Nomenclature A e jet exit area A o jet inlet area c speed of light E electric field e fundamental unit of charge F force h Planck constant ℏ reduced Planck constant M vehicle mass m propellant mass m o particle rest mass m ˙ propellant mass flow rate m ˙ e jet exit mass flow rate m ˙ o jet inlet mass flow rate N ˙ ph photon production rate N ˙ ph s photon production rate in source frame N ˙ pt particle production rate P power p momentum p e jet exit pressure p o jet inlet pressure T thrust t time u e propellant exhaust speed u o vehicle speed u ¯ o vehicle speed normalized by exhaust speed V vehicle speed V velocity V be breakeven vehicle speed β dimensionless speed ( v / c ) Γ volumetric fermion pair production rate γ Lorentz factor Δ v vehicle speed increment δ v particle speed increment λ wavelength ν frequency ν o frequency in observer frame ν s frequency in source frame

I. Introduction O n 12 January 1920, the New York Times ran a front-page story reporting on a Smithsonian press release about Robert Goddard’s rocket technology and his proposed application to send an instrumentation package to the moon. The very next day, the New York Times ran an unsigned editorial criticizing the proposal. Of particular interest in the editorial was a comment regarding Goddard’s understanding of basic physics. Specifically, the following: That Professor Goddard … does not know the relation of action and reaction, and of the need to have something better than a vacuum against which to react—to say that would be absurd. Of course he only seems to lack the knowledge ladled out daily in high schools [1]. Upon a first read, it might sound like a rational criticism; that is, until it is later clarified, within the same article that the author actually meant that the propellant (not the rocket) needed something to push against in order to generate thrust. So, although the rocket might be able to operate within the atmosphere, it would certainly not work in the vacuum of space. This misunderstanding of the application of Newton’s laws to rocketry was actually much more prevalent in Goddard’s day than simply a one-off editorial in a newspaper; in fact, Goddard was under constant criticism for his ideas: even from many within the scientific establishment. Now, one would hope that this rather ironic situation was the exception and not the rule, but it did demonstrate that sometimes the collective wisdom could in fact be wrong. Such, we claim, is the situation today with the idea of the “spacedrive.” By this, we mean a system that derives its propulsion in the vacuum of space without the use of propellant (i.e., propellantless). Often, such systems are incorrectly described as being reactionless, but this actually implies something that is much more problematic. Examples of the difference include terrestrial systems like cars, boats, and aircraft. These are propellantless in the sense that you do not preload them with some quantity of mass that is then expelled to generate thrust but, rather, they generate thrust by reacting against in situ resources (ground, water, air). Even in the case of the aircraft, although the fuel is indeed expelled, it is there primarily as an energy source, and not a reaction mass. None of these systems are reactionless. The debate about spacedrives is complicated because there is a large amount of “pseudoscience” on the topic. By pseudoscience, we are not actually referring to the application of ideas or theories that are not mainstream or “canon” but rather the application of any assumption or theory in the absence of the scientific method. This is a broad-reaching problem that goes both ways. There is the more recognizable group of individuals that gravitate toward alternative physics simply because it tends to be more accessible than the mainstream. Often, such theories have mechanical analogs that appeal to one’s sense of understanding in much same the way that quantum mechanics does not. Unfortunately, there is so much of this content available that it can be difficult to differentiate it from ideas that have stood up to the test of scrutiny. However, there is also the danger of going too far to the other extreme of the belief scale, where there is a tendency to discount any nonmainstream idea without giving it due consideration. This too is pseudoscience. The author of the editorial in the story above was guilty of this because he was blindly towing the line of what, in Goddard’s day, turned out to be a very common misconception of how momentum is conserved [1]. The story is, of course, quite apropos because the main (but, seemingly, not the only) criticism of spacedrives is that there can be no mechanism for them to conserve momentum in the vacuum of space. The purpose of this note is not to argue that any currently proposed technologies are actually manifesting the mechanisms discussed herein but simply to address this overly broad and unjustified criticism, as well as more recent criticisms that have emerged.

