Interstellar Travels On Board of Photon Rockets

By using Kinnersley's solution of general relativity, we model relativistic deep space flights and local celestial sphere on board of rockets that are thrusted by anisotropic absorption or emission of radiation. Relativistic kinematics of the photon rocket are obtained through energy-momentum conservation of the pointlike rocket and applied to light sails (beam-powered propulsion or absorption rockets) and blackbody rockets (i.e., emission rockets thrusted by anisotropic radiative cooling). A relativistic description of high-velocity trips toward Proxima Centauri on board of a light sail and an emission rocket in terms of acceleration, mass variation and time dilation shows that the true problem of interstellar travel is not the amount of propellant, nor the duration of the trip but rather its tremendous energy cost: while an energy of $10^{14}\rm J$ suffices for a flyby with an ultralight laser sail, about $15$ times the current world energy production would be required for sending a $100\rm T$ mission. We then study the propagation of light toward observers on board of radiation rockets with the null geodesics of Kinnersley spacetime in the Hamiltonian formulation. We apply this to the deformation of the local celestial sphere of the accelerated traveller under relativistic aberration and Doppler effect. A model of the traveller's local celestial sphere is built to produce striking illustrations of the change of the front and rear panoramas on-board of radiation spaceships heading toward star Alnilam. We also show how our results could interestingly be further applied to extremely luminous events like the large amount of gravitational waves emitted by binary black hole mergers.


I. INTRODUCTION
I would like to start this article by applying to Physics Albert Camus words from his essay "The Myth of Sisyphus".There is but one truly serious physical problem and that is interstellar travel.All the restwhether or not spacetime has four dimensions, whether gravity is emergent or can eventually be quantizedcomes afterwards.It is not that such questions are not important nor fascinating -and actually they could quite likely be related to the above-mentioned crucial problem -but interstellar travel is the most appealing physical question for a species of explorers such as ours.Interstellar travel, although not theoretically impossible, is widely considered as practically unreachable.This topic has also been often badly hijacked by science-fiction and pseudo-scientific discussion when not polluted by questionable works.
A simplistic view of the problem of interstellar travel is as follows.On one hand, the distance to the closest star system -Alpha Centauri -is roughly four orders of magnitude larger than the ∼ 4.5 billion kilometers toward planet Neptune.On the other hand, the highest velocity a man-made object has ever reached so far [27] is only of the order 10km/s.Consequently, it would roughly take more than one hundred of millennia to get there with the same technology.To cross interstellar distances that are of order of light-years (one light-year being ∼ 9.5 × 10 12 km) within human time scales, one must reach relativistic velocities, i.e. comparable to the speed of light.However, from the famous Tsiolkovsky rocket equation, (with ∆v is the variation of rocket velocity during ejection of gaz with exhaust velocity v e , m i,f being the initial and final total mass), any space vehicle accelerated by ejecting some mass that eventually reaches a final velocity ∆v of 10% of the speed of light c with an exhaust velocity v e of 1km/s, typical of a chemical rocket engine, would require an initial mass of propellant m i about 10 4000 larger than the payload m f .To reduce the initial mass m i to 100 times the mass of the payload m f with the same final velocity ∆v ∼ 0.1 × c requires exhausting mass at a speed of ∼ 6 × 10 3 km/s, about four orders of magnitude higher than with current conventional rocketry.With this quick reasoning, where Eq.( 1) is based on newtonian dynamics, one would conclude that interstellar travel would require either ten thousand times longer trips or ten thousand times faster propulsion.There have been many suggestions (sometimes based on questionable grounds) and engineering preliminary studies for relativistic spaceflight and we refer the reader to [1] and references therein for an overview.Recently, there has been some renewed interest on relativistic spaceflight using photonic propulsion, along with the Breakthrough Starshot project [2] aiming at sending nanocrafts toward Proxima Centauri by shotting high-power laser pulses on a light sail to which the probe will be attached.
According to us, such a quick analysis as above somehow hides the most important problem -the energy cost -really preventing interstellar travel to become a practical reality.Both issues of the duration of the trip and the propellant can be fixed, at least in principle.First, time dilation in an accelerated relativistic motion reduces the duration of any interstellar trip [3], though this requires delivering an enormous total amount of energy to maintain some acceleration all along the trip.Second, space propulsion can be achieved without using any (massive) propellant (and without any violation of Newton's third law): light and gravitational waves are carrying momenta and their emission exert some reaction recoil of the source.General relativity allows modelling the accelerated motion of a particle emitting or absorbing radiation anisotropically through a special class of exact solutions called the photon rocket spacetimes.
In 1969, Kinnersley published in [4] a generalization of Vaidya's metric, which is itself a generalization of the Schwarzschild exterior metric.Vaidya's solution [5] represents the geometry of spacetime around a pointlike mass in inertial motion and that is either absorbing or emitting radiation, through a null dust solution of Einstein's equation: (with R µν the Ricci tensor ; Φ a scalar field describing the radiation flux and k µ a null vector field).Kinnersley's solution [4] describes spacetime around such a radiating mass undergoing arbitrary acceleration due to anisotropic emission and contains four arbitrary functions of time: the mass of the point particle m and the three independent components of the four-acceleration a µ of the particle's worldline.It further appeared that Kinnersley's solution was only a special case of the most generic class of exact solutions of general relativity, the Robinson-Trautman spacetimes (cf.[6] for a review).The term "photon rockets" for such solutions was later coined by Bonnor in [7], although it would be preferable to use the term radiation rocket to avoid any confusion with quantum physics vocabulary [28].The literature on radiation rockets focused mainly on exploring this class of exact solutions or extending Kinnersley's solution ( [6,8,9] and references therein), notably through solutions for uniformly accelerated charged mass [10], gravitational wave-powered rockets [11], (anti-)de Sitter background [9] and photon rockets in arbitrary dimensions [12].Another interesting point around Kinnersley's solution is the fact that there is no gravitational radiation emission (at least at the linear level) associated to the acceleration of the point mass by electromagnetic radiation, as explained in [13].
In [12], the first explicit solutions of photon rockets were given as well as a detailed presentation of several useful background Minkowski coordinates for describing the general motion of the particle rocket.The solutions given in [12] for straight flight includes the case of hyperbolic motion ("constant acceleration") and another analytical solution, both in the case of an emitting rocket, for which the rocket mass decreases.The case of an absorbing rocket -a light sail -is not considered while, as we show here, can be described by the same tool.Geodesic motion of matter test particles is also briefly discussed in [12] to confirm the absence of gravitational aberration investigated in [17].
In the present paper, we apply Kinnersley's solution to the modelling of relativistic motion propelled either by anisotropic emission or absorption of radiation.This encompasses many photonic propulsion proposals like blackbody rockets [14] (in which a nuclear source is used to heat some material to high temperature and its blackbody radiation is then appropriately collimated to produce thrust) or light sails.Spacetime geometry in traveller's frame is obtained through the resolution of energy-momentum conservation equations, which results in decoupled rocket equations for the acceleration and the mass functions while standard approach using Einstein equations deals with a single constraint mixing together the radiation source characteristics and the kinematical variables.This approach also allows working directly with usual quantities such as emissivity, absorption coefficient and specific intensity of the radiation beam acting on the particle to derive the corresponding relativistic kinematics.We provide simple analytical solutions for the case of constant radiative power.We also rederive the relativistic rocket equation once obtained in [15] by a more simplistic analysis with basic special relativity.We then apply the radiation rocket kinematics to three detailed toy models of interstellar travel: (i) the starshot project: an interstellar fly-by of a gram scale probe attached to a light sail that is pushed away by ground-based laser ; (ii,iii) single and return trips with an emission radiative rocket of mass scale 100T.We show how the unreachable energy cost of the latter strongly speaks in favour of the former.Then, from the associated spacetime geometry around the traveller, we investigate light geodesics through hamiltonian formulation.Incoming and outgoing radial trajectories of photons are obtained and used to compute frequency shifts in telecommunications between the traveller, his home and his destination.We then investigate the deviation of the angle of incidence of incoming photons -as perceived by the traveller as it moves.We show how local celestial sphere deformation evolves with time in a different way for either an emission or an absorption rocket, and in a different way than in special relativity, establishing how this effect constitues a crucial question for interstellar navigation which cannot be captured by special relativity alone.
The layout of this paper is as follows.In section 2, we set the basics of radiation rockets in general relativity before we introduce our procedure to establish explicit solutions given incoming or outgoing radiation flux characteristics for an absorption/emission rocket (including light sails) respectively.Relativistic photon rocket equations, analytical solutions as well as numerical solutions for more realistic cases are derived.
We then apply these solutions to three toy models of interstellar travel: fly-by with a light sail, single and return interstellar trip with an emission rocket.Section 3 is devoted to the study of geodesics in Kinnersley spacetime using hamiltonian formulation.Relativistic aberration and Doppler effect for the accelerated observer are obtained and compared to the case of an observer moving at constant speed as described by special relativity.Visualisations of interstellar panoramas for accelerated travellers resulting from relativistic aberration and Doppler effect are presented.We finally conclude in section 4 by emphasising key implications of our results for the problem of interstellar travel as well as introducing another possible application of the present results to the astrophysical problem of gravitational wave recoil by binary black hole mergers.

