The Planck Photon Rocket

The Ultimate Limits of the Relativistic Rocket Equation
The Planck Photon Rocket
Espen Gaarder Haug ⇤
Norwegian University of Life Sciences
January 11, 2017
UNARMED SERVICES TECHNICAL INFORMATION AGENCY: This information must not be distributed
or leaked to hostile extraterrestrial life or organizations supporting these. If this information is leaked, then
hostile aliens could be here before we know it! People, entities, and leak-organizations even suspected for
misconduct will be prosecuted and potentially used as rocket fuel! For unclassified documentations see
Abstract
In this paper we look at the ultimate limits of a photon propulsion rocket. The maximum velocity
for a photon propulsion rocket is just below the speed of light and is a function of the reduced Compton
wavelength of the heaviest subatomic particles in the rocket. We are basically combining the relativistic
rocket equation with Haug’s new insight on the maximum velocity for anything with rest mass; see [1, 2, 3].
An interesting new finding is that in order to accelerate any subatomic “fundamental” particle to its
maximum velocity, the particle rocket basically needs two Planck masses of initial load. This might sound
illogical until one understands that subatomic particles with di↵erent masses have di↵erent maximum
velocities. This can be generalized to large rockets and gives us the maximum theoretical velocity of a
fully-e cient and ideal rocket. Further, no additional fuel is needed to accelerate a Planck mass particle
to its maximum velocity; this also might sound absurd, but it has a very simple and logical solution that
is explained in this paper.
Key words: Relativistic rocket equation, photon propulsion, rocket load, maximum speed rocket,
Planck mass, Planck length, reduced Compton wavelength, electron.
1 Introduction
Haug [3] has recently introduced a new maximum velocity for subatomic particles (anything with mass)
that is just below the speed of light given by
v max = c
r
where ¯ is the reduced Compton wavelength of the particle we are trying to accelerate and l p is the
Planck length, [4]. This maximum velocity puts an upper boundary condition on the kinetic energy, the
momentum, and the relativistic mass, as well as on the relativistic Doppler shift in relation to subatomic
⇤ e-mail espenhaug@mac.com. Thanks to Victoria Terces for helping me edit this manuscript. Also thanks to Alan Lewis,
Daniel Du↵y, ppauper and AvT for useful tips on how to do high precision calculations.
1
l 2 p
¯2
(1)
1

2
particles. Basically, no fundamental particle can attain a relativistic mass higher than the Planck mass,
and the shortest reduced Compton wavelength we can observe from length contraction is the Planck
length. In addition, the maximum frequency is limited to the Planck frequency, the Planck particle mass
is invariant, and so is the Planck length (when related to the reduced Compton wavelength).
Here we will combine this equation with the relativistic rocket equation in order to assess how much
fuel would be needed to accelerate an ideal particle rocket to its maximum velocity. We will also extend
this concept to look at the ultimate velocity limit for a macroscopic rocket traveling under ideal conditions
(in a vacuum).
2 The Limits of the Photon Rocket
The Ackeret [5] relativistic rocket equation is given by 1
and solved with respect to velocity we have
1+ v
c
m 0 = m 1
1
v
c
! c
2Isp
✓ ✓ ◆◆
Isp
v = c tanh
c ln mo
m 1
(2)
(3)
where I SP is the specific impulse, which is a measure of the e ciency of a “rocket”, m 1 is the final rest
mass of the rocket (payload), and m 0 is the initial rest mass of the rocket (payload plus fuel). We will
assume that the internal e ciency of the rocket drive is 100 percent, that is I SP
c
=1. Thisisbasically
equivalent to a rocket driven by photon propulsion, or a so-called photon rocket, see [11, 12, 13]. Next we
are interested in estimating the amount of fuel needed to accelerate a subatomic particle (using a photon
propulsion particle engine) to the Haug maximum velocity, and we get
1+ vmax
c
m 0 = m 1 v
1 max
c
0 r
m 0 = m 1
B
1+ c 1
r
@
c 1
1
0 q
m 0 = m 1
@ 1+ 1
q
1 1
l 2 p
¯2
c
l 2 p
¯2
c
! 1
2
l 2 p
¯2
l 2 p
¯2
1
C
A
1
2
1
A
1
2
(4)
when ¯ >>l
q p, as is the case for any observed fundamental particle, we can approximate with a series
l
expansion: 1
2 p
1 l
¯2 ⇡ 1
2 p
2 ¯2 , and we get
0
1
m 0 ⇡ m 1
@ 1+1 1 l 2 p
2 ¯2
A
1 1+ 1 l 2 p
2 ¯2
0 1
m 0 ⇡ m 1
@ 2 1 l 2 1
2
p
2 ¯2
A
0
m 0 ⇡ m 1
@ 4
1 l 2 p
2 ¯2
l 2 p
¯2
l 2 p
¯2
1
A
1
2
1
2
(5)
Since we assume that ¯ >>l p, then this can be further approximated quite well by
1 See also [6], [7], [8], [9] and[10].

