate or proper) always refer to some concrete frame! In summary, be aware of people saying that there are superluminal particles in our colliders or astrophysical processes. It is simply not true. Superluminal objects have observable consequences, and they have failed to be observed ( the last example was the superluminal neutrino affair by the OPERA collaboration, now in agreement with SR).

Remark (I): From the last table we observe that in SR, the rotation angle is imaginary. Therefore, we are forced to use this gadget of hyperbolic velocity in order to avoid “imaginary velocities”.

Remark (II): Hyperbolic velocities would become imaginary velocities if we used the imaginary formalism of SR, the infamous $ict=x_4$.

Remark (III): Hyperbolic velocities are not coordinate velocities, so they are not physical at all. They are just a tool to provide the right answers in terms of rapidities, or the hyperbolic angle, whose units are imaginary radians! Hyperbolic velocities are measured in imaginary units of velocity!

Remark (IV): About the imaginary issues you can have now. The spacetime separation formula $s^2=-c^2t^2+x^2+y^2+z^2$ means that the time t can often be treated mathematically as if it were an imaginary spatial dimension. That is, you can define $ct=iw$ so $-c^2t^2=w^2$, where $i$  is the square root of  -1, and $w$ is a “fourth spatial coordinate”. Of course it is not at all. It is only a trick to treat the problem in a clever way.  By the other hand, a Lorentz boost by a velocity $v$ can likewise be treated as a rotation by an imaginary angle. Consider a normal spatial rotation in which a primed frame is rotated in the $wx$-plane clockwise by an angle $\varphi$ about the origin, relative to the unprimed frame. The relation between the coordinates $(w',x')$ and $(w,x)$ of a point in the two frames is:

$\begin{pmatrix}w'\\ x'\end{pmatrix}=\begin{pmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{pmatrix}\begin{pmatrix}w\\ x\end{pmatrix}$

Now set $ct=iw$ and $\theta=i\varphi$, with $t,\theta$ both real. In other words, take the spatial coordinate $w$ to be imaginary, and the rotation angle $\varphi$ likewise to be imaginary. Then the rotation formula above becomes

$\begin{pmatrix}ct'\\ x'\end{pmatrix}=\begin{pmatrix}\cosh\theta & -\sinh\theta\\ -\sinh\theta & \cosh\theta\end{pmatrix}\begin{pmatrix}ct\\ x\end{pmatrix}$

This agrees with the usual Lorentz transformation formulat if the boost velocity $v$ and boost angle $\theta$ are related by the known formula $\tanh\theta=v/c=\beta$. We realize that if we identify the imaginary angle with the rapidity, we are back to Special Relativity. Indeed, it is only the rotations involving the time axis which can cause confusion because they are so different from our everyday experience. That is, we experience rotations along some direction in our daily experience, so we are familiarized with rotations and their (real) rotation angles. However, rotations along a time axis mixing space and time is a weird creature. It uses imaginary numbers or, if we avoid them, we have to use hyperbolic (pseudo)-rotations.

SUMMARY OF MAIN IDEAS

A) Lorentz factor $\gamma=\dfrac{E}{mc^2}$

$\boxed{\gamma \equiv \frac{dt}{d\tau}= \sqrt{1+\left(\frac{w}{c}\right)^2} = \frac{1}{\sqrt{1-(\frac{v}{c})^2}} = \cosh[\eta] \equiv \frac{e^{\eta} + e^{-\eta}}{2}}$

B) Proper-velocity or momentum per unit mass.

$\boxed{\frac{w}{c}\equiv \frac{1}{c} \frac{dx}{d\tau}=\frac{v}{c} \frac{1}{\sqrt{1-(\frac{v}{c})^2}}=\sinh[\eta]\equiv \frac{e^{\eta} - e^{-\eta}}{2} =\pm\sqrt{\gamma^2 - 1}}$

C) Coordinate velocity $v\leq c$.

$\boxed{\frac{v}{c} \equiv \frac{1}{c}\frac{dx}{dt}=\frac{w}{c}\frac{1}{\sqrt{1 + (\frac{w}{c})^2}} = \tanh[\eta] \equiv \frac{e^{2\eta} - 1} {e^{2\eta} + 1}= \pm \sqrt{1 - \left(\frac{1}{\gamma}\right)^2}}$

D) Hyperbolic velocity angle or rapidity.

$\boxed{\eta =\sinh^{-1}[\frac{w}{c}] = \tanh^{-1}[\frac{v}{c}] = \pm \cosh^{-1}[\gamma]}$

or in terms of logarithms:

$\boxed{\eta = \ln\left[\frac{w}{c} + \sqrt{\left(\frac{w}{c}\right)^2 + 1}\right] = \frac{1}{2} \ln\left[\frac{1+\frac{v}{c}}{1-\frac{v}{c}}\right] = \pm \ln\left[\gamma + \sqrt{\gamma^2 - 1}\right]}$

E) Warp speed (just for fun):

$\boxed{v=c\tanh\sinh^{-1}(W)=c\tanh\sinh^{-1}\left(\dfrac{w}{c}\right)}$

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?

LOG#043. Tachyons and SR (I).

“(…)Suppose that someon studying the distribution of population on the Hindustan Peninsula cockshuredly believes that there are no poeple north of the Himalayas, because nobody ca pass throught the mountain ranges! That would be an absurd conclusion. The inhabitants of Central Asia have been born there; they are not obliged to be born in India and tehn cross the mountain ranges. The same can be said about superluminal particles(…)”  This is a quote by George Sudharhan.