A. Deriving the Radiative Rocket Equations
In what follows, we build explicit solutions of the photon rocket spacetime.We first review some basics of so-called photon rockets in general relativity and refer the reader to [4,7,9,12,13] for alternative nice presentations.Our procedure to determine spacetime geometry around radiation rockets requires two frames and associated coordinates.The first observer O is located far from the radiation rocket so that the gravitational attraction of the (variable) rocket mass can be neglected and corresponds to an inertial frame with associated cartesian coordinates (X µ ) µ=0,••• ,3 = (cT, X, Y, Z), where c is the speed of light and is left to facilitate dimensional reasoning for the reader.The second observer O is the traveller embarked on the radiation rocket, using comoving spherical coordinates (x ν ) ν=0,••• ,3 = (c • τ, r, θ, ϕ), with τ is the proper time of the traveller.
We will assume all along this paper that the trajectory of the rocket follows a straight path along the direction Z in O coordinates so that the traveller's worldline L is given either by L ≡ (cT (τ ), 0, 0, Z(τ )) in O's coordinates or L ≡ (c • τ, r = 0) (and with any θ, ϕ) in O coordinates.Tangent vector field to the worldline L will be denoted by According to the equivalence principle, the accelerated traveller experiences a local gravitational field.The geometry of spacetime can be described in the comoving coordinates (c • τ, r, θ, ϕ) by the following metric: where the signs in the second line above stand for the cases of retarded or advanced time coordinates, respectively.One can indeed associate each point X µ of Minkowski space to a unique retarded or advanced point X µ L on the worldline L which is at the intersection of the past/future light cone of X µ and the worldline L. Retarded and advanced metric will be used in Section 3 for computing incoming and outgoing geodesics toward/from X µ L .The above metric is of course singular at the traveller's location r = 0, and the geometric quadratic invariants only depend on mass M (see [7]).Therefore, when M = 0 one retrieves Minkowski spacetime but viewed by an accelerated observer.Spacetime geometry Eq.(3) around the radiative rocket is totally specified by the two functions of proper time M (τ ) and α(τ ).The first function M (τ ) = 2Gm(τ )/c 2 , where G is Newton's constant, implements the gravitational effect of the inertial mass m of the radiative rocket.The second function α(τ ) is related to the four-acceleration of the radiative rocket in the following way [29].Let λµ = dλ µ /(cdτ ) be the four-acceleration of the worldline L and, since the unit tangent vector λ µ is timelike (η µν λ µ λ ν = 1), we have that λµ λ µ = 0 or in other words that λµ is a spacelike vector.The function α(τ ) is then given by α 2 = − λµ λµ = − λ µ λ µ ≥ 0• As we shall see below, both mass M and "acceleration" α functions will be linked together through the relativistic rocket equations.
One can move from traveller's comoving coordinates O to inertial coordinates O through the transformation (see also [7,9,12] for the retarded case): where the couple (T L (τ ), Z L (τ )) are the functions specifying the worldline L of the traveller in inertial coordinates O, ψ = ψ(τ ) is the rapidity, and where the case + (−) is for retarded (advanced) metric, respectively.It must be kept in mind that this transformation is only valid for negligible radiative mass m, or at infinite distance from the rocket, and reduces to Eq.(3) in the limit M → 0• Performing the transformation of metric Eq.( 3) with m = 0 to Minkowsky metric where a dot denotes a derivative with respect to c • τ .The tangent vector λ µ = dX µ /(cdτ ) ≡ Ẋµ in inertial coordinates is normalized: η µν λ µ λ ν = 1 and the norm of the four acceleration is given by λµ λµ = ψ2 • The usual approach to "photon rockets" is to pass by Einstein's equations, which reduces here to only one equation since the only non-vanishing component of the Ricci tensor for the metric Eq.( 3) writes down: and one can further verify that the scalar curvature R identically vanishes.Eq.( 8) therefore puts a single constraint between the radiation flux function Φ of Eq.(2) and the kinematical functions of the mass M , its derivative Ṁ = dM/(cdτ ) and the 4-acceleration α of the radiation rocket.A decoupled set of radiation rocket equations would be more suitable for physical modelling since we would like to derive directly the worldline L and the associated spacetime geometry (3) from ingoing and outgoing radiation characteristics.
To achieve this, we follow [13] and consider total energy-momentum conservation in inertial frame O which writes down: On the one hand, we have that for a point-particle: where X µ L (τ ) represents the location of the radiation rocket in spacetime coordinates of inertial frame O. On the other hand, radiation's stress-energy conservation defines the radiation reaction 4-force density acting on the rocket (see also [16]).The time component F 0 gives c −1 times the net rate per unit volume of radiative energy flowing into or escaping the particle while the spatial components F i give the thrust per unit volume that is imparted to the rocket.For a point-like radiative distribution, one can write down with f µ the radiation reaction 4-force, which has units power.With these definitions, conservation equation Eq.( 9) yield the relativistic radiation rocket equations Eqs.(10,11) where a dot now denotes a derivative with respect to τ and where we have used a contraction with λ µ together with λ µ λ µ = 1.The radiation reaction four-force f µ is given by (see also [16]): ) with ν the radiation frequency, dΩ = sin(θ) • dθ • dϕ the solid angle element, A the absorption coefficient (with dimensions of an area in m 2 ), I the specific intensity (with dimensions W/(m 2 Hz Ster)) and E the emission coefficient (with dimensions W/(Hz Ster)) and ) is a dimensionless null 4-vector with − → n (θ, ϕ) pointing outward the rocket, in the direction (θ, ϕ) of the unit 2-sphere: In Eq.( 11), the component f T gives the power that is either entering the rocket f T > 0 (for a light sail or in other words an absorption rocket) or fleeing the rocket f T < 0 (for an emission rocket) while the components f i are c times the thrust imparted by the radiation flux to the rocket.The kinematic part of radiation rocket equations, Eqs.(10), actually describes general relativistic motion in the presence of some 4-force field f µ (see also [3]).Therefore, specifying the radiation flux that is either emitted, reflected or absorbed by the radiative rocket and the associated momentum gained by the rocket allows one to solve the kinematical equations Eqs.(10), determining both the traveller's wordline L and the spacetime geometry through Eq.( 3).We now focus on explicit solutions in the next paragraph.

