Extreme Relativistic Rocketry

In Stephen Baxter’s “Xeelee” tales the early days of human starflight (c.3600 AD), before the Squeem Invasion, FTL travel and the Qax Occupation, starships used “GUT-drives”. This presumably uses “Grand Unification Theory” physics to ‘create’ energy from the void, which allows a starship drive to by-pass the need to carry it’s own kinetic energy in its fuel. Charles Sheffield did something similar in his “MacAndrews” yarns (“All the Colors of the Vacuum”) and Arthur C. Clarke dubbed it the “quantum ramjet” in his 1985 novel-length reboot of his novella “The Songs of Distant Earth”.

Granting this possibility, what does this enable a starship to do? First, we need to look at the limitations of a standard rocket.

In Newton’s Universe, energy is ‘massless’ and doesn’t add to the mass carried by a rocket. Thanks to Einstein that changes – the energy of the propellant has a mass too, as spelled out by that famous equation:

$$E=mc^2$$

For chemical propellants the energy comes from chemical potentials and is an almost immeasurably tiny fraction of their mass-energy. Even for nuclear fuels, like uranium or hydrogen, the fraction that can be converted into energy is less than 1%. Such rockets have particle speeds that max out at less than 12% of lightspeed – 36,000 km/s in everyday units. Once we start throwing antimatter into the propellant, then the fraction converted into energy goes up, all the way to 100%.

But… that means the fraction of reaction mass, propellant, that is just inert mass must go down, reaching zero at 100% conversion of mass into energy. The ‘particle velocity’ is lightspeed and a ‘perfect’ matter-antimatter starship is pushing itself with pure ‘light’ (uber energetic gamma-rays.)

For real rockets the particle velocity is always greater than the ‘effective exhaust velocity’ – the equivalent average velocity of the exhaust that is pushing the rocket forward. If a rocket energy converts mass into 100% energy perfectly, but 99% of that energy radiates away in all directions evenly, then the effective exhaust velocity is much less than lightspeed. Most matter-antimatter rockets are almost that ineffectual, with only the charged-pion fraction of the annihilation-reaction’s products producing useful thrust, and then with an efficiency of ~80% or so. Their effective exhaust velocity drops to ~0.33 c or so.

Friedwardt Winterberg has suggested that a gamma-ray laser than be created from a matter-antimatter reaction, with an almost perfect effective exhaust velocity of lightspeed. If so we then bump up against the ultimate limit – when the energy mass is the mass doing all the pushing. Being a rocket, the burn-out speed is limited by the Tsiolkovsky Equation:

$$V_f=V_e.ln\left(\frac {M_o}{M_i}\right)$$

However we have to understand, in Einstein’s Relativity, that we’re looking at the rocket’s accelerating reference frame. From the perspective of the wider Universe the rocket’s clocks are moving slower and slower as it approaches lightspeed, c. Thus, in the rocket frame, a constant acceleration is, in the Universe frame, declining as the rocket approaches c.

To convert from one frame to the other also requires a different measurement for speed. On board a rocket an integrating accelerometer adds up measured increments of acceleration per unit time and it’s perfectly fine in the rocket’s frame for such a device to meter a speed faster-than-light. However, in the Universe frame, the speed is always less than c. If we designate the ship’s self-measured speed as \(V’_f\) and the Universe measured version of the same, \(V_f\), then we get the following:

$$V’_f=V_e.ln\left(\frac {M_o}{M_i}\right)$$

[Note: the exhaust velocity, \(V_e\), is measured the same in both frames]

and…

$$V_f=V_e.\left(\frac {\left(\frac {M_o}{M_i}\right)^2-1}{\left(\frac {M_o}{M_i}\right)^2+1}\right)$$

To give the above equations some meaning, let’s throw some numbers in. For a mass-ratio, \(\left(\frac {M_o}{M_i}\right)\) of 10, exhaust velocity of c, the final velocities are \(V’_f\) = 2.3 c and \(V_f\) = 0.98 c. What that means for a rocket with a constant acceleration, in its reference frame, is that it starts with a thrust 10 times higher than what it finishes with. To slow down again, the mass-ratio must be squared – thus it becomes \(10^2=100\). Clearly the numbers rapidly go up as lightspeed is approached ever closer.

A related question is how this translates into time and distances. In Newtonian mechanics constant acceleration (g) over a given displacement (motion from A to B, denoted as S) is related to the total travel time as follows, assuming no periods of coasting at a constant speed, while starting and finishing at zero velocity:

$$S=\frac14 gt^2$$

this can be solved for time quite simply as:

$$t^2=\frac {4S}{g}$$

In the relativistic version of this equation we have to include the ‘time dimension’ of the displacement as well:

$$t^2=\frac {4S}{g}+\left(\frac {S}{c}\right)^2$$

This is from the reference frame of the wider Universe. From the rocket-frame, we’ll use the convention that the total time is \(\tau\), and we get the following:

$$\tau=\left(\frac {2c}{g}\right).arcosh\left(\frac {gS}{2c^2}+1\right)$$

where arcosh(…) is the so-called inverse hyperbolic cosine.