I had the honour to meet George Sudarshan ( his full name is Ennackal Chandy George Sudarshan, http://en.wikipedia.org/wiki/Sudarshan,_E._C._George) some years ago, in Jaca (Huesca), during the Sudarshanfest celebrating his 75th birthday. I also met some really cool people like Susumo Okubo ( yes, the man behind the Gell-Mann-Okubo mass formula for hadrons-see http://en.wikipedia.org/wiki/Gell-Mann-Okubo_mass_formula! ). I really enjoy knowing (japanese) scientists when they are really gentle and generous. Mr. Okubo indeed gave me a copy of his wonderful book about non-associative stuff. By the other hand, Indian scientists are also fascinating because they use to be people very uncommon and some of them, like Ramanujan in Mathematics, own exceptional gifts and talents.

George Sudarshan has made contributions in many branches of Physics. He originally proposed as well the V-A nature of the electroweak interactions that recover the Fermi theory of weak interactions in the low energy limit. In addition to this, he has developed in the field of optical coherence the so-called Sudarhsan-Glauber representation in Quantum Optics, he invented the theory of the Quantum Zeno effect, he has worked in open quantum systems, the relationship spin-statistics in general QFT contexts, and he is one of the defenders of the existence of tachyons, via an interpretation of relativity including those faster than light particles created by Feinberg and studied by E. Recami, Feinberg, M.Pavsic, Gregory Benford (yes, the writter and physicist/astronomer) and other physicists. His works about relativity with tachyons are usually labelled under the name “metarelativity”. I would like to dedicate him this post.

According to wikipedia, a tachyon or tachyonic particle is a hypothetical particle that, a priori, always moves faster than light. The word comes from the Greek word: ταχύς or tachys, meaning “swift, quick, fast, rapid”, and was coined by Gerald Feinbergin a 1967 paper.Feinberg proposed that tachyonic particles could be quanta of a quantum field with negative squared mass. However, it was soon realized that excitations of such imaginary mass fields do not in fact propagate faster than light,but instead represent an instability known as tachyon condensation.  Nevertheless, they are still commonly known as “tachyons”, and have come to play an important role in modern physics, for instance, their role in string theory is still being studied. I will not discuss about advanced topics like tachyon condensation in this post, I will expect to do it in the near future some day.

Superluminality and velocity are subtle concepts in SR. I have to discuss more about (apparent) superluminality in SR, but the goal of this post is somewhat more modest. I am going to introduce you tachyonic particles and some of its curious features.

First point. From the addition theorem of velocities in special relativity:

$V=\dfrac{v_1+v_2}{1+\dfrac{v_1 v_2}{c^2}}$

we can see that, a priori, the ranges of the velocities is not restricted from that concrete formula. the real issue with faster than light particles in special relativity comes from the relativistic expression of energy:

$E=Mc^2=m\gamma c^2=\dfrac{mc^2}{\sqrt{1-\dfrac{v^2}{c^2}}}$

This formula blows up at v=c for any finite value of the rest mass m! Indeed, we can say that particles slower than light, also called “tardyons”, have that energy-mass relation, while for photons ( or luxons), we do know that $E=pc$ and they are massless particles. What happens if we forget our prejudices and we allow for v>c velocities saving the SR formula for mass-energy? Well, it is easy to realize that E becomes an imaginary number! Imaginary numbers are complex numbers without real part squaring a negative value. For instance, the solution to the equation $x^2=-1$ is the imaginary number unit $x=i=\sqrt{-1}$. Don’t forget that complex numbers are more general numbers verifying $z=a+bi$, with magnitude $\vert z \vert ^2=a^2+b^2$.

If we plug v>c in the SR formula for mass-energy, we get a negative number inside the square root. After some easy algebra, for v>c we obtain:

$E=\dfrac{mc^2}{\sqrt{1-\dfrac{v^2}{c^2}}}=\dfrac{mc^2}{i\sqrt{\dfrac{v^2}{c^2}-1}}=\dfrac{\overline{m}c^2}{\sqrt{\dfrac{v^2}{c^2}-1}}$

and where we have defined the imaginary mass quantity $\overline{m}=\dfrac{m}{i}=-im$. We could also have defined the “barred” gamma factor as

$\boxed{\gamma \equiv -i\overline{\gamma}=\dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}}\leftrightarrow \boxed{\overline{\gamma}=\dfrac{1}{\sqrt{\dfrac{v^2}{c^2}-1}}}$

so, for tachyons,

$\boxed{E=\overline{m}\overline{\gamma}c^2=-im\overline{\gamma}c^2=\dfrac{\overline{m}c^2}{\sqrt{\dfrac{v^2}{c^2}-1}}}$

Moreover, with this imaginary gamma/boost factor, we would define metarelativity or Lorentz transformations for tachyons:

$x'=-i\overline{\gamma}(x-vt)$ $y'=y$ $z'=z$ $t'=-i\overline{\gamma}\left(t-\dfrac{v}{c^2}x\right)$

and so on with any other transformation including imaginary boosts. Thus, we have extended the special relativity to the imaginary realm and we call this theory metarelativity since it includes superluminal transformations. For superluminal transformations, we also have the same invariants than those for usual relativity. For instance:

$c^2t^{'2}-x^{'2}=c^2t^2-x^2=c^2\overline{t'}^2-\overline{x'}^2$

The prize is that we have imaginary position and time coordinates ( or imaginary boosts, if you prefer that idea).

A curious property of the relationships $\overline{m}=-im$ and $\overline{\gamma}=i\gamma$ is that

$\overline{m}\overline{\gamma}=m\gamma$

This equation shows that a tachyonic imaginary mass and a imaginary gamma factor are equivalent, when they are multiplied, to the real valued common relativistic expression for the relativistic mass. So, we could handle tachyons with imaginary masses and imaginary gamma factors, at least in principle, in the same operational way we handle normal particles in SR, excepting for the sign inside the square root and the strange inertial properties of those tachyonic particles.