B. Straight Accelerated Motion of Light Sails, Absorption and Emission Radiation Rockets
For a straight motion along the Z-axis, one has X = Y = λ X = λ Y = 0 together with Eqs.(5,6) so that the relativistic rocket equations Eqs.(10) reduce to The 3−velocity (i.e., velocity in the newtonian sense) of the radiation rocket with respect to the inertial observer O is given by V Z = dZ dT = c λ Z λ T = c tanh(ψ) while the newtonian 3-acceleration with respect to the inertial observer O is given by a = dV Z dT = c ψ/ cosh 3 (ψ) (a dotted quantity representing here the derivative of this quantity with respect to proper time τ ).
One can advantageously reformulate the system (12) in terms of dimensionless quantities by considering the characteristic scale for the proper time given by τ c = m 0 c 2 /|P | with m 0 the inertial mass of the radiation rocket at start and P is the scale of the power driving the rocket and that is either entering (P > 0) or leaving the rocket (P < 0).Using s = τ /τ c as a dimensionless time variable, one can rewrite Eq.( 12) as follows where a prime denotes a derivative with respect to s, M = m/m 0 , P(τ ) = f T /|P | is a dimensionless function describing the power either entering or leaving the rocket and T = f Z /f T = T (τ ).c/f T is a dimensionless function associated to the thrust T (in Newtons) driving the radiation rocket.
Actually, there are two types of radiation rockets: absorption rockets, for which P(τ ) > 0, and emission rockets, for which P(τ ) < 0• Light or solar sails belong to the former while a simple case of the latter consists of a system propelled by anisotropic radiative cooling (see also [14] for general introduction to the idea).If the thrust is directly proportional to the driving power, i.e.T = ±1, the first of the relativistic rocket equation Eq.( 13) reduces to Therefore, in the case of a purely absorbing rocket P(τ ) > 0, the rocket mass m is monotonically increasing while in the case of a purely emitting rocket P(τ ) < 0, the rocket mass m is monotonically decreasing.Furthermore, it can be shown from Eqs.( 13) with T = ±1 that the 3−velocity of the radiation rocket with respect to the inertial observer O, V Z = c tanh(ψ) verifies: which is nothing but a relativistic generalization of the Tsiolkovsky equation (1), as obtained for the first time by Ackeret in [15] from basic special relativity.
In the simplistic case where the driving power is constant, P(τ ) = ±1 and the thrust is directly proportional to this power, T (τ ) = ±1, one can easily obtain the following analytical solution of the system (13): where s 0 , ψ 0 the initial dimensionless time and rapidity respectively, P = ±1 (+1 for the absorption rocket and −1 for the emission one) and T = ±1 gives the direction of acceleration (+1 acceleration toward +Z and −1 backward Z).One can easily check that Eqs.(15,16) of course satisfy the Ackeret's equation (14).For an absorption rocket, the velocity will reach that of light c asymptotically and with an infinitely large inertial mass, while an emission rocket (P = −1) will reach the speed of light c after a finite proper time (s rel − s 0 ) = e −T ψ0 /2 where its inertial mass identically vanishes.
However, Eqs. (15,16) describe radiation rocket powered by a constant source, which is too simplistic.First, since the intensity of a light beam decreases with the inverse of the distance to the source, the feeding power of a light sail will decrease as P(τ ) ∼ Z −2 (τ ) unless the remote power source is increased accordingly, which is unpractical.Second, one could also consider an emission rocket propelled by directed blackbody radiation coming from the decay heat of some radioactive material, in which case the power source will decay as P(τ ) = exp(−s/S) with decay time S (in units of characteristic proper time τ c ).More realistic physical models of radiation rockets will therefore involve time dependance of the rocket's driving power and thrust P, T (τ ).Then kinematics must be obtained in general through numerical integration of Eqs.(13).
We can now give two couples of simple models of light sails and emission rockets.For the light sails, let us assume the following: (i) the power function is given by P(τ ) = (1 + Z(τ )/Z c ) −2 with Z c = c.τ c a characteristic scale and Z(τ ) is given by Eq.( 6) ; (ii) the thrust function T = ±(1 + ) where the sign gives the direction of acceleration and where is the reflectivity of the sail.A perfectly absorbing sail, a "black " one, exhibits = 0 while a perfectly reflecting "white"one has = 1• For the emission rockets, we can assume (i) that the output power is constant (see solutions Eqs.(15,16) with P = −1) or (ii) that the output power decays with proper time, P(τ ) = − exp (−s/S) with S the power decay time, as would happen if the emission rocket was powered by the radiative cooling of some radioactive material.In these cases, T = ±1• Figures 1-4 present the kinematics of these four radiation rockets.
The evolution of rockets velocity, starting from rest, and of the rockets inertial masses as traveller's proper time evolves are given in Figs. 1 and 2 respectively.The light sails will only reach some fraction of the speed of light c asymptotically while their inertial masses eventually freeze.Emission rocket with constant driving power reaches c after some finite proper time, at which its inertial mass vanishes.Once c is reached, the proper time of the traveller freezes and ceases to elapse.If the driving power is decaying exponentially with time, the emission rocket asymptotically reaches only some fraction of c while its inertial mass finally freezes.In the figures, we have used the following value of the power decay time S = 1/3• The acceleration of the rockets causes time dilation for the travellers as illustrated in Fig. 3. Light sails reach asymptotically a time dilation factor T /τ of around 4 for = 1 (white sail) and around 1.5 for = 0 (black sail).The emission rocket with constant power formally reach an infinite time dilation factor after some finite proper time since, as soon as it has reached c, proper time of the traveller freezes and the ratio T /τ → ∞• In the case of decaying internal power, the emission rocket finally experiences a frozen time dilation factor, around 1.15 for S = 1/3 as can be seen from Fig. 3. Finally, we give in Fig. 4 a Tsiolkovsky diagram showing the change in velocity with the variation of inertial mass of the radiation rockets.The black light sail with = 0 and the emission rocket all satisfy the relativistic Tsiolkovsky equation ( 14) while the Tsiolkovsky curve for the white sail with = 1 move from close to the purely absorbing case = 0 to the emission rocket case.It is important to bare in mind that this mass loss is a purely relativistic effect due to the interaction of the rocket with radiation and is therefore quite different than the reaction process due to the ejection of massive propellant.
Actually, there exists a well-known analytical solution that apply to radiation rockets as well.This special case is the so-called hyperbolic motion (cf.[3,12]), i.e. relativistic motion with a constant norm of the four-acceleration λµ , with units m −1 , λµ λµ = a 2 ≥ 0• For the characteristic time, we therefore choose τ c = 1/(c|a|) so that ψ = ψ 0 + d.f.(s − s 0 ) and M ≡ m/m 0 = exp (f.(s − s 0 )) with f = ±1 giving the rocket type (+1 absorption ; −1 emission) and d = ±1 so that the sign of the thrust is f.d•The corresponding four-force components can now be deduced from Eqs.( 12): f T = m 0 c 2 /τ c .f. exp(2f (s − s 0 )) and f Z = d.fT • In this solution, the speed of light c is reached asymptotically (when s → ∞) while the rocket's inertial mass becomes exponentially large (absorption rocket) or decays exponentially (emission rocket).In terms of the above examples, the emission rocket with a decay time S = 1/2 precisely corresponds to this solution of constant norm of four acceleration.
C. Applications to interstellar travels