II. Momentum Conservation The primary concern with spacedrives is the appearance of violating momentum conservation. For reference, the typical model of space propulsion systems includes an onboard propellent supply that is expelled to produce thrust. A simple application of momentum conservation under this model leads immediately to the well-known result M Δ V Δ t = Δ m Δ t u e ↦ T = M d V d t = d m d t u e = m ˙ u e (1) and the resulting power required (ignoring conversion efficiency) P = 1 2 m ˙ u e 2 = 1 2 T u e (2) This then results in the thrust-to-power ratio: T P = 2 u e (3) which is a common metric for evaluating the effectiveness of the technology. However, an already known technology that can achieve thrust without carrying and expelling mass is the photon rocket. Instead of massive particles, the momentum is carried by massless photons, with p = h / λ = h ν / c , which still allows for total system momentum to be conserved. In this case, T = N ˙ ph h ν c = P h ν h ν c = P c ↦ T P = 1 c (4) which as derived applies to the reference frame of the vehicle. The expression in the rest frame will be derived shortly. Because the thrust-to-power ratio for the photon rocket is so small ( 1 / c ), this particular approach does not trade well for most applications. However, because generating photons is such a commonplace phenomenon, the implications of what is actually happening are generally taken for granted. Here, we have an everyday example of generating momentum-carrying particles from the vacuum using only stored energy. As ordinary as it is, it is really a remarkable achievement from a propulsion standpoint. and the resulting power required (ignoring conversion efficiency)This then results in the thrust-to-power ratio:which is a common metric for evaluating the effectiveness of the technology. However, an already known technology that can achieve thrust without carrying and expelling mass is the photon rocket. Instead of massive particles, the momentum is carried by massless photons, with, which still allows for total system momentum to be conserved. In this case,which as derived applies to the reference frame of the vehicle. The expression in the rest frame will be derived shortly. Because the thrust-to-power ratio for the photon rocket is so small (), this particular approach does not trade well for most applications. However, because generating photons is such a commonplace phenomenon, the implications of what is actually happening are generally taken for granted. Here, we have an everyday example of generating momentum-carrying particles from the vacuum using only stored energy. As ordinary as it is, it is really a remarkable achievement from a propulsion standpoint. Quantum field theory (QFT) treats all particles, whether massless or massive, as excitations of their respective fields. The photon is an excitation of the electromagnetic field; similarly, the electron is an excitation in the electron–positron (e-p) field. Photons turn out to be easy to “make” in general because they typically contain so little energy. For instance, a photon in the visible spectrum contains only ∼ 2 eV of energy. By contrast, an electron–positron pair requires ∼ 1 MeV of energy to create, typically requiring nuclear-level processes. A natural example of particle creation is beta decay, where one of the downquarks (d) of a neutron (udd) decays via the weak force into an upquark (u), turning it into a proton (uud) and releasing an electron and an antineutrino in the process. The electron is never “inside” the neutron: it is generated from the e-p field. Whereas photons are required to travel at speed c , massive particles (such as e-p pairs) created from the vacuum travel at subluminal speeds: in principle, allowing for a higher T / P . The catch is, of course, having to supply the energy required to create the particles. Again, in the rest frame of the vehicle, T = N ˙ pt γ m o δ v = ( P γ m o c 2 ) γ m o δ v ↦ T P = δ v c 2 with γ = 1 1 − β 2 and β = δ v c (5) which is even worse than the photon rocket due to the energy required to create the particles. However, one could imagine a system that momentarily creates e-p pairs, accelerates them to generate a reaction force, allows them to annihilate back to the vacuum, and then somehow recycles that part of the energy that went into creating the particles initially. The momentum change of the vehicle will essentially be balanced by momentum flow into the underlying e-p field. In this case, the thrust to power becomes T = N ˙ pt γ m o δ v = ( P ( γ − 1 ) m o c 