In addition to this fact, the energy for tachyonic particles some has interesting properties:

1st. $E=\overline{m}\overline{\gamma}c^2$ decreases as v increases! That is, tachyons are less energetic and so “more stable” when they have higher velocities. This behaviour is very different from the common inertial properties of normal matter. Unlike ordinary particles, the speed of a tachyon increases as its energy decreases.

2nd. A bradyon, also known as a tardyon or ittyon, is a particle that travels slower than light. The term “bradyon”, from Greek word βραδύς (bradys, “slow”), was coined to contrast with the name of the tachyon. Just as bradyons are forbidden to break the light-speed barrier, so too are tachyons forbidden from slowing down to below c, because infinite energy is required to reach the barrier from either above or below.

3rd. Einstein, Tolman and others noted that special relativity allowing for tachyons/superluminal transmissions would imply that they could be used to send signals backwards in time. This tool or use of tachyon pulses is called the tachyonic antitelephone device.

An electric charged tachyon would loose energy as Cherenkov radiation just as ordinary charged particles do when they exceed the local speed of light in a medium. A charged tachyon traveling in a vacuum therefore undergoes a constant proper time acceleration and, by necessity, its wordline forms a hyperbola in space-time.

However,  if we reduce the tachyon’s energy, then it increases its speed, so that the single hyperbola formed is of two oppositely charged tachyons with opposite momenta (same magnitude, opposite sign) which annihilate each other when they simultaneously reach infinite speed at the same place in space. (At infinite speed the two tachyons have no energy each and finite momentum of opposite direction, so no conservation laws are violated in their mutual annihilation. The time of annihilation is frame dependent.)

Even an electrically neutral tachyon would be expected to loose energy via gravitational Cherenkov radiation, at least in theory, because it has a gravitational mass, and therefore increase in speed as it travels, as described above. However, we have not detected gravitational Cherenkov radiation, as far as I know. If the tachyon interacts with any other particles, it can also radiate Cherenkov energy into those particles. Neutrinos interact with the other particles of the Standard Model and, recently,  Andrew Cohen and Sheldon Glashow used this to argue that the neutrino anomaly seen by the OPERA experiment could not be explained by making neutrinos propagate faster than light. Indeed, we do know that neutrino have a non-zero REAL mass from neutrino oscillation experiments.

Coming back to tachyons, we have seen that SR allows then if you allow for “imaginary energies”. A tachyon has the strange feature that its mass has the value

$\mbox{TACHYON MASS}= \mbox{(SOMETHING)}\times i$

so, while I can weigth somethink like 70 kg, a tachyon clone of me would weight $70i$ kg. You can wonder what the imaginary mass means in terms of inertia with the SR equation above, but, of course, it is a weird result after all. And there are more “problems” and weird results for tachyons. For instance, tachyons, it they do gravitate according to Newton’s gravitation equation:

$F_N=-G\dfrac{M_1 M_2}{d^2}$

then they would experience “antigravitation”/antigravity. You can observe and “deduce” that the gravitational force between two tachyons with masses $M_1=M_2=i$ separated by a distance of 1m. Then, the gravitational pull between those tachyons would be repulsive, since the sign of the gravitational Newton force would be positive instead of negative! Is it not amazing? Yes, you can wonder about the Dark Energy enigma, mysterious stuff out there, but there are quantum problems related with “superluminal” tachyons. I will discuss them in the future, I promise. So, it is not easy at all to associate a tachyonic field/mass origin to the Dark Energy. And of course, this hypothesis of antigravitating tachyons face problems when we think about what an imaginary gravitational force between a tachyon and a tardyon would mean. It shows that the mysteries of tachyons are yet not completely understood, and they are connected with the theory of scalar fields and the phenomenon, previously commented, of tachyon condensation. Let me know if you understand them better!

Morever, the transversal length contraction of a tachyon, and the time dilation of a tachyon in metarelativity are imaginary quantities as well. It is an easy exercise to derive the following relationships:

$L_{\updownarrow}=iL_0\overline{\gamma}^{-1}$

$\tau=i\tau_0 \overline{\gamma}$

A second post about tachyons and metarelativity is coming, but before that, you will have to wait for a while. I have other topics in my current agenda to be published, previously, to more tachyonic posts. I suppose I am not being beamed with tachyons from the future.

Let the tachyons be with you ;)!

LOG#029. Interstellar trips in SR.

My final article dedicated to the memory of Neil Armstrong. The idea is to study quantitatively the relativistic rocket motion with numbers, after all we have deduced the important formulae, and we will explain what is happening in the two frames: S’-frame (in motion), S-frame (in rest on Earth). There are many “variations” of this problem, also called “Langevin’s paradox” or “the Langevin’s interstellar trip” problem by some authors. Here, we will follow the approach suggested in the book Gravitation, by Misner, Wheeler and Thorne, and we will study the interstellar trip (in the frame of special relativity) with the following conditions:

1st. Spacetime is locally minkovskian (i.e., spacetime is flat). The solution we will expose would not be valid in the case of an interstellar trip to a very far away quasar, or a very very long distance (about thousand millions of lightyears) where we should take into account the effect of the expansion of the Universe, i.e., that on large scales, spacetime is curved (in particular, accordingly to the current data, it is pseudoriemannian). Thus, we can use special relativity in order to calculate distances, velocities and accelerations from the purely kinematical sense. After all, it is logical, since General Relativity says that locally, in small enough regions of spacetime, spacetime is described by a minkovskian metric.

