# Yet another relativistic acceleration question

Hello. I have read many of the posts on this forum concerning relativistic accelerations, constant accelerations to exceed the speed of light, etc. and one thing is still bothering me about the concept: why can one not accelerate constantly to achieve any arbitrary velocity as calculated in their reference frame? I have been searching for a definitive answer to this question for a long time and have yet to find one. Allow me to construct the question using the following thought experiment.

I am traveling on a rocket that is powered by a nuclear reactor that works in such a way that it converts matter into energy at some efficiency. On board I have a normal bathroom scale, a clock, and a calculator. The engine is constantly emitting energy in such a way that the ship is constantly accelerated at about 10 m/s2 (1 g). From my perspective aboard the ship, physics should be working exactly as they would on earth due to the general equivalence principle. Since I know that I am in a rocket ship, however, and I can calculate my acceleration by dividing my weight by my mass (which I assume is remaining constant in my reference frame). I can also record the time that has elapsed since I began to accelerate using my clock (the proper time of my reference frame). Though I have no actual velocity in my constantly accelerating frame, using my calculator I should be able to calculate an "extrapolated instantaneous velocity", v, that some distant observer would see, were I not warping spacetime by accelerating. After accelerating at 10 m/s2 for about 350 days I should finally reach an extrapolated velocity of 300,000,000 m/s (speed of light) and can even exceed this to reach any arbitrary velocity, provided I have enough fuel.

Since we assume the laws of relativity work on Earth they should also hold in this spaceship, so how can I have apparently exceeded the speed of light? In my reference frame my mass remains constant so it would never take an infinite amount of energy to cause me to accelerate from my point of view. In fact, even from a distant observer's point of view, even if my mass goes to infinity as my velocity approaches c, the mass of my fuel is also going to infinity. Because the ship converts the fuel's mass into energy to provide acceleration, I would thus have an infinite amount of energy to accelerate my infinitely massive ship. So, regardless of who's reference frame you're in, there does not seem to be any reason why I cannot reach the speed of light, or any velocity for that matter, as long as I am bringing enough fuel with me. How is this possible?

Thanks,
Rexx

You are asking what happens to a rocket with constant trust. Here is a breakdown of this:
http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html [Broken]

If we had a rocket that could turn the CMBR into an accelerate of 1g we could reach the Andromeda Galaxy in 28 years, 14 years if we don't plan to stop there.

(Maybe reflecting light from the brightest stars in our neighborhood might be a better option, it might be hard to find a thermal sink cooler then the CMBR :)

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This explanation is still based on the perspective of a distant observer, stationary with respect to the spaceship. As I stated before, from his point of view the rocket would indeed constantly become more massive as it's velocity approached c, and would thus need a constantly increasing energy source in order to propel it further. However, since the fuel's energy is based on its mass, which is also increasing, the ship would have a constantly increasing energy source, and thus the ship would continue to be able to accelerate indefinitely. So why can we not do this?

Dale
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Your "extrapolated instantaneous velocity" is not your velocity in any reference frame. It is not limited to c as velocity is. Rapidity is a similar quantity which has dimensions of L/T but is not speed and is not limited to c.

Actually rapidity is arctanh(v/c), which is still limited to velocities below c. The extrapolated velocity is not a real velocity in any reference frame because in this situation we are accelerating, so there is no inertial reference frame. This quantity represents an assumed velocity that one would have in their own reference frame were they to stop accelerating at that instant. In any accelerating reference frame you would calculate your instantaneous velocity in the exact same way. Regardless of whether this velocity is real or not, the real point of this question is what would a distant stationary observer see at any point in time after the spaceship's proper time has indicated that the ship's "velocity" has exceeded the speed of light?

Dale
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the real point of this question is what would a distant stationary observer see at any point in time after the spaceship's proper time has indicated that the ship's "velocity" has exceeded the speed of light?
An inertial observer, regardless of distance, would not see anything unusual. They would see the rocket going at some v<c as indicated by the relativistic rocket equation that utesfan100 linked to earlier.

Cleonis
Gold Member
Hello. I have read many of the posts on this forum concerning relativistic accelerations, constant accelerations to exceed the speed of light, etc. and one thing is still bothering me about the concept: why can one not accelerate constantly to achieve any arbitrary velocity as calculated in their reference frame?

In relativistic physics, when dealing with acceleration, it's very unfruitful to try and think in terms of the instantaneously co-moving frame.