2 ) γ m o δ v ↦ T P ≈ 2 δ v (6) which is approximately that of Eq. ( u e replaced by the net change in particle speed ( δ v ) given to the particle (or particle pair) after creation and before annihilation. The mechanisms by which the creation, acceleration, and postannihilation energy recovery might be engineered are likely nontrivial. A. Frame Dependence A subtle but important point is that all three of the preceding examples (onboard propellant, photons, massive vacuum particles) assume that the origin frame of the momentum-carrying particles coincides with the rest frame of the vehicle. For the onboard propellant, this is a given. For the photons, we know that they have no rest frame and that their speed is always the same: independent of the chosen frame. However, their frequency (and therefore energy) is frame dependent; in Eq. (4), the frame is assumed to be that of the vehicle. For the massive vacuum particles, it can be the case that their rest frame will not necessarily correspond to that of the vehicle. However, according to the currently accepted QFT, the vacuum fields are assumed to be Lorentz invariant, meaning that they look and act the same in all inertial reference frames. So, whatever the mechanism in the vehicle frame that produces the particles from the vacuum, it will likely manifest them in the same frame. That is not to say that the particles will be at rest upon creation, but their center of mass will coincide with the reference frame of the vehicle. However, one can also consider the limiting case of the reaction mass originating at rest in the observer frame. By analogy, this is the case of the atmosphere used by a jet aircraft in flight. In its most general form, the force that a jet propulsion system is able to provide to an aircraft is F = m ˙ e u e − m ˙ o u o + p e A e − p o A o (7) where thrust is derived from both momentum flux and pressure forces. However, if the inlet and exit pressures ( p o = p e ), the inlet and exit mass flow rates ( m ˙ o = m ˙ e ), and the inlet and exit nozzle areas ( A o = A e ) are assumed to be equal, Eq. ( F = m ˙ o ( u e − u o ) (8) where thrust is derived from both momentum flux and pressure forces. However, if the inlet and exit pressures (), the inlet and exit mass flow rates (), and the inlet and exit nozzle areas () are assumed to be equal, Eq. ( 7 ) simplifies to the following: As can be seen from the simple force equation, in contrast to Eq. (1), the magnitude of the force goes to zero as the forward velocity of the craft u o approaches the exit velocity of the propulsion pod u e . In this situation, as the craft transits through the air, each differential volume of air at rest with respect to the observer frame enters the inlet of the jet engine with a velocity of u e relative to the craft frame. This same differential volume of air then exits the jet engine with the same relative velocity u e as seen by the craft. From the perspective of the observer frame, this differential volume does not change state in any way as it enters and exits the jet engine. It is as though the craft picked it up and put it back in its original state completely undisturbed relative to the observer. This is, of course, why the magnitude of the force goes to zero as the forward velocity of the craft reaches the exit velocity of the jet engine. A direct analogy to this in space would be the implementation of the Bussard ramjet [2], whereby interstellar hydrogen is scooped up, compressed to a fusion state, and then expelled to act as propellant. In both of these cases, the initial resting state of the reaction mass in the observer frame has performance implications relative to the case of the reaction mass starting at rest in the vehicle frame. B. Particle Pair Production Pulling particles from the vacuum is analogous to scooping up a bucket of water from the side of a boat on a lake and then subsequently tossing it out the back. In fact, the analogy is actually quite good. The vacuum, as it is referred to, is often described as a “sea of virtual particles” that are thought to be continuously popping in and out of existence due to quantum uncertainty. However, this description is actually quite misleading. According to the currently accepted model of the vacuum (QFT), virtual particles are not localized in the sense that we think of particles, where we can draw a nice finite box around them and write down a list of their properties, such as mass, charge, spin, etc. Virtual particles are more like the small, nondescript undulations on the surface of the