Acceleration phase of a light sail
We start by applying our modelling to the starshot project [2] in which tiny probes attached to light sail will be accelerated by high power laser shots from ground to reach a relativistic velocity after a short acceleration phase before heading to Proxima Centauri for a flyby.We assume here a mass of 10g for the probe and the sail, a power of the laser beam at source of P = 100GW, decaying with distance as the inverse of the distance to the source as in the previous section.Fig. 5 gives the evolution of the 3-velocity (with newtonian result V = P/(cm 0 ).(1 + )τ indicated by dashed lines), the 3-acceleration (the last two with respect to inertial observer) and the mass with respect to the proper time τ during about one hour of continuous push by the lasers.Two different values of the reflexivity = 0 and = 1 are indicated, to give an idea of the spread of these kinematical variables with the reflexivity.The total energy cost of the mission roughly corresponds to the amount of energy spend by the power source during the whole acceleration phase, E = P × τ (for constant power of the source), which corresponds to 100GW × 4000s ≈ 10 14 J• In about an hour of continuous propulsion, the 10g probe reaches a velocity between 0.3 and ∼ 0.6 × c for corresponding acceleration decreasing from the range [6000 ; 3000] × g to [2000 ; 1000] × g• This decrease of the acceleration is a purely relativistic effect.Finally, the mass relative variation of the probe lies in the range 15-35%, which is non-negligible for performing corrections of trajectory with the embarked photon thrusters.

Travelling to Proxima Centauri with an emission radiation rocket
The next application of our former results is a simple modelling of interstellar travels to Proxima Centauri, located about 4 light-years away, with large emission radiation rockets.This example is purely illustrative, and we will not list the numerous engineering challenges that must be overcome in order to even start thinking about such mission, yet it will clearly show the major impediment of interstellar travel: the energy cost.
Let us consider a model of a rocket propelled by the redirection of the blackbody radiation emitted by a large hot surface in radiative cooling.The power driving the rocket is therefore given by P = σAT 4 where σ = 5.670373×10 −8 W/(m 2 .K 4 ) is the Stefan-Boltzmann constant, A the surface of the radiator at temperature T • To fix the ideas, we choose the total mass of the rocket is m 0 = 100T• We also assume a decay time of S = τ c with τ c = m 0 c 2 /P the characteristic time scale.kinematical quantities for both a return and a single trip at a distance of ∼ 4 light-years.The presented single trip is done with a radiator of A = 1km 2 at temperature T ∼ 8.4 × 10 3 K (and a total driving power P ∼ 3 × 10 2 TW) while the presented return trip is done with A = 100km 2 at T = 3000K for a power source of P ∼ 4 × 10 2 TW• Those parameters have been chosen for illustrative reasons and do not pretend to be feasible, simply remember that the characteristic power of a civil nuclear reactor is of order 1GW.In the top left plot of Fig. 6, one can see the velocity pattern of the trajectories.
In the single trip, the rocket first accelerates at a speed of ∼ 0.94 × c before it must be turned upside down for deceleration after about 8 months, and finally arrives at destination in about 3 years.The acceleration with respect to inertial observers are not shown but are not above 1.25 × g• In the case of the return trip, the rocket reaches about 0.97 × c after 5 months of acceleration before flipping for deceleration and reaches destination after ∼ 2.7 years.However, the rocket does not stop and immediately goes back toward its departure location.
After having reached a return velocity of about −0.97 × c at mission time ∼ 3 years, the returning rocket will have to flip for deceleration and finally arrive back at home at rest after ∼ 5.4 years.However, the total duration of the return trip from the point of view of the inertial observer stayed at home is ∼ 10.6 years, as a consequence of time dilation (see Fig. 6 lower right panel).Similarly, the single trip has been performed in about 3.5 years from the point of view of the traveller but about 6 years from the point of view of his home.The accelerations underwent by the return rocket are less than 2g's.In the upper right panel of Fig. 6, one can see the mass decrease of the rockets as a function of mission time.The flips of the rockets have been assumed instantaneous, which explains the shape of the curves around the flips.Worldlines of the interstellar travels in inertial coordinates shown in the lower left panel of Fig. 6, the worldline of an observer stayed at home corresponds to the (Z = 0, T ) vertical line.The events of departure and arrival of the return rocket are given by the intersection of the traveller's wordline and the vertical axis.
We finish this section by echoing our introduction.Emission radiation rockets do not involve new physics and actually are propelled by the least noble sort of energy which is heat.By maintaining acceleration throughout the trip, one can significantly reduces its duration by comparison of the one measured by an observer stayed at departure location.However, the major impediment is that such rockets needs to embark an extreme power source, of scale 100TW, within the lowest mass possible and make it work all along the several years of travel duration.By integrating f T (τ ) all along the trips, one finds an estimation of the total energy cost E of an interstellar mission to Proxima Centauri with a 100T scale spaceship is the unbelievable figure of E ∼ 9 × 10 21 J.To fix the ideas, this amount of energy corresponds to about 15 times the world energy production in 2017... Needless to say that no one can (presently?)afford such interstellar travel and that one should instead rely on another scheme, like the starshot concept [2], to physically investigate nearby star systems.