How are you going to express the current velocity? If I understand you correctly you are proposing to use what amounts to newtonian dead reckoning: "If I have sustained acceleration so-and-so for such-and-such duration as measured by my own clocks, then my accumulated velocity must exceed c."

Now, your point of departure is some solar system, and you have a velocity relative to your point of departure. Conversely, your point of departure has a velocity relative to you. Given the current relative velocity the distance between you and your point of departure must be corrected for relativistic length contraction, and there is a corresponding correction for the relative velocity.

I think it's interesting to focus on the difference between newtonian physics and relativistic physics here.
The common element is that if you build up enough velocity you can travel from one end of the Galaxy to the other in decades, even though the same journey takes light thousands of years.
- In terms of newtonian physics you cut the cross-Galaxy journey time down by exceeding the speed of light.
- In terms of relativistic physic the time it takes to travel to the other side of the Galaxy is to be accounted for by time dilation and length contraction.

It's not that you cannot apply the counterpart of dead reckoning: you can. But it has to be relativistically correct dead reckoning. You must extrapolate the location of your point of departure, including correction for length contraction.

Hi Cleonis, thanks for your response. I couldn't agree more about the serious lack of fruit present when it comes to using instantaneous comoving frames in relativistic acceleration, which is where my whole problem with this problem lies. According to the link that was previously put in this post concerning the equations of a relativistic rocket, acceleration is defined exactly in this fashion, i.e. a combination of velocities as measured by a stationary observer (which will suffer from the gamma factor) and times measured by an inhabitant of the relativistic rocket (which will not). In my proposed thought experiment acceleration is measured directly within the reference frame of the rocket, by using a=F/m, not a = dv/dt, as v is essentially meaningless in one's own reference frame. The problem that I have with the problem of the rocket not really starting out at rest because whatever solar system it's starting out in is already moving is that the rocket is not starting with a velocity of 0, just some arbitrary velocity, and is accelerating constantly to achieve another arbitrary velocity that is c more m/s than the original one was. In this way, the important value is not your initial or final "absolute velocity" as compared with some "absolute reference frame" (which Einstein showed could not exist), but it is the amount of (proper) time one has been accelerating for that is important. Once this value reaches the speed of light (which is constant in any reference frame), I would think that something unusual would happen, because one would calculate themselves to have exceeded this fundamental velocity in their own reference frame.

Dale
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Once this value reaches the speed of light (which is constant in any reference frame), I would think that something unusual would happen
Nothing unusual would happen at this point in any reference frame.

The problem that I have with the problem of the rocket not really starting out at rest because whatever solar system it's starting out in is already moving is that the rocket is not starting with a velocity of 0, just some arbitrary velocity, and is accelerating constantly to achieve another arbitrary velocity that is c more m/s than the original one was. In this way, the important value is not your initial or final "absolute velocity" as compared with some "absolute reference frame" (which Einstein showed could not exist), but it is the amount of (proper) time one has been accelerating for that is important. Once this value reaches the speed of light (which is constant in any reference frame), I would think that something unusual would happen, because one would calculate themselves to have exceeded this fundamental velocity in their own reference frame.

Hi,

Here are all the equations you are looking for. Nothing "unusual" happens when v->c as long as v never equals c. No massive object can reach the speed of light. It is easy to show that the mechanical work required to do that would be infinite.

These are a more general form of the same equations posted before, which still define acceleration in a way differently than I have. In this set of equations acceleration is being measured by a distant observer which can then not be used in conjunction with the proper time of the traveller, as they do not refer to the same reference frame. If I were to look at the clock on the ship and then multiply it by the acceleration I directly observe the ship to have, these equations would be correct. However, in order to actually calculate the ships "velocity" in the same way that a traveller in the ship would, it would have to be based on the mass of the traveller (remember, this is how the traveller determined his ship's acceleration). Because the traveller's relativistic mass would be multiplied by a factor of gamma from my point of view, the relativistic acceleration would be multiplied by a factor of 1/gamma because we are using a = F/m. This is the correct acceleration. We can would multiply this by the proper time (which is again multiplied by a factor of gamma from our stationary POV) to find that the gammas cancel, and we have a velocity of simply a*tau. Also, how does one easily show that the mechanical work required for an object to accelerate itself to the speed of light is infinite?

Nothing unusual would happen at this point in any reference frame.