lake. Furthermore, the spontaneous conversion of these surface undulations into a discrete bucket-sized volume of water is similarly unlikely to spontaneously creating a real particle/antiparticle pair out of the vacuum. The probability of sufficient energy localizing itself, even invoking the uncertainty principle, is just vanishingly small. In the case of the lake, the mechanism for supplying the energy to localize the water is the act of scooping it into the bucket. The discrete collection of water (now at an energy state above that of the lake surface) can be accelerated to the rear of the boat, where it will then merge back into the lake, leaving no visible trace. To push the analogy further, if the craft was a catamaran, the water could be extracted, accelerated, and returned to the lake, all within the physical confines of the boat, with no external evidence that anything occurred other than the acceleration of the catamaran. Returning again to particle pairs, if sufficient energy can be localized, real particle pairs can indeed be generated. This is routinely done in particle accelerators, where a sufficiently high energy is localized by colliding real particle pairs to generate showers of other real particle pairs. Likewise, such a process can be realized with sufficiently high-energy photon pairs (soft gamma rays), as first calculated by Landau and Lifshitz in 1934 [3]. Alternatively, a sufficiently high density of less energetic photons (i.e., an intense electric field) is also capable of electron–positron pair production. The scale of the field required was first derived by Sauter [4] in 1931; however, the leading-order nonlinear corrections for the rate at which the vacuum decayed (pair production) was formalized by Schwinger [5] in 1951, and both the mechanism and the required field strength bear his name today. Later evaluation of Schwinger’s results, such as that of Nikishov [6] in 1970, showed that the actual rate of fermion pair production per unit volume was given by the leading-order term alone: Γ = ( e E ) 2 4 π 2 exp ( − π m 2 e E ) ↦ ( e E ) 2 4 π 2 ℏ 2 c exp ( − π m 2 c 3 e E ℏ ) (9) where the first expression is given in natural units ( ℏ = c = 1 ), and the second is given in more standard units. From the exponential term, it can be seen that there is a characteristic electric field, typically taken to be m 2 c 3 / e ℏ . This is known as the Schwinger limit, and it has a value of 1.3 ( 10 18 ) V / m . The prefactor is also quite large, so the overall rate becomes of order unity at around 2.7 ( 10 16 ) V / m , which is still quite significant. In a later analysis, Popov [ λ = 0.1 nm and a pulse duration of τ = 0.1 ps , generating a total pulse energy of 1 kJ ( P ≈ 4 ( 10 16 ) W ) would be just sufficient to generate one e + e − pair per pulse. where the first expression is given in natural units (), and the second is given in more standard units. From the exponential term, it can be seen that there is a characteristic electric field, typically taken to be. This is known as the Schwinger limit, and it has a value of. The prefactor is also quite large, so the overall rate becomes of order unity at around, which is still quite significant. In a later analysis, Popov [ 7 ] provided an example of a possible implementation using a superposition of high-energy x-ray laser pulses. At a wavelength ofand a pulse duration of, generating a total pulse energy of 1 kJ () would be just sufficient to generate onepair per pulse. Although the technical challenges are many, the main takeaway is that the possibility of a momentum-conserving propellantless (but not reactionless) spacedrive interacting with the quantum vacuum is perfectly consistent within the currently accepted framework of QFT. which is even worse than the photon rocket due to the energy required to create the particles. However, one could imagine a system that momentarily creates e-p pairs, accelerates them to generate a reaction force, allows them to annihilate back to the vacuum, and then somehow recycles that part of the energy that went into creating the particles initially. The momentum change of the vehicle will essentially be balanced by momentum flow into the underlying e-p field. In this case, the thrust to power becomeswhich is approximately that of Eq. ( 3 ), withreplaced by the net change in particle speed () given to the particle (or particle pair) after creation and before annihilation. The mechanisms by which the creation, acceleration, and postannihilation energy recovery might be engineered are likely nontrivial.