III. PROPAGATION OF LIGHT TOWARD OBSERVERS ON BOARD OF RADIATION ROCKETS
A. Geodesics in the hamiltonian formalism We now investigate spacetime geometry around the radiation rockets through characterizing null geodesics, which are nothing but the trajectories of light.Let us first remind Kinnersley's metric Eq.( 3) in the comoving coordinates of the radiation rocket where the upper (lower) sign is for retarded (advanced) traveller's proper τ and where M (τ ) = 2Gm(τ )/c 2 (with dimension length) and α = − ψ/c (with dimension inverse of length) are functions of proper time τ associated respectively to the mass and to the four-acceleration (i.e., g µν λ µ λ ν = α 2 ) of the rocket.Light rays incoming toward (or outgoing from) the traveller must be computed from the retarded (advanced) metric.Any geodesic curve G is specified in these coordinates by the following set of functions G = (c.τ(σ), r(σ), θ(σ), ϕ(σ)) with σ some affine parameter on the geodesic.These functions are solutions of the geodesic equation: but those equations are difficult to handle in the metric (3), as can be seen in [12,17].This is why we prefer here to proceed with the so-called hamiltonian formulation of geodesics [18].
Geodesic equations ( 17) are Euler-Lagrange equations for the action of a point-like particle S = m.ds(with ds the line-element in spacetime) but also of the following lagrangian L = 1/2g µν ẋµ ẋν with ẋµ = dx µ dσ • Introducing canonical momenta as usual by p α = ∂L ∂ ẋα , one can introduce an associated hamiltonian Then, instead of solving Euler-Lagrange equations (17), one can advantageously solve rather their hamiltonian counterparts: The contravariant metric components g αβ read where the + (−) sign is for incoming (outgoing) geodesics.According to this, the (co-)geodesic equations ( 19) can now be written down (c = 1, a dot indicating a derivative w.r.t.τ ): with p ϕ a constant of motion, since the metric does not explicitly depend on ϕ (axial symmetry).The corresponding hamiltonian, which is also a constant of motion dH/dσ = 0, is given by For light rays, or null geodesics, H identically vanishes while for matter geodesics H < 0 and both types of geodesics obey the same set of ODEs (19).
A first trivial particular solution is given by constant τ , θ and ϕ, while r ∼ σ (p r = p θ = p ϕ = 0 p τ = cst), which shows that τ is indeed a null coordinate.Special relativity describes motion at constant velocity and vanishing mass corresponding to the special case M = α = 0, which yields a second class of particular solutions: these are null geodesics, p ϕ = p θ = 0, p r = 2p τ = 0 and therefore τ = τ 0 ∓ 2r (the case of vanishing p r is the previous trivial solution).
In the general case, spacetime geometry around the photon rocket is ruled by the two functions M and α, which can be obtained by solving the relativistic rocket equations given the radiation reaction four-force (see section II).However, the mass function can be safely neglected in any physical situation except those of huge "luminosity" of the rocket.To see this, one can simply rewrite the metric Eq.( 3) with the characteristic units introduced before, by setting s = τ /τ c and R = r/(cτ c ) with τ c = m 0 c 2 /|f T | the characteristic time scale of the photon rocket physical system.Doing so, the mass term M/r in Eq.( 3) reduces to M/R × G|f T |/c 5 (m = M.m 0 ) and that the product 5 .This means that one can safely neglect the mass effect carried by M in front of the acceleration effect due to α as long as where we have replaced |f T | by L, the luminosity driving the photon rocket.It is surprising to notice that it is not the rest mass m 0 but the luminosity L that matters for the photon rocket spacetime geometry [30].One can interestingly ask for which kind of physical phenomena the mass function M should not be neglected anymore.Well, a remarkable example is binary black hole mergers and their recoil (also dubbed black hole kick) through the anisotropic emission of gravitational waves during merging.For instance, in the very final moments of the GW150914 binary black hole merger event, the emitted power of gravitational radiation reached about 10 49 W [19], so that GL/c 5 ≈ 10 −3 .With such luminosity, the complete photon rocket metric, including M and α, should be considered in characterising the spacetime geometry around a black hole merger self-accelerated by its anisotropic emission of gravitational waves.As a matter of comparison, electromagnetic record luminosities are far lower: the most brilliant supernova reached a luminosity of "only" 10 38 W [20] (with GL/c 5 ≈ 10 −15 ) while the brightest quasar, 3C273, has luminosity of order 10 39 W [21] (yielding GL/c 5 ≈ 10 −14 ).It is also worth noticing that the luminosity L = c 5 /(2G) appears as an absolute upper bound build from dimensional considerations in general relativity by Hogan [22], and is dubbed Planck luminosity.
In what follows, we will apply photon rockets to model of interstellar travel and will assume "weak " luminosities in the sense of Eq.( 28) such that their mass function is negligible M 1. Future works should investigate further the applications of photon rocket spacetimes to the modelling of astrophysical events such as black hole merger recoil.