Why not? In my own reference frame I have just broken the light barrier, something that is supposed to be impossible. According to Einstein, all reference frames are equally valid, so I have definitely just gone faster than the speed of light in some way. Are you telling me that at the very least some hatch built into the universe filled with balloons and confetti wouldn't open up to congratulate me for achieving the physically impossible? But seriously, because of the gamma factor, if I have exceeded c in my reference frame, then this gives imaginary results to relativistic equations of motion in all other reference frames with a velocity less than mine. How is this not something unusual? What does a spaceship moving through imaginary space or time look like?

These are a more general form of the same equations posted before, which still define acceleration in a way differently than I have.

No, they don't. a=F/m. I can derive them for you in a few lines.

F=m*d/dt(v/sqrt(1-(v/c)^2)

so,

F/m=d/dt(v/sqrt(1-(v/c)^2)

Since F/m is constant, the above becomes a very simple differential equation with the solution:

v=at/sqrt(1+(at/c)^2)

For at<<c, you recover the Newtonian equation v=at

If you integrate one more time, you will get x as a function of a and t. Indeed:

dx/dt=at/sqrt(1+(at/c)^2)

x(t)=c^2/a*(sqrt(1+(at/c)^2)-1)

Again, for at<<c, you recover the Newtonian formula x(t)=at^2/2

Also, how does one easily show that the mechanical work required for an object to accelerate itself to the speed of light is infinite?

Work=Integral{Fdx}=Integral_0_to_c{m*d/dt(v/sqrt(1-(v/c)^2)*dx}=
=m*Integral_0_to_c{d/dt(v/sqrt(1-(v/c)^2)*vdt}=
=m*Integral_0_to_c{v/(sqrt(1-(v/c)^2))^3*dv}=oo

In my own reference frame I have just broken the light barrier,

No, you didn't. You need infinite energy to even reach light speed (see above).

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Dale
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Why not? In my own reference frame I have just broken the light barrier
No, in your own reference frame you are at rest. As I said before, the number you are calculating is not your velocity in any reference frame, so it is not limited to c.

According to Einstein, all reference frames are equally valid, so I have definitely just gone faster than the speed of light in some way.
No you have not. You have merely calculated a number that is greater than c, which is not prohibited by relativity.

because of the gamma factor, if I have exceeded c in my reference frame, then this gives imaginary results to relativistic equations of motion in all other reference frames with a velocity less than mine. How is this not something unusual? What does a spaceship moving through imaginary space or time look like?
Again, you are at rest in your reference frame, and you are not at v>c in any reference frame.

Janus
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In fact, even from a distant observer's point of view, even if my mass goes to infinity as my velocity approaches c, the mass of my fuel is also going to infinity. Because the ship converts the fuel's mass into energy to provide acceleration, I would thus have an infinite amount of energy to accelerate my infinitely massive ship.
Rexx

Not quite. Even if we accepted this as true, You haven't accounted for time dilation and other factors. Since from that distant observer's point of view time would be running slower and slower for you, you would be burning your fuel at a slower and slower rate.
This and the other factors mentioned would reduce your rocket's exhaust velocity as seen by this observer. For example, if your exhaust velocity was .5c as measured by you, then when you are moving at .75c relative to this observer he would measure your exhaust velocity as being .35c. At .9c it would drop to .172c, and at .99c it drops to .0192c.

Since it is the exhaust velocity that determines how efficient your engine is (how much change in velocity burning a given amount of fuel produces), Your outside observer will see the efficiency of your rocket falling off more and more as your velocity increases with respect to him. You'll never get that last bit of change in velocity to get you to c.

No, they don't. a=F/m. I can derive them for you in a few lines.

F=m*d/dt(v/sqrt(1-(v/c)^2)

There are several problems with this. First of all, whose reference frame are you talking about here? Is 'v' the velocity measured by the stationary observer? If so where did the gamma factor come from? Are you saying that the velocity of the spaceship will be observed to be v = gamma*v? Second, the equation to find the force in this situation is not correct from anyone outside of the spaceships point of view. One cannot use simply F = ma = dv/dt because this is assuming constant mass. Newton's real second law is F = dp/dt = d(mv)/dt = v(dm/dt) + m(dv/dt). Because the mass of the ship is changing (due to the expenditure of fuel) the simple F=ma equation does not work outside of the ship. This is the crux of my argument. Because the fuel being used to accelerate the ship is, itself, accelerating, it will undergo relativistic mass dilation just as the ship does. Because all fuels ultimately work by converting potential energy (in the form of mass) into kinetic energy, there will be an infinite amount of energy available to continue accelerating the ship to any velocity, provided there is enough fuel. This can be seen in your own equations by introducing gamma factors where they are missing (m = gamma*m0) and keeping track of whose reference frame you are talking about.