III. Energy Conservation A fairly recent critique of propellentless spacedrives was offered up in an analysis by Higgins‡ in 2015 and was independently mentioned in a presentation by Williams [8] in 2017, although no reference to the analysis appears in the 2017 paper. This analysis considers the possibility that such devices may actually violate energy conservation. The main points of the argument are summarized as follows. As was discussed in Eqs. (3–6), a common performance parameter in propulsion systems is the ratio of thrust to power. For typical systems using propellant, this ratio is a constant during operation [as Eq. (3) would imply], provided the propellant exit velocity is maintained at a constant value. A seemingly reasonable assumption would be to take this ratio as a fixed parameter for propellantless devices as well. By that reasoning, a constant thrust would develop constant acceleration; as a result, the velocity of the vehicle would increase linearly in time. This would further imply that the kinetic energy of the vehicle increased quadratically in time. However, referring back to the constant-power assumption, the rate of energy development in the system should only be linear in time, and not quadratic. The conclusion was that energy is not conserved. The analysis went on to equate the accumulated input energy to the instantaneous kinetic energy to derive what was identified as a breakeven speed that the vehicle could never exceed. This speed is given as V be = 2 P T (10) which happens to be equivalent to Eq. ( which happens to be equivalent to Eq. ( 3 ) if the system had been expelling propellant. The rationale for there being a maximum speed is that, as long as the kinetic energy has not exceeded the total input energy, then at least energy is not being created from nothing. However, the model as derived does not include any mechanism for the instantaneous energies to ever be unequal and, in fact, they must be equal at all times. So, there is a problem with the model. The problem can be seen by again considering a boat on a lake, but now you have a more conventional pair of oars at your disposal. As you row, assume that you apply the same amount of force during each stroke so that the thrust is constant, at least on average. As the boat goes faster, the rate that you have to move the oars to maintain that constant force increases. In other words, the rate of power generation must increase. This is, of course, well known by the relationship for instantaneous power delivered to a body: P = F ⋅ V . So, the assumptions of constant thrust and constant power cannot both be true. Taking the power to be the constant, the thrust must drop off as 1 / V . The change in speed and subsequent energy balance are therefore given by d V = F M d t = P M V d t ↦ M V d V = P d t ↦ 1 2 M V 2 = P t (11) where it is seen that the kinetic energy increases linearly with time, as it must. So, there is no implicit violation of energy conservation in the operation of a propellantless spacedrive. A. Massive Propellants in the Vehicle Frame This result then begs the question of why propellant-based rocket systems can then generate a constant thrust for a constant-power input. Along the same lines as Eq. (1), if one writes down the expression in an inertial frame for how the kinetic energy (KE) of the vehicle and the propellant change over an interval Δ t , this results in Δ KE Δ t = M V Δ V Δ t − Δ m Δ t V u e + 1 2 Δ m Δ t u e 2 ↦ P = M V d V d t + 1 2 m ˙ u e ( u e − 2 V ) (12) Taken in the rest frame of the vehicle ( V = 0 ), this reduces to the familiar result P = ( 1 / 2 ) m ˙ u e 2 , where the input power is equivalent to the jet power. However, remaining in the rest frame and rearranging Eq. ( P = 1 2 m ˙ u e 2 + V ( M d V d t − m ˙ u e ) = 1 2 m ˙ u e 2 (13) where the result at the far right comes from substitution of the momentum equation [Eq. ( V . More important, the increasing rate at which power goes into accelerating the vehicle is seen to be offset by the decreasing rate at which power goes into the propellant stream. As a result, both constant thrust and constant input power can be maintained over time for propellant-based systems. Taken in the rest frame of the vehicle (), this reduces to the familiar result, where the input power is equivalent to the jet power. However, remaining in the rest frame and rearranging Eq. ( 12 ) results inwhere the result at the far right comes from substitution of the momentum equation [Eq. ( 1 )] into the parentheses. So, the input power is equal to the jet power in all frames, regardless of. More important, the increasing rate at which power goes into accelerating the vehicle is seen to be offset by the decreasing rate at which power goes into the propellant stream. As a result, both constant thrust and constant input power can be maintained over time for propellant-based systems. B. Massive Propellants in the Rest Frame Consider again the case of the jet aircraft, where the propellant is an in situ resource residing in the rest frame. The power consumed can be expressed in the jet frame as P = 1 2 m ˙ o ( u e 2 − u o 2 ) (14) or alternatively in the observer frame as P = T u o + 1 2 m ˙ o ( u e − u o ) 2 (15) with both expressed under the same simplifying assumptions as Eq. ( m ˙ = ρ u o A o ) relative to the rest frame of the particles. Because both power and thrust are proportional to the mass flow rate, this difference is not relevant to the T / P ratio, which is then found to be T P = 2 u e + u o (16) or alternatively in the observer frame aswith both expressed under the same simplifying assumptions as Eq. ( 8 ). These are easily shown to be equivalent. For the jet operating in the atmosphere, the mass flow rate can be independent