3
m 0 ⇡
s
4
m 1
l 2 p
¯2
2¯
m 0 ⇡ m 1 =2m p
l p
(6)
This means that in order to accelerate any particle (an electron, for example) to its maximum velocity
we need a particle rocket with two Planck masses of fuel, 2m p ⇡ 4.35302 ⇥ 10 08 kg. The velocity of the
electron will then be
✓ ✓ ◆◆ 2mp
v max = c tanh ln
m e
= c
⇣ ⌘ 2 2mp
m e
1
⇣ ⌘ 2
⇡ c⇥0.999999999999999999999999999999999999999999999124
2mp
m e
+1
(7)
The Einstein relativistic mass 2 of the electron is then equal to the Planck mass. This is the same
maximum velocity as given by [1, 2]. These calculations require very high precision and were calculated
in Mathematica. 3
Remarkably, the concept of two Planck masses being used as fuel to reach the maximum velocity for
a subatomic particle holds for any particle. Naturally, this can only work because the maximum velocity
of heavier particles is lower than that of lighter particles.
The equation 6 above is only a good approximation as long as ¯ >>l p, which is the case for all
observed subatomic particles so far. In the special case, we initially have a payload equal to the Planck
mass particle with m 1 = m p we must have ¯ = l p, so we need to use the equation as it was before we
applied the series approximation expansion
0 q
m 0 = m 1
@ 1+ 1
q
1 1
0 r
1+ 1
m 0 = m p
B
@
r
1 1
l 2 p
¯2
l 2 p
¯2
l 2 p
l 2 p
l 2 p
l 2 p
1
A
1
C
A
1
2
1
2
m 0 = m p
✓ 1
1
◆ 1
2
= mp (8)
In other words, as we accelerate the Planck mass particle to its maximum velocity we will need no
extra mass as fuel. At first this may seem absurd, as we will always need some energy for the acceleration.
However, the solution is simple; as [1] has shown, the Planck mass particle must always be at rest when
observed from any reference frame, the Planck mass particle and the Planck length are remarkably
invariant entities. The maximum velocity of a Planck mass particle is
r s
lp
v max = c 1
2 = c lp
1
2 =0 (9)
¯2
The Planck mass particle is the very turning point of light. What is the velocity of light at the precise
instant when it changes direction? According to Haug, at this very instant it will be at rest. In the very
next instant, the Planck particle will be dissolved into energy.
3 Relation to the Tipler Factor of 2
It is also worth noting that our result of two Planck masses as the initial q mass needed to accelerate a
payload for any 4 l
fundamental particle to its maximum velocity, v max = c 1
2 p
¯2 ,seemstobeconsistent
2 See [14] and[1].
We used several di↵erent set-ups in Mathematica; here is one of them: N[Sqrt[1 (1616199 ⇤ 10^( 41))^2/(3861593 ⇤ 10^( 19))^2], 50],
where 1616199 ⇤ 10^( 41) is the Planck length and 3861593 ⇤ 10^( 19) is the reduced Compton wavelength of the electron.
An alternative way to write it is: N[Sqrt[1 (SetPrecision[1.616199 ⇤ 10^( 35))^2, 50]/(SetPrecision[3.861593 ⇤ 10^( 13))^2, 50]], 50].
4 Except the Planck mass particle itself.
l 2 p