B. Application: deformation of the interstellar traveller's local celestial sphere
In this section, we model the local celestial sphere of the interstellar traveller, which is affected by mainly two phenomena: relativistic aberration and Doppler effect.We assume the traveller undergoes an accelerated trajectory, starting from rest at τ = 0 and Z(0) = T (0) = ψ(0) = 0 such that the coordinates (θ, ϕ) at start τ = 0 are usual spherical coordinates (see Eqs.( 4) and [7]) that can be used to map the reference celestial sphere.This reference celestial sphere also corresponds to the one of the inertial observer whose worldline is tangent to the traveller's worldline at departure τ = 0. We are interested in the trajectories of light rays between departure τ = 0 up to their reception by the interstellar traveller at some proper time τ = τ R , since the paths of light rays before traveller's departure τ < 0 are not affected by its motion (the traveller stayed at rest at home at τ < 0).We also assume here that M = 0, since we are not considering extreme luminosities as mentioned above, and therefore Eq.( 3) will describe Minkowski flat spacetime (see also [7]) but from the point of view of the accelerated traveller.In this accelerated frame, light rays will undergo angular deviation, leading to relativistic aberration, and frequency shifts (Doppler effect) which are different from those described by special relativity with motion at constant velocity.Both effects are of crucial importance for the interstellar traveller since this affect not only its telecommunications but also its navigation by modifying positions and colour of the guiding stars.As we shall see below, these effects both depend on the trajectory followed by the traveller.Indeed, light rays trajectories are solutions of the geodesic equations Eqs.(20)(21)(22)(23)(24)(25)(26) with a null value of the Hamiltonian Eq.( 27) and these solutions depends on the time variation of the acceleration function α(τ ) (M can be safely neglected unless one faces extreme luminosities).Here, we will solve the geodesic equations Eqs.(20)(21)(22)(23)(24)(25)(26)) by integrating numerically backward in time, from the reception of the light ray by the traveller at (τ = τ R , r = 0, θ = θ R , ϕ = ϕ R ) back to the time of traveller's departure τ = 0 at which the light ray was emitted by the reference celestial sphere at τ = 0, r = r E , θ E , ϕ E • In order to compute the local celestial sphere of the traveller, we are interested in incoming light rays that have a null impact parameter, so that p ϕ = 0 (so that ϕ R = ϕ E ) and that arrive radially p θ (τ R ) = 0• Since τ R and θ R are considered as free parameters, this leaves only two initial conditions, p r (τ R ), p τ (τ R ), to be determined.From Eqs.(20-26) we can set, without loss of generality, p r (τ R ) = 1 so that τ ≈ σ at reception (the affine parameter is then simply scaled by choice to proper time at reception).p τ (τ R ) must then be obtained by solving the Hamiltonian constraint H = 0 Eq.( 27) with respect to p τ , given all the other initial conditions at τ = τ R .This achieves fixing our set of initial conditions at given τ R • Then, integrating backward the geodesics equation Eqs.(20)(21)(22)(23)(24)(25)(26) until the rays were emitted from the reference celestial sphere at τ = 0, one can compute the angular deviation and the frequency shifts of light between the reference celestial sphere at τ = 0 and the local celestial sphere of the traveller at τ = τ R .
To compute relativistic aberration for the accelerated traveller, one needs to account for two contributions.The first is the angular coordinate change θ R = θ E at both ends of the null geodesic (remember that p ϕ = 0 so that ϕ R = ϕ E ).The second input comes from the fact that the angular coordinate θ R does not correspond anymore to the usual spherical coordinate at τ = τ R and ψ R = 0• To find the corresponding angle Θ on the local celestial sphere of the traveller, one has to move back to the instantaneous rest frame of the traveller.This is done by imposing 4), where ρ and Θ are local spherical coordinates [31].Doing so, we can write down the correspondance relation between both angles of incidence Θ in the local traveller's frame and the angular coordinate θ R at reception with β = tanh ψ R • One can check that Eq.( 29) is identical to the formula of relativistic aberration in special relativity as obtained by Einstein in [3,23] for motion at constant velocity for which θ R is then the angle of incidence as measured by the observer at rest.The aberration angle for the accelerated observer is therefore given by Θ = Θ(θ E ) with Θ given by Eq.( 29) in which θ R is a function of θ E as obtained by the backward integration of null geodesic equations.
Of course, in the case of motion at constant velocity ψ = cst, α = 0 and null geodesics are given by the trivial solution p ϕ = p θ = 0, p r = 2p τ = 1, τ = τ 0 ∓ 2r, such that one has θ R = θ E leading to the special relativistic aberration described by Eq.( 29).Quite interestingly, among all possible traveller's wordlines, there is a non trivial one for which there is also no angular deviation θ(σ) = cst, or in other words θ R = θ E and the relativistic aberration of the accelerated traveller reduces to the one of special relativity and motion at a constant velocity.This is the case when α(τ ) = cst, which corresponds to the hyperbolic motion of Section II.Indeed, setting dθ/dσ = 0 in Eq.( 22), one obtains that Then, since α = Ṁ = 0 (α = cst, M = 0), Eq.( 24) yields that p τ = cst whose value can be obtained from the Hamiltonian constraint H = 0. Solving Eq.( 27) with respect to p τ and assuming p r = 0, one finds that Putting Eqs.(30,31) and p ϕ = 0 into Eqs.(21) and ( 25), we obtain Finally, one can use Eqs.(30,32,33), p ϕ = 0 and dθ/dσ = 0 to retrieve Eq.( 26), showing that α = cst implies dθ/dσ = 0• Let us now focus on the Doppler effect for accelerated traveller on board of photon rockets.The frequency shift can be obtained as follows.The energy of the photon measured at spacetime event e by some observer is given by E e = h.νe = (p µ λ µ ) e with λ µ the unit tangent vector to the observer O s worldline, h is Planck's constant and ν e the measured frequency of the photon.In this application, the receiver is the traveller, with wordline (r = 0, τ ) in his local coordinates, so that the received frequency of the photon is p τ (τ R )• At start τ = 0, we consider an emitter on the reference celestial sphere that has no proper motion with respect to home position, corresponding to a worldline given by fixed inertial coordinates (X, Y, Z) τ =0 = (X E , Y E , Z E ) and proper time T .This models a fictitious star located at (X E , Y E , Z E ) with assumed no proper motion at the position (θ E , ϕ E ) on the reference celestial sphere and whose light frequencies of the emitted light rays are those observed by the inertial observer stayed at home.The components of the tangent vector to the emitter's worldline are given by dx µ /dT = d(τ E , r E , θ E , ϕ E )/dT (and dϕ E /dT = 0) and must be computed from Eqs.(4).
Fortunately, since at start we have ψ E = 0, this yields simply From these relations, one can compute the unit tangent vector of the emitter at time of emission: where is the norm of the tangent vector λ µ = dx µ /dT • The photon energy at emission e = E is therefore • Finally, the frequency shift is simply given by E R /E E , the ratio of the received frequency ν R over the emitted one ν E .
For a specific wordline L of the traveller, the particular α(τ ) will yield to different null geodesics between τ R and τ = 0 and therefore different aberration angles and frequency shifts.We now illustrate some deformations of the traveller's celestial sphere with the following three types of photon rockets, all starting at rest at τ = 0 and then accelerating monotonically.The first type is the hyperbolic motion, given by ψ = s (α = −dψ/ds = −1 in characteristic units) and M = e ±s with s = τ /τ c , τ c being the characteristic time defined in Section II.This case of hyperbolic motion is a critical point for which aberration of the accelerated traveller is the one described by special relativity.For reasons that will appear clearly below, we choose to consider also a perfect light sail (see section II) for which α monotonically increases from α (0) = −2 to α → 0 when s → ∞ and an emission rocket with constant output power Eq.(15,16) in which α ≤ −1.ward the observer at (τ = τ R , r = 0 and V Z (τ R ) = 0.9•c) as plots of the proper time τ (σ) along the null geodesics as functions of r •cos θ for the three different photon rockets mentioned above (emission rocket, hyperbolic motion and light sail).The different curves corresponds to different initial conditions θ R • One can clearly see the Minkowskian regime τ ∼ 2r of the null geodesics as r(σ) → 0 and the metric (3) becomes close to the case of motion at constant velocity α = 0 (remind that M = 0 here).Also shown in Fig. 7 is a consistency check through the violation of the hamiltonian constraint H = 0 along the null geodesics: this constraint is found pretty stable, the value of H is kept around the order of magnitude of the numerical integrator tolerance.In establishing the following results, we have always monitored this hamiltonian constraint, which has been found quite robust and always controlled by the numerical integration tolerance.Fig. 8 presents the relativistic aberration for the accelerated traveller as proper time evolves on board of an emission photon rocket (top panel), a rocket with α = cst (central panel) and a light sail (bottom panel).In the first case, one has that the received angle of incidence Θ gets smaller and smaller than the angle of incidence at start θ E as the traveller accelerates and gains velocity.One can see from Fig. 8 (top panel) that this effect is quantitatively stronger than the aberration in special relativity for the emission photon rocket since in this case Θ < Θ SR given by Eq.( 29) with θ R = θ E .The case of hyperbolic motion gives rise to the same relativistic aberration as if the traveller was in motion at constant velocity, Θ = Θ SR as explained above, (see Fig. 8, central panel) and is shown for reference of the two other cases.The case of the sail in Fig. 8 (bottom panel), the angle of incidence as measured by the accelerated traveller Θ is greater than the case of special relativity: Θ > Θ SR .Hence, we have shown that relativistic aberration of an accelerated observer depends on the type of photon rocket.Fig. 9 presents the Doppler effect for the accelerated traveller on board of the three different photon rockets discussed here, through the ratio ν R /ν E as a function of the direction of reception Θ.The unit circle marks the transition from redshifts ν R /ν E < 1 (for directions of reception Θ far from zero) to blueshifts ν R /ν E > 1 (for directions of reception Θ ≈ 0 close to that of motion).The emission rocket presents Doppler shifts that are found close to those described by special relativity through the formula (cf.[3]): with β = tanh ψ (Eq.(34) for β = 0.95 is shown as dashed lines in Fig. 9).The departures from the special relativistic case is however stronger with higher velocities.For hyperbolic motion and light sails, the Relativistic aberration for a motion at constant velocity of V Z /c = 0.95 in special relativity is given in black dashed line Doppler effect for accelerated travellers departs further and further from the special relativistic value Eq.(34) as velocity increases.However, it must be noticed that on-axis Doppler shifts (for Θ = 0; π) are given by the formula from special relativity Eq.(34).Indeed, for θ = 0; π we have that p θ is conserved (since sin θ = 0 in Eq.( 22)) and therefore θ E = θ R = 0; π and, from Eq.( 29), Θ = θ R = 0; π• Finally, let us build a model of the deformation of the local celestial sphere of the accelerated traveller during his trip toward a distant star under the combined effects of aberration and Doppler frequency shifts.Our reference celestial sphere will be given by data from the fifth edition of the Yale Bright Star Catalogue [24] in which we choose some star for the traveller's destination and map through appropriate axis rotations the right ascension and declination coordinates onto spherical coordinates (θ E , ϕ) at τ = 0 with the axis Z pointing toward destination star.For each star in the catalog, we can also obtain the temperature from its B-V magnitude from the results in [25].With this temperature in hand, we have a blackbody spectrum for each star in the catalog and from this spectrum we can associate a specific colour from colorimetric considerations [26].For aesthetic reasons, we choose the destination star as Alnilam, at the center of the Orion belt and will show only some field of view centered around the front and rear directions of the interstellar rockets.To reconstruct a local view of the accelerated traveller at proper time τ R , we will loop on each star in the catalog and compute both its local position and colour taking into account relativistic aberration and Doppler effect as follows.
From the angle of incidence θ E of a given star in the catalog, we can obtain the observed angle of incidence Θ from our previous results in Fig. 8 and hence the associated position on the traveller's local celestial sphere.The observed angle of incidence Θ of the star will also determine its frequency shift from relations shown in Fig. 9 and a rendering of some star's observed colour by applying the associated Doppler shift to the star's blackbody spectrum to obtain a RBG triplet from our colorimetric functions.Finally, each star lying in the field of view is plotted as a sphere located at the found position, with colour associated to the shifted blackbody spectrum and with a size inversely proportional to its visual magnitude on the reference celestial sphere.The accelerated observer is located at the center of the local celestial sphere looking either in the front (Fig. 10) or the rear (Fig. 11) directions of motion.
The evolution of the traveller's celestial sphere heading toward Alnilam as its velocity increases is given in Figures 10 and 11 for an emission rocket and a light sail.Remember that the acceleration modifies both relativistic aberration and Doppler effect compared to their description in special relativity.Doing so, the relativistic beaming is observed in the front view (Fig. 10) but it is stronger for the emission rocket and weaker for the light sail than what is predicted by special relativity.One can sees this by looking at how the asterism of the Winter Hexagon shrinks more rapidly on board of an emission rocket than on board of a light sail as velocity increases.One can also see how the Big Dipper in the upper left appears earlier in the field of view for the emission rocket than for the light sail (Fig. 10, central panel).The Doppler effect is responsible for the nice reddening of stars outside of some cone centered on destination while stars that are observed close to the front direction appear bluer than they are  and Delphinus (on the left).Then, as velocity increases, this rear view is emptied of stars more rapidly on board of the emission rocket than on board of a light sail.This can be seen while looking at how Scorpius and Aquila are leaving the field of rear view.This difference is due to the stronger relativistic aberration for the emission rocket.The Doppler shift on the rear is in both cases very close to the one predicted by special relativity for angles of incidence above 120 degrees, as can also be seen from Fig. 9.

