What Are the Limits of Acceleration and Energy in Relativity?

[Frogeyedpeas]

So I have some unresolved issues with relativity in general (as in both SR and GR) which I would like to bring up and then discuss:

First, Cosmic Limit: It is logical to find that c = maximum velocity, but what is the maximum possible acceleration? No form of non asymptotic acceleration function a(t) can exist in this world because that would suggest that for some time interval [0,q]

\int^{Q}_{0} a(t) dt ≥ C which is simply not possible...

So does that mean that one can generalize the lorentz transformation to acceleration the way it is also used for velocity? What about for the next derivative, Jerk, and so on and so forth... If there does exist a lorentz transform for each of this then that means that one could potentially find a limit as to how high these numbers get, which could build the framework for an extention of SR to study notions such as the acceleration/deceleration of time for observers etc...

Second, Energy to Mass Equivalence: As of right now the most conclusive relationship for energy I have found is as follows:

E² = (M₀C²/((1-v²/c²)^1/2))² + (M₀VC/((1-v²/c²)^1/2))²

Where:

E = total energy
M₀ = Rest Mass
v = Velocity
C, c = speed of light


(The equation was too complicated for Latex to load)

Why is this equation once again limited to only velocity? Why is acceleration not a measure of energy, suppose two objects are of equal mass, equal velocity, but of different accelerations at that exact moment... they clearly have different energies and therefore (now extending to GR) will have a different gravitational pulls as well (and the consequences keep going from there on)

 

[PAllen]

You should get any book on special relativity to answer these questions in a systematic way. There are actually several ways force has been generalized. Most typical is to define 4-force as the derivative of 4-momentum, which is just mass time 4-velocity. Proper acceleration is the norm of 4-acceleration which is the derivative of 4-velocity. There is no limit to the magnitude of 4-velocity. I'm sure you've hear that no matter how great the force, an object never reaches c. Well the acceleration experience by it is proportional to the force, irrespective of the fact that its speed is changing only slightly.

All of the derivatives are with respect to proper time on an object's world line. 4-velocity is just (dt / d tau, dx/d tau, dy / dtau, dz /d tau) in some chosen coordinate system, where tau is proper time.

Given, where you are coming from, a short answer won't help you much. You should get a book.

 

[Bill_K]

And first, frogeyedpeas, you should get a book on classical mechanics.

suppose two objects are of equal mass, equal velocity, but of different accelerations at that exact moment... they clearly have different energies

The energy of a particle does not depend on its acceleration.

 

[Frogeyedpeas]

So Bill, is that to say that if I had a rocket at 1000 mph with mass R which would be 2000 mph within the next second (assuming it was using a fuel whose waste was of equal mass)

Versus, a bullet at 1000 mph with mass R where acceleration is 0

both would have the same energy?

from my viewpoint there is more force being acted on the rocket, which is being produced by the rocket, thereby it has more potential energy as well as (over the interval) more energy in general. However despite the fact it has more energy (by this definition of mine which may or may not be accurate) according to SR both those objects are of equal energy at that given moment in time.

 

[PAllen]

So Bill, is that to say that if I had a rocket at 1000 mph with mass R which would be 2000 mph within the next second (assuming it was using a fuel whose waste was of equal mass)

Versus, a bullet at 1000 mph with mass R where acceleration is 0

both would have the same energy?

Clearly there is more force being acted on the rocket, which is being produced by the rocket, thereby it has more potential energy as well as (over the interval) more energy in general. However despite the fact it has more energy (by this definition of mine which may or may not be accurate) according to SR both those objects are of equal energy at that given moment in time.

Bill's point, which I missed, is forget SR. Your belief here is completely wrong in classical (Newtonian) mechanics. This shows, as Bill said, it would be hopeless for you to study SR without reviewing classical mechanics.

The kinetic energy of a particle (classical or SR) depends only on its current speed. Force, per se, does not produce potential energy. Potential energy is associated with position in a conservative field, which can be represented by a scalar potential. But really, you need more than a few sentences. Just read any elementary book on classical mechanics (you don't need something advanced like Goldstein for this purpose).