Reasons why lightspeed travel is impossible

[Passionflower]

It's not actually trivial that constant proper acceleration is the same as constant F, since there are two different notions of "force" in relativity, as I mentioned earlier, and you seem to be talking about "force" as defined by the derivative w/respect to coordinate time of mass*coordinate velocity*gamma, as opposed to the four-force which is the derivative w/respect to proper time of the energy-momentum four-vector. Apparently it does work out that the notion of force you're using will be constant in the case of constant proper acceleration, despite the fact that this force involves coordinate velocity and time in a single inertial frame whereas proper acceleration deals with the coordinate acceleration in a series of instantaneously comoving frames (and the proper acceleration can also be understood as the magnitude of the acceleration four-vector), but this is a nontrivial fact which requires some proof.

These statements surprise me.
Isn't this basic special relativity?

 

[JesseM]

These statements surprise me.
Isn't this basic special relativity?

I'm not saying the proof would be particularly complicated, but I don't think it's so trivial that it requires no proof, or that someone who was familiar with basic SR but had never seen such a proof would find it immediately obvious how to prove it or even find it obvious that it's true in the first place.

 

[starthaus]

It's not actually trivial that constant proper acceleration is the same as constant F, since there are two different notions of "force" in relativity, as I mentioned earlier, and you seem to be talking about "force" as defined by the derivative w/respect to coordinate time of mass*coordinate velocity*gamma, as opposed to the four-force which is the derivative w/respect to proper time of the energy-momentum four-vector. Apparently it does work out that the notion of force you're using will be constant in the case of constant proper acceleration, despite the fact that this force involves coordinate velocity and time in a single inertial frame whereas proper acceleration deals with the coordinate acceleration in a series of instantaneously comoving frames (and the proper acceleration can also be understood as the magnitude of the acceleration four-vector), but this is a nontrivial fact which requires some proof.

It is very easy to show that $$\frac{dp}{dt}=F$$ where F is the 3-force I'm using throughout this thread and t is coordinate time is the same as $$\frac{dp}{d\tau}=F_M$$ where $$F_M$$ is the Minkowski force and $$\tau$$ is proper time. I did a short writeup that you can read https://www.physicsforums.com/blog.php?b=1911 .

 

[Austin0]

Classically, the rocket equation is:

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

With Vf being the final velocity of the rocket,
Ve the exhaust velocity,
Mf the final mass of the rocket
and
Mi the intial mass of the rocket.

Factoring in Relativity gives us:

$$V_f = c \tanh \left (\frac{Ve}{c} \ln \left(\frac{M_i}{M_f}\right) \right )$$

Which gives us the result that any ratio of initial mass to final mass less than infinite results in a final velocity of less than c.

A rocket accelerates forward by throwing mass backwards, the efficiency of this process depends on the exhaust velocity.

So if I'm in the "stationary" frame watching the rocket, the efficiency of the rocket will be determined by the difference between the rocket's velocity and the exhaust velocity.

Assume that the rocket is already moving at 0.9c relative to me and the exhaust velocity relative to the ship is 0.1c. Relative to me, the exhaust velocity is

$$\frac{0.9c-0.1c}{1- \frac{0.9c(0.1c)}{c^2}} =$$ 0.879c

and difference between rocket and exhaust velocity is 0.021c

When the rocket reaches 0.95c, the exhaust velocity with relative to me will be 0.939c for a difference of 0.011c , almost half that of the difference at 0.9c.

At a rocket velocity of 0.99c the difference drops off to 0.002c

From my frame, the rocket's efficiency drops off as it approaches c and the rocket's acceleration decreases the nearer it gets to c.

AM I correct in assuming that constant proper acceleration implies constant thrust and fuel consumption as measured within the accelerating system?
SO if propulsion was ion dirve of some kind and the thrust was measured by some finite number of ions per unit time, then it would seem to mean that not only would the momentum of the ions [per particle] decrease as you have shown here, but the number of ions/dt would also decrease by the gamma factor due to the internal dilation of clocks in the accelerated system.
SO the total combined observed decrease in thrust would be even greater. Is this at all correct??
I have just learned of the 2/$$\gamma$$³ coordinate acceleration falloff rate and am trying to understand why this would be the case.
Thanks