2nd. We select a 4 stage accelerated motion with our rocket. We will assume that our rocket is 100% efficient in the sense it uses photons as propellant particles. The four stages are: acceleration from rest to g (acceleration step 1), decceleration to rest with -g until we approach the destination (that would be the one-way trip), acceleration with -g (seen from the S-frame) and decceleration with +g ( with respect to the S-frame) to reach Earth again in rest at the end of the round-way trip. Schematically, we can draw a spacetime sketch of this 4 stage journey:

Please, note the symmetry of the procedure and the different travel steps.

3rd. We set g=9.8m/s², or, as we saw in one of our previous posts, so we use g=1.03lyr/yr² (in units where c=1).

4th. We can not refuel during the travel.

5th. In the case we return to Earth, we proceed to come back after we stop at the destination inmediately. In this case, there is no refuel of the starship or rocket in any point of the trip.

6th. We neglect any external disturbance which can stop us or even destroy us, e.g., micrometeorites, cosmic radiations, comets, and any other body that could alter our route. Indeed, this kind of stuff has to be seriously considered in any realistic travel, but we want to solve an ideal problem consisting in an ideal interstellar trip according to the current knowledge.

The main quantities we have to compute are:

1. In the S’-frame of the rocket $\tau, 2\tau,4\tau$, in years. They are, respectively, the time we are accelerating with +g, the time we are deccelerating with -g, and the total time accelerating supposing we return to earth inmediately from our destination target.

2. Distance in which we are accelerating (x, in lightyears), seen from the S-frame (we will not discuss the problem in the S’-frame, since it involves lenght contraction and it is more subtle in the calculations). According to our previous studies, we have

$\boxed{x=\dfrac{c^2}{g}\left(\cosh \left[\dfrac{g\tau}{c}\right]-1\right)}$

3. Maximum depth in space D=2x. This allows us to select our target or destination to set the remaining parameters of the trip. Due to the symmetry of our problem, we have

$\boxed{D=2x=\dfrac{2c^2}{g}\left(\cosh \left[\dfrac{g\tau}{c}\right]-1\right)}$

4. Total length travelled by the spaceship/rocket (in the S’frame) in the roundtrip ( to the destination and back).

$\boxed{L=4x=\dfrac{4c^2}{g}\left(\cosh \left[\dfrac{g\tau}{c}\right]-1\right)}$

5. Maximum speed to an specific destination, after accelerating in a given proper time. It reads:

$\boxed{V=c\tanh \left(\dfrac{g\tau}{c}\right)}$

Indeed, the relativistic gamma factor is $\gamma =\cosh \dfrac{g\tau}{c}$

6. S-frame duration t(years) of the stage 1 (acceleration phase 1). It is

$\boxed{t=\dfrac{c}{g}\sinh \left(\dfrac{g\tau}{c}\right)}$

7. One way duration of the trip (according to the S-frame):

$\boxed{t'=2t=\dfrac{2c}{g}\sinh \left(\dfrac{g\tau}{c}\right)}$

8. Total duration of the trip in the S-frame ( round way trip):

$\boxed{t''=4t=\dfrac{4c}{g}\sinh \left(\dfrac{g\tau}{c}\right)}$

9. Mass ratio of the final mass with the initial mass of the spaceship after 1 stage, 2 stages ( one way trip) and the 4 stages total round way trip:

$\boxed{R=\dfrac{M_f}{M_i}=\exp \left(-\dfrac{g\tau}{c}\right)}$

$\boxed{R=\dfrac{M_f}{M_i}=\exp \left(-\dfrac{2g\tau}{c}\right)}$

$\boxed{R=\dfrac{M_f}{M_i}=\exp \left(-\dfrac{4g\tau}{c}\right)}$

10. Fuel mass vs. payload mass ratio (FM/PM) after 1 stage, 2 stages (until the destination) and the total trip:

$\boxed{R=\dfrac{M}{m}=\exp \left(\dfrac{g\tau}{c}\right)-1}$

$\boxed{R=\dfrac{M}{m}=\exp \left(\dfrac{2g\tau}{c}\right)-1}$

$\boxed{R=\dfrac{M}{m}=\exp \left(\dfrac{4g\tau}{c}\right)-1}$

We can obtain the following data using these expressions varying the proper time:

Here in the last two entries, data provided to be out of the limits of our calculator ( I used Libre Office to compute them). Now, we can make some final observations:

1st. We can travel virtually everywhere in the observable universe in our timelife using a photon rocket, a priori. However, it shows that the mass-ratio turns it to be theoretically impossible.

2nd. Remember that in the case we select very long distances, these calculations are not valid since we should use General Relativity to take into account the expansion of spacetime.

3rd. Compare $\tau$ with t, $2\tau$ with $2t$, $4\tau$ with $4t$. For instance, for $4\tau=40$ then we get 57700 years, and v=0.99999999773763c, FM/PM=30000 (multiplying por m-kilograms gives the fuel mass).

4th. In SR, the technological problems are associated to the way photon rockets have, the unfavorable mass ratios (of final mass with initial mass) and fuel mass/payload rations neccesary in the voyage (supposing of course, current physics and that we can not refuel during the trip).

5th. You can choose some possible distance destinations (D=2x) and work out the different parameters of the travel with the above equations.

In this way, for instance, for some celebrated known space marks, arriving at complete stop ( you can make other assumptions and see how the answer changes varying g or varying the arrival velocity too), we can easily get:

Example 1: 4.3 ly Nearest star (Proxima Centauri) in $t=2\tau=3.6$ yrs (S’-frame).

Example 2: 27 ly Vega (Contact movie and book) star in $t=2\tau=6.6$ yrs (S’-frame).