of the flight speed due to the compressor drawing the flow into the inlet even when the craft is at rest. In space, without assuming any particular mechanism for acquiring the reaction mass, one could consider as one limiting case the same situation as the jet, whereby the reaction mass availability is not a limiting factor. One might also assume as another limiting case that the mass flow rate will result from the motion of the vehicle and instead be proportional to the flight speed () relative to the rest frame of the particles. Because both power and thrust are proportional to the mass flow rate, this difference is not relevant to theratio, which is then found to be If the mass flow rate is not limited, the thrust to power for the system will start at a peak value of 2 / u e (the same as the propellant-carrying rocket) when the craft is stationary and eventually reach a minimum value of 1 / u e as the system reaches a maximum cruise speed of u e . However, if the mass flow rate is dependent upon the vehicle speed, this has important consequences that are not captured in the T / P ratio alone. Writing the mass flow rate explicitly in terms of the vehicle speed results in P = 1 2 ρ A o u e 2 u ¯ o ( 1 − u ¯ o 2 ) = 27 8 P max u ¯ o ( 1 − u ¯ o 2 ) (17) where P max is implicitly defined. One can easily verify that maximum power consumption occurs when the vehicle speed is one-third of the exhaust speed; when plugged back into Eq. ( P = P max , justifying the nomenclature. A similar analysis can be done for the thrust, which can then be seen to reach a maximum when the vehicle speed is half of the exhaust speed. For the performance analysis that follows, it is convenient to express the thrust in terms of the maximum power: T = 27 4 P max u e u ¯ o ( 1 − u ¯ o ) (18) whereis implicitly defined. One can easily verify that maximum power consumption occurs when the vehicle speed is one-third of the exhaust speed; when plugged back into Eq. ( 17 ), it results in, justifying the nomenclature. A similar analysis can be done for the thrust, which can then be seen to reach a maximum when the vehicle speed is half of the exhaust speed. For the performance analysis that follows, it is convenient to express the thrust in terms of the maximum power: Note that, under this model, the thrust starts at zero when the vehicle speed is zero, and then it returns to zero upon the vehicle achieving the exhaust speed. This would mean that the vehicle would need some other form of acceleration until a nonzero speed relative to the rest frame was achieved. Also note that the T / P ratio does not go to zero despite the thrust going to zero because the thrust power approaches a value of zero at exactly the same rate. This is an important consideration when applying Eq. (16) to performance evaluation in this case because it masks that there is an upper limit to the achievable flight speed. C. Massless Propellants So, where does this leave the photon rocket? Unlike a system that uses a massive, subluminal propellant, the velocity of the propellant in the photon rocket is always the same ( c ) in every frame of reference. Moreover, Eq. (4) leaves one with the impression that the ratio T / P is independent of velocity, which again poses a problem for energy conservation. Using a more precise treatment of the relativistic momentum conservation equation in the rest (observer) frame, applying thrust to a vehicle with momentum of p = γ M V , T = d p d t = M d ( γ V ) d t = M γ 3 d V d t = N ˙ ph h ν o c (19) and likewise, a more precise treatment of energy conservation in the rest frame shows P = d d t ( KE ) + N ˙ ph h ν o = d d t ( γ − 1 ) M c 2 + N ˙ ph h ν o = M γ 3 V d V d t + N ˙ ph h ν o (20) and likewise, a more precise treatment of energy conservation in the rest frame shows Substituting elements of Eq. (19) into Eq. (20) results in P = N ˙ ph h ν o ( 1 + β ) = T c ( 1 + β ) ↦ T P = 1 c + V (21) where, again, it is seen that, at constant power generation, the effective thrust drops off as the velocity increases, although the effect is small at nonrelativistic speeds. One final loose end is to look at the power consumed on board the spacecraft as seen from the rest frame. Transforming between the rest frame and the spacecraft frame requires accounting for both the Doppler shift of the photons and the rate at which the photons are generated due to clocks in each frame running at different rates. Starting from the spacecraft frame, the power is going entirely into generating just the stream of photons. Transforming then to the rest frame P = N ˙ ph s h ν s = ( N ˙ ph 1 − β 2 ) h ν o 1 + β 1 − β = N ˙ ph h ν o ( 1 + β ) = T c ( 1 + β ) = T ( c + V ) (22) results in the same expression relating power to thrust that was derived in Eq. ( where, again, it is seen that, at constant power generation, the effective thrust drops off as the velocity increases, although the effect is small at nonrelativistic speeds. One final loose end is to look at the power consumed on board the spacecraft as seen from the rest frame. Transforming between the rest frame and the spacecraft frame requires accounting for both the Doppler shift of the photons and the rate at which the photons are generated due to clocks in each frame running at different rates. Starting from the spacecraft frame, the power is going entirely into generating just the stream of photons. Transforming then to the rest frameresults in the same expression relating power to thrust that was derived in Eq. ( 21 ). Energy is shown to be conserved across all frames of reference. where it is seen that the kinetic energy increases linearly with time, as it must. So, there is no implicit violation of energy conservation in the operation of a propellantless spacedrive.