4
with an interesting result presented by Tipler in 1999, see [16]. Tipler discussed what he called “ultrarelativisitic
rockets,” these are rockets basically moving at a speed very close to that of light. Tipler has
shown that for a photon rocket, the total initial mass-energy that is needed to accelerate the payload to
a velocity very close to that of light only is twice the payload energy-mass as measured from the Earth
(the relativistic mass of the payload when the payload has reached its final velocity, very close to the
speed of light). The initial mass is still enormous compared to the rest-mass of the payload, but is only 2
in relation to the final relativistic payload mass. We also get the same result from the maximum velocity
formula and analysis above. The relativistic mass of any fundamental particle traveling at its maximum
velocity is
q
c 1
m 1
=
vmax
2
c 2
c
s
1
m 1
l
r
l 2 ! 2
= m 1
p¯
p
c 1 ¯2
c 2
= mp (10)
and we find that to accelerate any particle (payload) to this velocity we need two Planck masses of
initial mass (energy). In more general terms, we can write this as
r
1
m 0
m 1
v2 max
c 2
= 2mp
m 1
l p¯
=
2m p
v
m 1
0 s
l 2 12
p
u @c 1 A
t ¯2
c 1
c 2
= 2mp
m 1
l p¯
=2 (11)
Our result can be seen as the subatomic derivation/connection to the same result as given by Tipler.
However, our result derived for subatomic particles seems to provide additional insight. According to our
theory, the Tipler result of factor 2 is also valid at rocket velocities considerably below c, ifthesubatomic
fundamental particles accelerated are super heavy, that is significant relative to the Planck mass 5 . And
again in the special case of a Planck mass we actually have a maximum velocity of zero and no additional
fuel is needed to accelerate the object that is at rest. Yet, this can only hold true if the Planck mass
particle only lasts for an instant.
4 Maximum Velocity of Rocket Ship
The maximum velocity of any composite object (even a nucleus) is likely to be limited by the fundamental
particle with the shortest reduced Compton wavelength from which it is constructed. In other words, the
speed limit of a rocket is limited by the heaviest subatomic “fundamental” particle it is built from. When
this particle reaches its maximum velocity, that is given by
v max = c
r
it will first turn into a Planck mass particle and then will burst into energy. If this type of fundamental
particle is a significant part of the macroscopic object (spaceship) we are traveling in, then the whole ship
is likely to be destroyed at the moment we reach this velocity. If the proton was a fundamental particle,
then the maximum velocity of a rocket traveling under ideal conditions (in a vacuum) would be 6
v = c
s
1
1
l 2 p
¯2
(12)
l 2 p
¯2
P
= c ⇥ 0.99999999999999999999999999999999999999705 (13)
For comparison, at the Large Hadron Collider in 2008, the team talked about the possibility of
accelerating protons to the speed of 99.9999991% of the speed of light [15]. When the Large Hadron
Collider went full force in 2015, they increased the maximum speed slightly above this (likely to around
99.99999974% of the speed of light). In reality, if a proton consists of a series of other subatomic particles,
then the speed limit given above for a proton will not be very accurate. Alternatively, we could have
looked at the reduced Compton wavelength of the quarks that the standard model claims make up the
proton.
5 The Ultimate Limitations on Nano-scale Bullets
The analysis above also gives some ultimate limitations on small weaponry. One could think of the
limitation on an electron rocket gun, for example. Electrons accelerated to their maximum velocity
5 In other words for subatomic particles with much shorter reduced Compton wavelength than the ones observed so far.
6 Here assuming l p =1.616199 ⇥ 10 35 and ¯P =2.10309 ⇥ 10 16 .

5
would have enormous kinetic energy compared to today’s primitive small arms, but even so, they would
have less kinetic energy than one might expect, based on the content in certain physics textbooks that
give no limits on how close v can get to c, see[17]. The kinetic energy from a single bullet would be
approximately
k e =
q
1
m e
v 2 max
c 2
✓ ◆
m ec 2 1 1
=¯hc
⇡ m pc 2 ⇡ 1, 956, 149, 410 Joule,
l p
¯e
while (for example) the kinetic energy from a 44 magnum revolver with a 240-grain (16 g) bullet
traveling at 500 m/s is only k e,44m ⇡ 1 2 0.016 ⇥ 5002 =2, 000 Joule. This means that the photon rocket
gun has about 978,075 times more impact energy than a 44 magnum; that is an enormous di↵erence. One
of the additional benefits of such a gun is the massive quantity of bullets it could carry, each bullet having
a rest mass of only 2m p ⇡ 0.000044g. (When someone comes with a 44 magnum, show them your Planck
0.000044 magnum gun and tell them size does not necessarily matter!). One could potentially think of
each bullet as one Planck mass of matter and one Planck mass of antimatter (plus an electron?). When
the antimatter and matter Planck mass are united, this could convert the fuel into acceleration photon
energy (photon rocket fuel), at least hypothetically.
Such a rocket-gun would likely never run out of bullets, a 50-gram magazine would give 1,148,628
bullets. Of course, one would need containers for the fuel and so forth. Hopefully such a gun will never
come into existence, but as with many new inventions (and equations), the emerging capabilities can be
used for good and for bad. If such advanced technology were built into our i-Phones, for example, it
could potentially be used as an energy generator to provide heat and electricity for an entire house for
months, if not years. And in case of an alien invasion, one would be happy to have a seemingly innocent
energy-generating device that could convert itself into a very powerful gun.
6 Summary and Conclusion
The maximum amount of fuel needed for any fully-e cient particle rocket is equal to two Planck masses.
This amount of fuel will bring any subatomic particle up to its maximum velocity. At this maximum
velocity the subatomic particle will itself turn into a Planck mass particle and likely will explode into
energy. Interestingly, we need no fuel to accelerate a fundamental particle that has a rest-mass equal to
Planck mass up to its maximum velocity. This is because the maximum velocity of a Planck mass particle
is zero as observed from any reference frame. However, the Planck mass particle can only be at rest for
an instant. The Planck mass particle can be seen as the very turning point of two light particles; it exists
when two light particles collide 7 . Haug’s newly-introduced maximum mass velocity equation seems to be
fully consistent with application to the relativistic rocket equation and it gives an important new insight
into the ultimate limit of fully-e cient particle rockets.
7 See [1] foradetaileddiscussionandpresentationofanatomismparticlemodel.