Example 3: 30000ly, our galactic center, in $t=2\tau=20$ yrs (S’-frame).

Example 4: 2000000ly , the Andromeda galaxy, in $t=2\tau=28$ yrs(S’-frame).

Example 5: Generally, you can travel n ly anywhere (neglecting curved spacetime) in $t=2\tau=1.94 \cosh^{-1}(n/1.94+1)$ years (S’-frame).

LOG#028. Rockets and relativity.

The second post in this special thread of 3 devoted to Neil Armstrong memory has to do with rocketry.

Firstly, for completion, we are going to study the motion of a rocket in “vacuum” according to classical physics. Then, we will deduce the relatistic rocket equation and its main properties.

CLASSICAL NON-RELATIVISTIC ROCKETS

The fundamental law of Dynamics, following Sir Isaac Newton, reads:

$\mathbf{F}=\dfrac{d\mathbf{p}}{dt}$

Suppose a rocket with initial mass $M_i$ and initial velocity $u_i=0$. It ejects mass of propellant “gas” with “gas speed” (particles of gas have a relative velocity or speed with respect to the rest observer when the rocket move at speed $\mathbf{v}$) equals to $u_0$ (note that the relative speed will be $u_{rel} and the propellant mass is$latex m_0\$. Generally, this speed is also called “exhaust velocity” by engineers. The motion of a variable mass or rocket is given by the so-called Metcherski’s equation:

$\boxed{M\dfrac{d\mathbf{v}}{dt}=-\mathbf{u_0}\dfrac{dM}{dt}+\mathbf{F}}$

where $-\mathbf{u_0}=\mathbf{v_{gas}}-\mathbf{v}$. The Metcherski’s equation can be derived as follows: the rocket changes its mass and velocity so $M'=M+dM$ and $V'=V+dV$, so the change in momentum is equal to $M'V'=(M+dM)(V+dV)$, plus an additional term $v_{gas}dv_{gas}$ and $-mV$. Therefore, the total change in momentum:

$dP=Fdt=(M+dM)(V+dV)+v_{gas}dv_{gas}-mV$

Neglecting second order differentials, and setting the conservation of mass (we are in the non-relativistic case)

$dM+dm_{gas}=0$

we recover

$MdV=v_{rel}dM+Fdt$

that represents (with the care of sign in relative speed) the Metcherski equation we have written above.

Generally speaking, the “force” due to the change in “mass” is called thrust.  With no external force, from the remaining equation of the thrust and velocity, and it can be easily integrated

$u_f=-u_0\int_{M_i}^{M_f}dM$

and thus we get the Tsiolkowski’s rocket equation:

$\boxed{\mathbf{u_f}=\mathbf{u_0}\ln \dfrac{M_i}{M_f}}$

Engineers use to speak about the so-called mass ratio $R=\dfrac{M_f}{M_i}$, although sometimes the reciprocal definition is also used for such a ratio so be aware, and in terms of this the Tsiokolski’s equation reads:

$\boxed{\mathbf{u_f}=\mathbf{u_0}\ln \dfrac{1}{R}}$

We can invert this equation as well, in order to get

$\boxed{R=\dfrac{M_f}{M_i}=\exp\left(-\dfrac{u_f}{u_0}\right)}$

Example: Calculate the fraction of mass of a one-stage rocket to reach the Earth’s orbit. Typical values for $u_f=8km/s$ and $u_0=4km/s$ show that the mass ratio is equal to $R=0.14$. Then, only the $14\%$ of the initial mass reaches the orbit, and the remaining mass is fuel.

Multistate rockets offer a good example of how engineer minds work. They have discovered that a multistage rocket is more effective than the one-stage rocket in terms of maximum attainable speed and mass ratios. The final n-stage lauch system for rocketry states that the final velocity is the sum of the different gains in the velocity after the n-th stage, so we can obtain

$\displaystyle{u_f=\sum_{i=1}^{n}u_i^f=u_1^f+\cdots+u_n^f}$

After the n-th step, the change in velocity reads

$u_i^f=c_i\ln \dfrac{1}{R_i}$

where the i-th mass ratios are definen recursively as the final mass in the n-th step and the initial mass in that step, so we have

$\displaystyle{u_f=\sum_i c_i\ln \dfrac{1}{R_i}}$

and we define the total mass ratio:

$\displaystyle{R_T=\prod_i R_i}$

If the average effective rocket exhaust velocity is the same in every step/stage, e.g. $c_i=c$, we get

$\displaystyle{u_f=c\ln \left( \prod_{i=1}^{n} R_i^{-1}\right)}$

or

$\displaystyle{u_f=c \ln \left[ \left(\dfrac{M_0}{M_f}\right)_1\left(\dfrac{M_0}{M_f}\right)_2\cdots \left(\dfrac{M_0}{M_f}\right)_n\right]=c\ln \left[\left(\dfrac{M_0}{M_f}\right)_T\right]}$

The influence of the number of steps, for a given exhaust velocity, in the final attainable velocity can be observed in the next plots:

RELATIVISTIC ROCKETS

We proceed now to the relativistic generalization of the previous rocketry. An observer in the laboratory frame observes that total momentum is conserved, of course, and so:

$M'du'=-u'_0dM'$

where $du'$ is the velocity increase in the rocket with a rest mass M’ in the instantaneous reference frame of the moving rocket S’. It is NOT equal to its velocity increase measured in the unprimed reference frame, du. Due to the addition theorem of velocities in SR, we have

$u+du=\dfrac{u+du'}{1+\dfrac{udu'}{c^2}}$

where u is the instantenous velocity of the rocket with respect to the laboratory frame S. We can perform a Taylor expansion of the denominator in the last equation, in order to obtain:

$u+du=(u+du')\left(1-\dfrac{udu'}{c^2}\right)$

and then

$u+du=u+du'\left(1-\dfrac{u^2}{c^2}\right)$

and finally, we get

$du'=\dfrac{du}{1-\dfrac{u^2}{c^2}}=\gamma^2_u du$

Plugging this equation into the above equation for mass (momentum), and integrating

$\displaystyle{\int_{0}^{u_f}\dfrac{du}{1-\dfrac{u^2}{c^2}}=-u'_0\int_{M'_0}^{M'_f}dM}$

we deduce that the relativistic version of the Tsiolkovski’s rocket equation, the so-called relativistic rocket equation, can be written as:

$\dfrac{c}{2}\ln \dfrac{1+\dfrac{u_f}{c}}{1-\dfrac{u_f}{c}}=u'_0\ln\dfrac{M'_i}{M'_f}$

We can suppress the primes if we remember that every data is in the S’-frame (instantaneously), and rewrite the whole equation in the more familiar way:

$\boxed{u_f=c\dfrac{1-\left(\dfrac{M_f}{M_0}\right)^{\frac{2u_0}{c}}}{1+\left(\dfrac{M_f}{M_0}\right)^{\frac{2u_0}{c}}}=c\dfrac{1-R^{\frac{2u_0}{c}}}{1+R^{\frac{2u_0}{c}}}}$

where the mass ratio is defined as before $R=\dfrac{M_f}{M_i}$. Now, comparing the above equation with the rapidity/maximum velocity in the uniformly accelerated motion:

$u_f=c\tanh \left(\dfrac{g\tau}{c}\right)$

we get that relativistic rocket equation can be also written in the next manner:

$u_f=c\tanh \left[ -\dfrac{u_0}{c}\ln \left(\dfrac{1}{R}\right)\right]$

or equivalently

$u_f=c\tanh \left[ \dfrac{u_0}{c}\ln R\right]$

since we have in this case

$\dfrac{g\tau}{c}=-\dfrac{u_0}{c}\ln \left(\dfrac{1}{R}\right)=\dfrac{u_0}{c}\ln R$

and thus

$R^{\frac{u_0}{c}}=\left(\dfrac{M_f}{M_i}\right)^{\frac{u_0}{c}}=\exp \left(-\dfrac{g\tau}{c}\right)$

If the propellant particles move at speed of light, e.g., they are “photons” or ultra-relativistic particles that move close to the speed of light we have the celebrated “photon rocket”. In that case, setting $u_0=c$, we would obtain that:

$\boxed{u_f=c\dfrac{1-\left(\dfrac{M_f}{M_0}\right)^{2}}{1+\left(\dfrac{M_f}{M_0}\right)^{2}}=c\dfrac{1-R^{2}}{1+R^{2}}=c\tanh \ln R}$

and where for the photon rocket (or the ultra-relativistic rocket) we have as well

$\dfrac{g\tau}{c}=-\ln \left(\dfrac{1}{R}\right)=\ln R$

Final remark: Instead of the mass ratio, sometimes is more useful to study the ratio fuel mass/payload. In that case, we set $M_f=m$ and $M_0=m+M$, where M is the fuel mass and m is the payload. So, we would write

$R=\dfrac{m}{m+M}$

so then the ratio fuel mass/payload will be

$\dfrac{M}{m}=R^{-1}-1=\exp \left(\dfrac{g\tau}{c}\right)-1$

We are ready to study the interstellar trip with our current knowledge of Special Relativity and Rocketry. We will study the problem in the next and final post of this fascinating thread. Stay tuned!

LOG#027. Accelerated motion in SR.

Hi, everyone! This is the first article in a thread of 3 discussing accelerations in the background of special relativity (SR). They are dedicated to Neil Armstrong, first man on the Moon! Indeed,  accelerated motion in relativity has some interesting and sometimes counterintuitive results, in particular those concerning the interstellar journeys whenever their velocities are close to the speed of light(i.e. they “are approaching” c).

Special relativity is a theory considering the equivalence of every  inertial frame ( reference frames moving with constant relative velocity are said to be inertial frames) , as it should be clear from now, after my relativistic posts! So, in principle, there is nothing said about relativity of accelerations, since accelerations are not relative in special relativity ( they are not relative even in newtonian physics/galilean relativity). However, this fact does not mean that we can not study accelerated motion in SR. The own kinematical framework of SR allows us to solve that problem. Therefore, we are going to study uniform (a.k.a. constant) accelerating particles in SR in this post!

First question: What does “constant acceleration” mean in SR?   A constant acceleration in the S-frame would give to any particle/object a superluminal speed after a finite time in non-relativistic physics! So, of course, it can not be the case in SR. And it is not, since we studied how accelerations transform according to SR! They transform in a non trivial way! Moreover, a force growing beyond the limits would be required for a “massive” particle ( rest mass $m\neq 0$). Suppose this massive particle (e.g. a rocket, an astronaut, a vehicle,…) is at rest in the initial time $t=t'=0$, and it accelerates in the x-direction (to be simple with the analysis and the equations!). In addition, suppose there is an observer left behind on Earth(S-frame), so Earth is at rest with respect to the moving particle (S’-frame). The main answer of SR to our first question is that we can only have a constant acceleration in the so-called instantaneous rest frame of the particle.  We will call that acceleration “proper acceleration”, and we will denote it by the letter $\alpha$. In fact, in many practical problems, specially those studying rocket-ships, the acceleration is generally given the same magnitude as the gravitational acceleration on Earth ($\alpha=g\approx 9.8ms^{-2}\approx 10 ms^{-2}$).