IV. Bounding Spacedrive Performance What can now be said about the performance of a spacedrive? The assessment of terrestrial propellantless propulsion systems might be considered overly simplistic for some of the proposed spacedrive models. Nevertheless, it appears that there are three regimes of operation, all energy and momentum conserving, but with different implications toward performance. The first is having a reaction mass that manifests in the vehicle reference frame. This applies to both the classic propellant-carrying spacecraft as well as to the creation of particles (massive or massless) from the vacuum in the vehicle rest frame. In this case, the thrust and power can independently remain constant as power transfer shifts from reaction mass to vehicle mass with changing vehicle speed. The second case is where the reaction mass manifests in the observer frame or, equivalently, the rest frame of the vehicle at the start of its journey. This would apply to utilization of existing mass in the interplanetary or interstellar medium (a la Bussard), or alternatively as a limiting case when particle creation from the vacuum does not occur in the vehicle frame. The last case is where there is perhaps no identifiable reaction mass but instead an accelerating force that originates in the observer frame. This case arises from the discussion of energy conservation considerations and does not refer to any specific mechanism. It is simply the observation that, at constant power input, an externally supplied force would have to drop off as 1 / V as the vehicle speed increased. For the following comparison, a nominal vehicle equipped with an unspecified spacedrive technology and a powerplant capable of producing 2 MW will be assumed. The spacecraft mass is taken to be 90 tonnes, and the thrust to power at the onset of thrusting is 0.1 N / kW . It should be noted that the actual purpose of the article by Williams [8] was to present a mission trade study comparing spacedrives to conventional electric propulsion systems for an assumed Earth–Mars cargo mission. Although, for this mission, his conclusion was that a spacedrive with the same T / P as we assume here did not trade favorably, he also offered threshold performance values for power and propulsion system advances where the results would shift in favor of a spacedrive-equipped system. For higher Δ v missions than the one chosen for his analysis, the spacedrive would already be favored. In the current analysis, we merely present a comparison of the T / P , Δ v , and acceleration profiles under the various assumptions for how the reaction force might be generated. The thrust to power as a function of mission elapsed time is presented in Fig. 1. In this figure, the vehicle frame (VF) line depicts the performance where a vacuum-derived reaction mass is generated in the moving frame of the vehicle and the energy required to create that mass is continuously recycled. The thrust-to-power ratio is seen to be constant and is given by Eq. (6), where δ v is selected as 2 ( 10 4 ) m / s to meet the assumed peak value of T / P . Fig. 1 Thrust-to-power curves for spacedrive-equipped vehicles. The observer frame (OF) lines represent examples of the reaction mass manifesting in the observer frame and not the vehicle frame. The OF-0, OF-0-2X, and OF-0-INF lines are examples where (similar to the jet aircraft) the mass flow rate is not tied to the vehicle motion. The OF-0 case is where the mass flow rate is assumed fixed at the initial value of 0.01 kg / s , and power consumption of the propulsion system drops as the vehicle speed increases. The OF-0-2X case is where the mass flow is allowed to increase such that the propulsion system is consuming the full 2 MW of available power until the mass flow reaches 0.02 kg / s , arbitrarily chosen to be twice the initial rate. Beyond this point, the mass flow is assumed to remain constant and power consumption of the propulsion system again drops as the vehicle speed increases. The OF-0-INF is a limiting case where the mass flow is allowed increase without bound such that the propulsion system draws the maximum 2 MW input power at all times. The OF-1 and OF-100 lines represent cases where the availability of the reaction mass is determined by the speed of the vehicle, with two different assumptions of the initial speed when the drive is brought online. In the case of OF-1, this initial speed is only 1 m / s , whereas in the case of OF-100, the initial speed is 100 m / s . The T / P is seen to