IV. CONCLUSION
Interstellar travel at relativistic velocity (V c) is not forbidden by any physical laws.Even better, it could be done in principle without the need of any massive propellant, without invoking any speculative physics and with trip duration significantly reduced by relativistic time dilation.Indeed, energy-momentum conservation in relativity allows propulsion using anisotropic emission or absorption of radiation, leading to accelerated trajectories on which time is slowed down as it is in the equivalent gravity field.These principles are actually well known but are often not correctly dealed with in discussions on this unfortunately somewhat controversial subject.Deep space propulsion by the reaction of radiation emission or absorption are at the basis of many plausible and working devices such as solar or laser-pushed sails and radiative cooling rockets.Indeed, radiative rockets could even be propelled by the least noble type of energy which is heat, through collimating the blackbody radiation of some hot radiator.It is the energy cost of interstellar travel that really prevents it from becoming a practical reality.
In addition, what was missing so far was a rigorous physical modelling of radiation rockets in the framework of general relativity, which is unavoidable when one deals with accelerations, and this is the contribution of the present paper.Kinnersley's solution of general relativity gives a pointlike description of a photon rocket, although Einstein's equations reduce in this case to a single relation between the two functions of acceleration and mass in the metric and the (incoming and outgoing) radiation flux.To disentangle this problem, we use the energy-momentum conservation, which leads to the usual relativistic kinematics of the point particle, and derive specific models for light sails and radiative cooling rockets.We then applied these models to the practical example of interstellar trips to the Proxima Centauri star system, deriving important physical quantities for the acceleration, the variation of the rocket's inertial mass and the time dilation on-board of the rockets.It is shown how the strategy of ultralight laser sails is far more plausible than a manned radiative cooling rocket, notably from the point of view of the energy cost.Indeed, while the former would require about a year of operation of a single nuclear reactor the latter would require about 15 times the world energy production... for a single mission!Among the (numerous) technological challenges to achieve such an interstellar mission, there are the questions raised by telecommunication, course correction, navigation and imaging at destination.All these issues depend on how light rays are perceived by the traveller, and this depends on its past acceleration.By using hamiltonian formulation of geodesic flow, we have computed the trajectories of the incoming light rays for various types of photon rockets and derive the relativistic aberration (angular deviation of null geodesics) and Doppler effect (frequency shifts) experienced by the accelerated travellers.Our results extend the predictions of special relativity that are only valid for motion at constant velocity.It was also established analytically that, in the case of hyperbolic motion with constant norm of four-acceleration, the aberration is strictly the same as in special relativity but not the Doppler effect.In general, different acceleration histories leads to stronger or weaker relativistic aberrations and Doppler shifts.We also built visualisations for the traveller's local celestial sphere that accounts for the modified aberration and Doppler effects found and show what panoramas on-board of an accelerated spaceship heading toward star Alnilam would look like.
The mass function of the Kinnersley metric was neglected while computing the modifications of relativistic beaming and Doppler effect mentioned above.This is a rather safe assumption for the case of interstellar travels on board of photon rockets since we shew that the effects of acceleration largely dominates those of rocket inertial mass when the luminosity that powers the photon rocket is much less than the huge value c 5 /G ≈ 10 52 , sometimes referenced as Planck luminosity.Quite interestingly, the extreme amount of energy lost in gravitational radiation by binary black hole mergers would constitute an astrophysical application of Kinnersley metric where one could not neglect the effect of the mass function anymore.In particular, further studies should interestingly investigate the impact of the mass function on the Doppler effect and relativistic beaming associated to the radiation recoil of the merger.This can be done by using the co-geodesic equations derived here and apply them to mass and acceleration functions modelling the merger.
It is often (naively?!) hoped that moving to other star systems will be our only escape if one day this planet becomes inhospitable.But actually, developing interstellar travel might well precipitate the exhaust of our planet resources.In our view, it is crucial that the difficulties and implications of interstellar travel are correctly taught, based on rigorous scientific argumentation.In addition, it seems to us that time has come for starting the development of a technology demonstrator for high-velocity photon rocket in the Solar system.The results of this paper are of direct application for the computation of the trajectory, the input-ouput transmissions to the probes, the relativistic aberration aberration effects of image capture during a flyby and course corrections of such a high-velocity demonstrator.We can now pave the way for interstellar exploration with photon rockets, beyond the engineering sketches and theoretical exploratory works done so far.Hopefully, we will at last leave ourselves to this intimate experience common to all those who go out stargazing: the call of the stars.