Second question: What are the observed acceleration in the different frames? If the instantaneous rest frame S’ is an inertial reference frame in some tiny time $dt'$, at the initial moment, it has the same velocity as the particle (rocket,…) in the S-frame, but it is not accelerated, so the velocity in the S’-frame vanishes at that time:

$\mathbf{u}'=(0,0,0)$

Since the acceleration of the particle is, in the S’-frame, the proper acceleration, we get:

$\mathbf{a}'=(a'_x,0,0)=(\alpha,0,0)=(g,0,0)=\mbox{constant}$

Using the transformation rules for accelerations in SR we have studied, we get that the instantaneous acceleration in the S-frame is given by

$\mathbf{a}=(a_x,0,0)=\left(\dfrac{g}{\gamma^3},0,0\right)$

Since the relative velocity between S and S’ is always the same to the moving particle velocity in the S-frame, the following equation holds

$v=u_x$

We do know that

$a_x=\dfrac{du_x}{dt}=\left(1-\dfrac{u_x^2}{c^2}\right)^{3/2}g$

Due to time dilation

$dt'=dt/\gamma$

so in the S-frame, the particle moves with the velocity

$du_x=\left(1-\dfrac{u_x^2}{c^2}\right)^{3/2}g dt$

We can now integrate this equation

$\int_0^{u_x}\dfrac{1}{(c^2-u_x^2)^{3/2}}du_x=\dfrac{g}{c^3}\int_0^t dt$

The final result is:

$\boxed{u_x=\dfrac{g t}{\sqrt{1+\left(\dfrac{g t}{c}\right)^2}}}$

We can check some limit cases from this relativistic result for uniformly accelerated motion in SR.

1st. Short time limit: $gt<< c\longrightarrow u_x\approx gt=\alpha t$. This is the celebrated nonrelativistic result, with initial speed equal to zero (we required that hypothesis in our discussion above).

2nd. Long time limit: $t\rightarrow \infty$. In this case, the number one inside the root is very tiny compared with the term depending on acceleration, so it can be neglected to get $u_x\approx \dfrac{gt}{gt/c}=c$. So, we see that you can not get a velocity higher than the speed of light with the SR framework at constant acceleration!

Furthermore, we can use the definition of relativistic velocity in order to integrate the associated differential equation, and to obtain the travelled distance as a function of $t$, i.e. $x(t)$, as follows

$u_x=\dfrac{dx}{dt}=\dfrac{gt}{\sqrt{1+\left(\dfrac{g t}{c}\right)^2}}$

$\int_0^x dx=\int_0^t\dfrac{gt dt}{\sqrt{1+\left(\dfrac{g t}{c}\right)^2}}=\int_0^t\dfrac{ctdt}{\sqrt{\dfrac{c^2}{g^2}+t^2}}$

We can perform the integral with the aid of the following known result ( see,e.g., a mathematical table or use a symbolic calculator or calculate the integral by yourself):

$\int \dfrac{ctdt}{\sqrt{\left(\dfrac{c}{g}\right)^2+t^2}}=c\sqrt{\left(\dfrac{c}{g}\right)^2+t^2}+\mbox{constant}=c\sqrt{\left(\dfrac{c}{g}\right)^2+t^2}+C$

From this result, and the previous equation, we get the so-called relativistic path-time law for uniformly accelerated motion in SR:

$x=c\sqrt{\left(\dfrac{c}{g}\right)^2+t^2}-\dfrac{c^2}{g}$

or equivalently

$\boxed{x=x(t)=\dfrac{c^2}{g}\left(\sqrt{1+\left(\dfrac{gt}{c}\right)^2}-1\right)}$

For consistency, we observe that in the limit of short times, the terms in the big brackets approach $1+\frac{1}{2}\left(\frac{gt}{c}\right)^2$, in order to get $x\approx \frac{1}{2}gt^2$, so we obtain the nonrelativistic path-time relationship $x=\frac{1}{2}gt^2$ with $g=a_x$. In the limit of long times, the terms inside the brackets can be approximated to $gt/c$, and then, the final result becomes $x\approx ct$. Note that the velocity is not equal to the speed of light, this result is a good approximation whenever the time is “big enough”, i.e., it only works for “long times” asymptotically!

And finally, we can write out the transformations of accelaration between the two frames in a explicit way:

$a_x=\left[1-\dfrac{\left(\dfrac{gt}{c}\right)^2}{1+\left(\dfrac{gt}{c}\right)^2}\right]^{3/2}g$

that is

$\boxed{a_x=\dfrac{1}{\left[1+\left(\dfrac{gt}{c}\right)^2\right]^{3/2}}g}$

Check 1: For short times, $a_x\approx g=\mbox{constant}$, i.e., the non-relativistic result, as we expected!

Check 2: For long times, $a_x\approx \dfrac{c^3} {g^2t^3}\rightarrow 0$. As we could expect, the velocity increases in such a way that “saturates” its own increasing rate and the speed of light is not surpassed. The fact that the speed of light can not be surpassed or exceeded is the unifying “theme” through special relativity, and it rest in the “noncompact” nature of the Lorentz group due to the $\gamma$ factor, since it would become infinity at v=c for massive particles.