drop off over time in all of the OF cases as the speed increases, in accordance with Eq. (16). Because the power drawn by the propulsion system approaches zero as the vehicle speed approaches the exit speed, the limiting value of T / P in all cases is 0.05 N / kW , or half of the initial value. This limiting value is approached asymptotically by all of the reaction mass flow rate-limited cases but is reached in a finite time (150 days) for the OF-0-INF case. Although an unbounded generation of reaction mass is likely not achievable, any assumed limit at this stage is arbitrary. The last case is that of the externally applied force, represented by the EX line. Because the force varies as 1 / V , it will in theory exceed the assumed initial value at the start of the mission, so it has been artificially capped at this value. This is the best that can be assumed without specifics of the implementation. At approximately 55 days, the thrust begins to drop below this initial value and then continues to drop over the course of the mission. Figure 2 shows the mission Δ v curves for the same scenarios. For the constant T / P case (VF), the velocity is seen to increase linearly in time because the spacecraft mass is constant and the thrust is constant, producing a constant acceleration. The next best performer is the externally supplied force (EX), which matches initially until the T / P starts to drop off as its speed-limited value falls below the maximum initial value. The three cases of observer frame reaction mass asymptotically approach their maximum achievable speed of 2 ( 10 4 ) m / s , but they are seen to get within 1% of this maximum speed at different times. Fig. 2 Mission Δ v curves for spacedrive-equipped vehicles. Although the T / P curves provide some insight into the performance of a hypothetical spacedrive system, it is perhaps more useful to consider the acceleration of the vehicle over time. The acceleration reflects not only the implications of thrust to power for a particular case but also factors in how much power the spacedrive draws over time. In some of the cases, the power consumed by the propulsion system is 2 MW at all times of the mission, whereas in other cases, the power consumed goes to zero over time. The acceleration curves are depicted in Fig. 3. Fig. 3 Acceleration curves for spacedrive-equipped vehicles. Here, the OF-0 curve is seen to start at a peak value of 0.0022 m / s 2 and decrease asymptotically to zero. The OF-1 and OF-100 curves start close to zero, reach a peak value of ∼ 0.0019 m / s 2 at different times, and then again decrease asymptotically to zero. The peak acceleration for each case corresponds to the section of highest rate of change in mission Δ v in Fig. 2. The VF case stays fixed at a constant acceleration of 0.0022 m / s 2 . The OF-0-2X and OF-0-INF follow a similar decreasing value until the OF-0-2X case reaches the mass flow rate limit of 0.02 kg / s , and then it decreases to zero over time similarly to the OF-0 case. The EX case maintains the initial acceleration of 0.0022 m / s 2 for the first 55 days, and then the acceleration begins to decrease to zero over time.

V. Conclusions The purpose of this analysis has been to convey that the idea of a spacedrive (a propellantless propulsion system) is in fact permitted under the currently accepted models of how the universe works. The generic ideas discussed were not meant to offer an engineering solution but were meant to demonstrate that discounting the idea of the spacedrive out of hand due to the loose application of the conservation laws is counterproductive. This is not to say that any currently proposed technologies should be accepted without scrutiny but that scrutiny involves as much justifying one’s criticisms as one’s support. A variety of momentum- and energy-conserving scenarios were presented, assessing how the power, thrust, and T / P expressions varied in the rest frame of an observer and the moving frame of the vehicle while remaining self-consistent, as they must. The implication of accumulating reaction mass in the rest frame versus creating it for free in the vehicle frame was also considered and presented as possible performance bounds for spacedrive systems.

A. D. Ketsdever Associate Editor

‡ Higgins, A. J., “Reconciling a Reactionless Propulsive Drive with the First Law of Thermodynamics,” 2015.

Acknowledgment The authors would like to thank NASA for organizational and institutional support allowing for the generation and publication of this Note.