6
Appendix
This shows a slightly di↵erent and slightly more complex way to derive the same result. In the case of a
photon rocket, when combined with Haug’s maximum velocity for subatomic particles, we have
✓
artanh
✓ ✓
mo
v max = c tanh ln
v max
c
vmax
c
◆
✓
= tanh ln
✓ ◆
mo
= ln
m 1
vmax
e
artanh(
c ) = mo
m 1
m 0 = m 1e
m 0 = m 1e
◆◆
m 1
✓ ◆◆
mo
m 1
vmax
artanh(
c )
0 s
l 2 1
p
c 1
artanhB
¯2
C
@ c A
m 0 = m 1e artanh r
m 0 = m 1e
Further, when ¯ >>l p we can use a series approximation,
m 0 ⇡ m 1e
m 0 ⇡ m 1e
0
m 0 ⇡ m 1
@ 2 1
2
m 0 ⇡ m 1
v uuut
4
1
0 s
1
2 ln 1+ 1
B s
@
1 1
q
1
0
1
2 ln 1+1 1 l 2 1
p
B 2 ¯2
@
1 1+ 1 l 2 C
A
p
2 ¯2
0
1
2 ln 2 1 l 2 1
p
B 2 ¯2
@ l
1
2 C
A
p
2 ¯2
1 l 2 p
2 ¯2
l 2 p
¯2
1
A
1
2
l 2 !
p
¯2
l 2 p
¯2
l 2 p
¯2
1
C
A
l 2 p
¯2 ⇡ 1
Further, when ¯ >>l p, then then this can be very well-approximated by
l 2 p
¯2
l 2 p
¯2
1 l 2 p
2 ¯2 ,thisgives
(14)
(15)
m 0 ⇡
s
4
m 1
l 2 p
¯2
2¯
m 0 ⇡ m 1 =2m p
l p
(16)
This is the same result as we obtained in the main part of the paper using a slightly easier derivation.

7
References
[1] E. G. Haug. The Planck mass particle finally discovered! Good bye to the point particle hypothesis!
http://vixra.org/abs/1607.0496, 2016.
[2] E. G. Haug. A new solution to Einstein’s relativistic mass challenge based on maximum frequency.
http://vixra.org/abs/1609.0083, 2016.
[3] E. G. Haug. The gravitational constant and the Planck units. A simplification of the quantum realm.
Physics Essays Vol 29, No 4, 2016.
[4] M. Planck. The Theory of Radiation. Dover 1959 translation, 1906.
[5] J. Ackeret. Zur theorie der racketen. Helvetica Physica Acta, 19:103–2509,1946.
[6] W. L. Bade. Relativistic rocket theory. American Journal of Physics, 310(21),1953.
[7] K. B. Pomeranz. The relativistic rocket. American Journal of Physics, 565(34),1966.
[8] G. Vulpetti. Maximum terminal velocity of relativistic rocket. Acta Astronautica, 85(2):81–90,1985.
[9] U. Walter. Relativistic rocket and space flight. Acta Astronautica, 59(6),2006.
[10] A. F. Antippa. The relativistic rocket. European Journal of Physics, 30:605–613,2009.
[11] G. G. Zelkin. A photon rocket. Priroda (Nature), 11:1–9,1960.
[12] V. Snilga. There will be no photon rocket. Bulletin of the Astronomical Institute of Czechoslovakia,
7:31–33, 1960.
[13] P. Burcev. On the mechanics of photon rockets. Bulletin of the Astronomical Institute of Czechoslovakia,
15:79–81,1964.
[14] A. Einstein. Zur elektrodynamik bewegter körper. Annalen der Physik, (17),1905.
[15] G. Brumfiel. LHC by the numbers. News: Briefing, Nature, Published online 9 September, 2008.
[16] F. Tipler. Ultrarelativistic rockets and the ultimate future of the ultimate future of the universe.
NASA Breakthrough Propulsion Physics Workshop Proceedings, 1999.
[17] E. G. Haug. Modern physics’ incomplete absurd relativistic mass interpretation. and the simple
solution that saves Einstein’s formula. http://vixra.org/abs/1612.0249, 2016.