It is inevitable: as time passes, a relativistic treatment is indispensable, as the next figures show

The next table is also remarkable (it can be easily built with the formulae we have seen till now with any available software):

Let us review the 3 main formulae until this moment

$\boxed{a_x=\dfrac{1}{\left[1+\left(\dfrac{gt}{c}\right)^2\right]^{3/2}}g}$ $\boxed{u_x=\dfrac{\alpha t}{\sqrt{1+\left(\dfrac{g t}{c}\right)^2}}}$ $\boxed{x=x(t)=\dfrac{c^2}{g}\left(\sqrt{1+\left(\dfrac{gt}{c}\right)^2}-1\right)}$

We have calculated these results in the S-frame, it is also important and interesting to calculate the same stuff in the S’-frame of the moving particle. The proper time $\tau=t'$ is defined as:

$\boxed{d\tau=dt\sqrt{1-\left(\dfrac{u_x}{c}\right)^2}}$

Therefore,

$d\tau=dt\left[1-\dfrac{\left(\dfrac{gt}{c}\right)^2}{1+\left(\dfrac{gt}{c}\right)^2}\right]^{1/2}$

We can perform the integral as before

$\displaystyle{\int_0^\tau d\tau=\int_0^t\dfrac{dt}{\sqrt{1+\left(\dfrac{gt}{c}\right)^2}}}$

and thus

$\tau=\dfrac{c}{g}\int_0^\tau\dfrac{dt}{\sqrt{\left(\dfrac{c}{g}\right)^2+t^2}}=\dfrac{c}{g}\ln \left(\dfrac{gt}{c}+\sqrt{\left(\dfrac{gt}{c}\right)^2+1}\right)\bigg|_0^t$

Finally, the proper time(time measured in the S’-frame) as a function of the elapsed time on Earth (S-frame) and the acceleration is given by the very important formula:

$\boxed{\tau=\dfrac{c}{g}\ln \left(\dfrac{gt}{c}+\sqrt{1+\left(\dfrac{gt}{c}\right)^2}\right)}$

And now, let us set $z=gt/c$, therefore we can write the above equation in the following way:

$\dfrac{g\tau}{c}=\ln \left( z+\sqrt{1+z^2}\right)$

Remember now, from our previous math survey, that $\sinh^{-1}z=\ln \left( z+\sqrt{1+z^2}\right)$, so we can invert the equation in order to obtain t as function of the proper time since:

$\boxed{\tau=\dfrac{c}{g}\sinh^{-1}\left(\dfrac{gt}{c}\right)}$

$\boxed{t=\dfrac{c}{g}\sinh \left(\dfrac{g\tau}{c}\right)}$

Inserting this last equation in the relativistic equation path-time for the uniformly accelerated body in SR, we obtain:

$x=x(\tau)=\dfrac{c^2}{g}\left(\sqrt{1+\sinh^2\left(\dfrac{g\tau}{c}\right)}-1\right)$

i.e.,

$\boxed{x=x(\tau)=\dfrac{c^2}{g}\left[\cosh \left(\dfrac{g\tau}{c}\right)-1\right]}$

Similarly, we can calculate the velocity-proper time law. Previous equations yield

$u_x=\dfrac{c\sinh\left(\dfrac{g\tau}{c}\right)}{\sqrt{1+\sinh^2\left(\dfrac{g\tau}{c}\right)}}=\dfrac{c\sinh \left(\dfrac{g\tau}{c}\right)}{\cosh \left(\dfrac{g\tau}{c}\right)}$

and thus the velocity-proper time law becomes

$\boxed{u_x=c\tanh \left(\dfrac{g\tau}{c}\right)}$

Remark: this last result is compatible with a rapidity factor $\varphi= \left(\dfrac{g\tau}{c}\right)$.

Remark(II): $a_x=\dfrac{du_x}{dt}=\left(1-\dfrac{u_x^2}{c^2}\right)^{3/2}g=\left(1-\tanh^2\left(\dfrac{g\tau}{c}\right)\right)^{3/2}g=\dfrac{1}{\cosh^3\left(\dfrac{g\tau}{c}\right)}g$. From this, we can read the reason why we said before that constant acceleration is “meaningless” unless we mean or fix certain proper time in the S’-frame since whenever we select a proper time, and this last relationship gives us the “constant” acceleration observed from the S-frame after the transformation. Of course, from the S-frame, as this function shows, acceleration is not “constant”, it is only “instantaneously” constant. We have to take care in relativity with the meaning of the words. Mathematics is easy and clear and generally speaking it is more precise than “words”, common language is generally fuzzy unless we can explain what we are meaning!

As the final part of this log entry, let us summarize the time-proper time, velocity-proper time, acceleration-proper time-proper acceleration and distance- proper time laws for the S’-frame:

$\boxed{t=\dfrac{c}{g}\sinh \left(\dfrac{g\tau}{c}\right)}$ $\boxed{u_x=c\tanh \left(\dfrac{g\tau}{c}\right)}$ $\boxed{a_x=\dfrac{1}{\cosh^3\left(\dfrac{g\tau}{c}\right)}g}$ $\boxed{x=x(\tau)=\dfrac{c^2}{g}\left[\cosh \left(\dfrac{g\tau}{c}\right)-1\right]}$

My last paragraph in this post is related to express the acceleration $g\approx 10ms^{-2}$ in a system of units where space is measured in lightyears (we take c=300000km/s) and time in years (we take 1yr=365 days). It will be useful in the next 2 posts:

$g=10\dfrac{m}{s^2}\dfrac{1ly}{9.46\cdot 10^{15}m}\dfrac{9.95\cdot 10^{14}s^2}{1yr^2}=1.05\dfrac{lyr}{yr^2}\approx 1\dfrac{lyr}{yr^2}$

Another election you can choose is

$g=9.8\dfrac{m}{s^2}=1.03\dfrac{lyr}{yr^2}\approx 1\dfrac{lyr}{yr^2}$

so there is no a big difference between these two cases with terrestrial-like gravity/acceleration.