There are several problems with this. First of all, whose reference frame are you talking about here? Is 'v' the velocity measured by the stationary observer?

If so where did the gamma factor come from?

From F=dp/dt and p=mv/sqrt(1-(v/c)^2)

This is basic relativity, did you ever take a class?

Are you saying that the velocity of the spaceship will be observed to be v = gamma*v?

No, see above.

Second, the equation to find the force in this situation is not correct from anyone outside of the spaceships point of view. One cannot use simply F = ma = dv/dt because this is assuming constant mass.

Again, you are assuming wrong. See above. You should really consider taking a class before making all these allegations.

Fredrik
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why can one not accelerate constantly to achieve any arbitrary velocity
Because the work required to accelerate to speed v goes to infinity as v goes to c.

Also, how does one easily show that the mechanical work required for an object to accelerate itself to the speed of light is infinite?
See this post.

I am traveling on a rocket that is powered by a nuclear reactor that works in such a way that it converts matter into energy at some efficiency. On board I have a normal bathroom scale, a clock, and a calculator. The engine is constantly emitting energy in such a way that the ship is constantly accelerated at about 10 m/s2 (1 g). From my perspective aboard the ship, physics should be working exactly as they would on earth due to the general equivalence principle.
What you're describing is constant proper acceleration, not constant coordinate acceleration. You will have the same coordinate acceleration in every co-moving inertial frame, but that's a different frame at every point on your world line. Your world line, as described using an inertial coordinate system in which you started out at rest, is a hyperbola. (One of the curves in the spacetime diagram here).

Since I know that I am in a rocket ship, however, and I can calculate my acceleration by dividing my weight by my mass (which I assume is remaining constant in my reference frame).
The result is your proper acceleration, not your coordinate acceleration in the inertial coordinate system in which you started out at rest (or with some specified velocity v).

I can also record the time that has elapsed since I began to accelerate using my clock (the proper time of my reference frame).
Yes, that's the proper time, a coordinate independent property of the curve that represents your motion.

Though I have no actual velocity in my constantly accelerating frame, using my calculator I should be able to calculate an "extrapolated instantaneous velocity", v, that some distant observer would see, were I not warping spacetime by accelerating.
What you're doing has no effect at all on the geometry of spacetime.

After accelerating at 10 m/s2 for about 350 days I should finally reach an extrapolated velocity of 300,000,000 m/s (speed of light) and can even exceed this to reach any arbitrary velocity, provided I have enough fuel.
If you're going to calculate velocity as "acceleration*time", you have to use the coordinate acceleration and the coordinate time of a specific inertial frame, and what you're plugging into the calculation is the proper acceleration and the proper time. Two wrongs don't always make a right.

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Dale
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This is the crux of my argument.
No, the crux of your argument is a simple mistake. The number you are calculating is not velocity as you claim.

Perhaps something is happening to your clock that makes you arrive at this wrong conclusion? Perhaps your sense of length when you have relative motion to something, is different to when you when you were at rest with it? Wait a sec, doesn't the theory of Special Relativity say something about the non absoluteness of space and time? Might be worth looking into ;)

Because the work required to accelerate to speed v goes to infinity as v goes to c.

What you're doing has no effect at all on the geometry of spacetime.

I know the work would go to infinity. I was simply playing Devil's Advocate to generate some dialogue, not looking for a link. My point is that though the work would appear to go to infinity, so would the mass of fuel in my spaceship because of m=ym0. Since my engine (and any engine really) converts a part of the mass of fuel to the mechanical work of accelerating my spaceship, as the apparent velocity of my spaceship approaches c, it takes an infinite amount of work/energy to accelerate it, but I happen to have just that much energy because of my infinite fuel mass/energy. I'm saying that the argument of an infinite amount of energy needed to accelerate a ship to c does not work if the ship is propelling itself (i.e. converting its own mass into energy). Also, since when does acceleration not effect the geometry of spacetime? According to the principle of general relativity there is no way to locally distinguish an accelerating RF from a gravitational field, and I'm pretty sure that somewhere in GR Einstein might have mentioned that gravitational fields effect the geometry of spacetime.