Converting between the two differing time-frames is the Lorentz-factor or gamma, which relates the two time-flows – primed because they’re not the total trip-times used in the equation above, but the ‘instantaneous’ flow of time in the two frames – like so:

$$\gamma=\left(\frac {t’}{\tau’}\right)=\left(\frac {1}{1-\left(\frac {V}{c}\right)^2}\right)$$

For a constant acceleration rocket, its \(gamma\) is related to displacement by:

$$\gamma=\left(\frac {gS}{2c^2}+1\right)$$

For very large \(\gamma\) factors, the rocket-frame total-time \(\tau\) simplifies to:

$$\tau\approx\left(\frac {2S}{c}\right).ln\left(2\gamma\right)$$

The relationship between the Lorentz factor and distance has the interesting approximation that \(\gamma\) increases by ~1 for every light-year travelled at 1 gee. To see the answer why lies in the factors involved – gee = 9.80665 m/s2, light-year = (c) x 31,557,600 seconds (= 1 year), and c = 299,792,458 m/s. If we divide c by a year we get the ‘acceleration’ ~9.5 m/s2, which is very close to 1 gee.

This also highlights the dilemma faced by travellers wanting to decrease their apparent travel time by using relativistic time-contraction – they have to accelerate at bone-crushing gee-levels to do so. For example, if we travel to Alpha Centauri at 1 gee the apparent travel-time in the rocket-frame is 3.5 years. Increasing that acceleration to a punishing 10 gee means a travel-time of 0.75 years, or 39 weeks. Pushing to 20 gee means a 23 week trip, while 50 gee gets it down to 11 weeks. Being crushed by 50 times your own body-weight works for ants, but causes bones to break and internal organs to tear loose in humans and is generally a health-hazard. Yet theoretically much higher accelerations can be endured by equalising the body’s internal environment with an incompressible external environment. Gas is too compressible – instead the body needs to be filled with liquid at high pressure, inside and out, “stiffening” it against its own weight.

Once that biomedical wonder is achieved – and it has been for axolotls bred in centrifuges – we run up against the propulsion issue. A perfect matter-antimatter rocket might achieve a 1 gee flight to Alpha Centauri starts with a mass-ratio of 41.

How does a GUT-drive change that picture? As the energy of the propellant is no longer coming from the propellant mass itself, the propellant can provide much more “specific impulse”, \(I_{sp}\), which can be greater than c. Specific Impulse is a rocketry concept – it’s the impulse (momentum x time) a unit mass of the propellant can produce. The units can be in seconds or in metres per second, depending on choice of conversion factors. For rockets carrying their own energy it’s equivalent to the effective exhaust velocity, but when the energy is piped in or ‘made fresh’ via GUT-physics, then the Specific Impulse can be significantly different. For example, if we expel the propellant carried at 0.995 c, relative to the rocket, then the Specific Impulse is ~10 c.

$$I_{sp}=\gamma_e.V_e=c.\sqrt{\gamma_e^2-1}$$

…where \(\gamma_e\) and \(V_e\) are the propellant gamma-factor and its effective exhaust velocity respectively.

This modifies the Rocket Equation to:

$$V’_f=I_{sp}.ln\left(\frac {M_o}{M_i}\right)$$

Remember this is in the rocket’s frame of reference, where the speed can be measured, by internal integrating accelerometers, as greater than c. Stationary observers will see neither the rocket or its exhaust exceeding the speed of light.

To see what this means for a high-gee flight to Alpha Centauri, we need a way of converting between the displacement and the ship’s self-measured speed. We already have that in the equation:

$$\tau=\left(\frac {2c}{g}\right).arcosh\left(\frac {gS}{2c^2}+1\right)$$

which becomes:

$$\frac {g\tau}{2c}=arcosh\left(\frac {gS}{2c^2}+1\right)$$

As \(V’_f=\left(\frac {g\tau}{2c}\right)\) and \(\left(\frac {gS}{2c^2}+1\right)=\gamma\), then we have

$$V’_f=I_{sp}.ln\left(\frac {M_o}{M_i}\right)=arcosh\left(\gamma\right)$$

For the 4.37 light year trip to Alpha Centauri at 50 gee and an Isp of 10 c, then the mass-ratio is ~3. To travel the 2.5 million light years to Andromeda’s M31 Galaxy, the mass-ratio is just 42 for an Isp of 10c.

Of course the trick is creating energy via GUT physics…

Author: Adam

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