FIG. 3 :
FIG. 1: Evolution of the 3-velocity V Z w.r.t. to inertial observer O with the traveller's proper time τ for absorption (left) and emission (right) rockets

FIG. 4 :
FIG. 4: Tsiolkovsky diagram of the variation of velocity as a function of the rocket mass

1 FIG. 5 :
FIG.5: Acceleration phase of a laser-pushed light sail, with a beam power at source of 100GW and a mass of 10g for reflexivity = 0 and = 1.Upper plots: velocity (left) and acceleration (right) with respect to inertial observer ; bottom plot: mass variation

FIG. 6 :
FIG. 6: Single and return trips (respectively straight and dashed lines) toward any destination located about 4 lightyears away, like the Alpha Centauri star system, with an emission radiation rocket.Upper left: velocity as a function of traveller's proper time τ ; Upper right: mass variation of the photon rocket during the trip ; Lower left: worldline L of the trips in inertial coordinates ; Lower right: time dilation for the traveller = p r α (p θ cos(θ) + p r r sin(θ)) − p 2 ϕ cos(θ) r 2 sin 3 (θ)(26)

FIG. 7 :
FIG. 7: Convergence of null geodesics toward the traveller at (r = 0, τ = τR) and tanh(ψR) = 0.9 for different values of the initial emission angle θE (upper left panel: emission rocket with constant driving power ; upper right panel: hyperbolic motion with α = 1 ; lower left panel: light sail and lower right panel : Hamiltonian for the case of hyperbolic motion and a tolerance of the integrator of 10 −12 , exact value for H is zero for null geodesics)

FIG. 8 :
FIG.8: Relativistic aberration for the accelerated traveller through the angle of incidence at emission θE as a function of the received angle of incidence θR for the emission rocket (top), hyperbolic motion (center) and light sail (bottom) at various velocities (V Z /c = 0.2, 0.39, 0.57, 0.76, 0.95).Relativistic aberration for a motion at constant velocity of V Z /c = 0.95 in special relativity is given in black dashed line

FIG. 10 :
FIG. 10: Front views for travellers on board of photon rockets during their trip toward star Alnilam, at the center of the field of view, for increasing velocities (left: emission rocket ; right: light sail)

FIG. 11 :
FIG. 11: Rear views for travellers on board of photon rockets during their trip toward star Alnilam as their velocity increases (left: emission rocket ; right: light sail)