Not quite. Even if we accepted this as true, You haven't accounted for time dilation and other factors. Since from that distant observer's point of view time would be running slower and slower for you, you would be burning your fuel at a slower and slower rate.
This and the other factors mentioned would reduce your rocket's exhaust velocity as seen by this observer. For example, if your exhaust velocity was .5c as measured by you, then when you are moving at .75c relative to this observer he would measure your exhaust velocity as being .35c. At .9c it would drop to .172c, and at .99c it drops to .0192c.

Since it is the exhaust velocity that determines how efficient your engine is (how much change in velocity burning a given amount of fuel produces), Your outside observer will see the efficiency of your rocket falling off more and more as your velocity increases with respect to him. You'll never get that last bit of change in velocity to get you to c.

Actually time dilation is the only way that my scenario can make sense. It is the fact that time dilation cancels out with mass dilation which makes the efficiency of the rocket constant in any reference frame. The efficiency of the engine mentioned in this thought experiment is not based on exhaust velocity, but on how well the reactor on the ship will convert mass into energy. Because the engine is a nuclear reactor, the "exhaust" being produced are high energy photons, which go out of the back of the ship at the speed of light, regardless of who's reference frame you're in, so the efficiency of the engine cannot be measured in this way. Because the rate of production of the photons is based on how rapidly mass is being converted into energy, the efficiency of the ship is based on dm/dt. If you are on board the ship, the efficiency will be dm0/dtau, m0 is the proper or rest mass, and tau is the proper time. From the point of view of a distant stationary observer, the efficiency will be dm/dt. Remembering that m=ym0 and t=ytau, we see that from the stationary observer's POV, the efficiency of the spaceship's engine is actually dm/dt = (ydm0)/(ytau) = dm0/tau, which is the same as is observed onboard. Therefore the spaceship's engine appears to have the exact same efficiency regardless of who's reference frame one is in.

In fact, even if the efficiency were to be measured in the manner that you described, with massive exhaust particles, the thrust isn't just a function of the exhaust velocity, but of the exhaust momentum. Therefore the mass of the exhaust particles must be taken into account as well. Even though they appear to be coming out slower from a distant observer's point of view, they are also becoming much more massive, and the ship is becoming less massive, allowing the ship to continue to accelerate with the same efficiency even from anybody's point of view.

Janus
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Actually time dilation is the only way that my scenario can make sense. It is the fact that time dilation cancels out with mass dilation which makes the efficiency of the rocket constant in any reference frame. The efficiency of the engine mentioned in this thought experiment is not based on exhaust velocity, but on how well the reactor on the ship will convert mass into energy. Because the engine is a nuclear reactor, the "exhaust" being produced are high energy photons, which go out of the back of the ship at the speed of light, regardless of who's reference frame you're in, so the efficiency of the engine cannot be measured in this way. Because the rate of production of the photons is based on how rapidly mass is being converted into energy, the efficiency of the ship is based on dm/dt. If you are on board the ship, the efficiency will be dm0/dtau, m0 is the proper or rest mass, and tau is the proper time. From the point of view of a distant stationary observer, the efficiency will be dm/dt. Remembering that m=ym0 and t=ytau, we see that from the stationary observer's POV, the efficiency of the spaceship's engine is actually dm/dt = (ydm0)/(ytau) = dm0/tau, which is the same as is observed onboard. Therefore the spaceship's engine appears to have the exact same efficiency regardless of who's reference frame one is in.
You are neglecting the fact that from the stationary observer's POV, those photons undergo a Relativistic Doppler Shift. For example, for a ship moving at 0.9c, they will be red shifted to a frequency 0.23 of that of their emission frequency.(compared to 0.53 according to classical Doppler shift). Since the momentum of these photons is related to their frequency, the stationary observer does see a drop off in the efficiency of the rocket.
In fact, even if the efficiency were to be measured in the manner that you described, with massive exhaust particles, the thrust isn't just a function of the exhaust velocity, but of the exhaust momentum. Therefore the mass of the exhaust particles must be taken into account as well. Even though they appear to be coming out slower from a distant observer's point of view, they are also becoming much more massive, and the ship is becoming less massive, allowing the ship to continue to accelerate with the same efficiency even from anybody's point of view.

Surely you are familiar with the rocket equation:

$$\Delta v = V_e \ln(MR)$$

With MR being the Mass Ratio of ship mass to fuel mass. Since your argument is that Ship mass and fuel Mass undergo the same mass dilation, then the exhaust velocity becomes the deciding factor.

If you take all the relativistic effects into account, this equation changes to:

$$\Delta V = c \tanh \left (\frac{V_e}{c} \ln(MR)\right )$$

which, in turn, becomes

$$\Delta V = c \tanh (\ln(MR))$$

Note that no matter how high the mass ratio, your final velocity remains below c.

Another way to show that the ships speed will never reach c, even according to the ship observer, is to use the addition of velocities theorem:

$$w = \frac{u+v}{1+\frac{uv}{c^2}}$$

Imagine that, as your ship accelerates, it drops off markers that retain the ship's velocity at the point at which they are dropped off.

Thus, after reaching 0.1c relative to your starting point, you drop off a marker. You then accelerate an additional 0.1c relative to this marker, and then check your relative speed to the starting point. You will, by the above formula, not get 0.2c, but 0.198c. If you drop off another marker and accelerate to 0.1c relative to it, you will now find that you have a relative velocity of 0.292c with respect to your starting point. No matter how many times you do this, accelerating to 0.1c relative to the last marker, you will never achieve a relative velocity of c with respect to your starting point.

You are neglecting the fact that from the stationary observer's POV, those photons undergo a Relativistic Doppler Shift. For example, for a ship moving at 0.9c, they will be red shifted to a frequency 0.23 of that of their emission frequency.(compared to 0.53 according to classical Doppler shift). Since the momentum of these photons is related to their frequency, the stationary observer does see a drop off in the efficiency of the rocket.

A very good point. However, you are still neglecting the fact that because the ship is accelerating, the fuel mass is being dilated. Because the usable energy of the fuel is some fraction of E = m0c2, and the resulting photons have energy E = hv (where v is the frequency, we have that (for the sake of argument we will say that the engine works at 100% efficiency and converts matter completely into photons) E = m0c2 = hv, or m0=hv/c2. Because of mass dilation, a distant stationary observer will see that the mass becomes m=ym0 at increased velocity. This means that the energy of the photons that the observer will see are also increased by the gamma factor, and will have energy of ym0=yhv/c2. Because h and c are constants, this implies that observer is measuring the frequency of the emitted photons as v=yv0, i.e. the frequency increases with the velocity, maintaining the original efficiency. I'll address the second part of your post in a little while.

As a side note, I'd like to thank everyone, including yourself, Janus, who is making the effort to actually have a meaningful dialogue with me concerning these topics instead of attempting to insult me or questioning my educational background (which includes a double major in physics and mathematics, and many "classes" btw, starthaus). The reason I have questions about the basics of relativity is because I am questioning the basics of relativity, something many physicists are not willing to do. I don't in any way think that relativity is wrong, simply incomplete. If I am wrong it should not be difficult to counter each one of my arguments with sound physical reasoning from traditional relativistic principles.

A very good point. However, you are still neglecting the fact that because the ship is accelerating, the fuel mass is being dilated. Because the usable energy of the fuel is some fraction of E = m0c2, and the resulting photons have energy E = hv (where v is the frequency, we have that (for the sake of argument we will say that the engine works at 100% efficiency and converts matter completely into photons) E = m0c2 = hv, or m0=hv/c2. Because of mass dilation, a distant stationary observer will see that the mass becomes m=ym0 at increased velocity.

No, it doesn't. For one, photons have zero rest mass.
Secondly, you have your frequency shift formula all wrong.
Thirdly, it is already known that energy is frame dependent, the correct question
is : "Is the rocket thrust frame dependent"? Janus already gave you the correct answer to this question. He also explained to you why your rocket can never reach c.

This means that the energy of the photons that the observer will see are also increased by the gamma factor, and will have energy of ym0=yhv/c2. Because h and c are constants, this implies that observer is measuring the frequency of the emitted photons as v=yv0,

You got your frequency shift all wrong.

i.e. the frequency increases with the velocity, maintaining the original efficiency.

This is also wrong as Janus already pointed out. Energy is frame dependent and frequency is not only frame dependent but also direction dependent. Indeed, if the rocket moves away from you, the perceived frequency is shifted DOWN (and so is the energy $$h \nu$$) because of the relativistic Doppler formula:

$$\nu= \nu_0 (1-v/c)* \gamma$$

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