Relative Maxiumum Velocity

[arkain101]

I can never let this relativity stuff down it seems. There is always some aspect that keeps popping back up. :naughty:

 

 

For a moment lets exclude the Special Relativity (SR) Equations and Theory and use only classical sense and formula.

 

 

Firstly, to define velocity, one could say it is the motion of an object relative to an observation frame.

 

So the question is, how fast can an object go relative to an observation frame?

 

 

To keep this simple, lets assume we have a hypothetical force that moves at velocity "C" speed of light.

 

If we place this force between any two objects in space regardless of their mass and use it between the two objects; each object will receive an exertion of a force.

 

 

In example 1:

 

We have Two objects (A and positioned with zero distance between them, they have equal masses. Also, we have an observation frame that is at rest relative to the intial state of A and B. Then, we apply this force between the two objects. They each take 50% of the velocity and fly apart 1/2 of C.

 

(this is just a simple example sharing the obvious behavior with the given forces)

 

 

 

The next question is, what is the fastest possible speed we can obtain for "one" of the objects if we adjust the mass of each object using the same hypothetical force?

 

Example 2:

(there are other ways to express an object accelerating, but I used this as the most simplistic example to support my point)

 

We give object A, Mass "mA" and object B, 1% of mass A. Now when we arange the objects side by side, and allow the force to accelerate them apart; according to the observer, object B will be traveling some 0.99 % the speed of light, while object A , accelerates, but very little.

 

(I will not bother doing calculations of actual velocities as it is not required for my main point)

 

 

A conclusion can be made that no matter how much energy is put between two objects, an object can never reach C because the force will be devided between the two objects. This is exactly the same conclusions that special relativity makes, however here we have reached it using a different process of reasoning than special relativity requires.

 

Is it not a fact that prior to even using the theory of special relativity, classical physics will demonstrate the impossiblity of C?

 

I think this example deserves its attention because when we hear hear some of the laymans explanations for special relativity we hear things like (as I heard on many physics tv programs and dvds) "As you keep putting more and more energy into an object and still more energy, where does that energy go? It has to go somewhere.. and it turns out Einsteins theory demonstrated that it went into the objects mass". These kinds of examples that you find quite regularly, makes no mention of the magical force used to accomplish such a task, or the classical laws that would govern such an event, as I have just brought forward. And I find That a poor form of explanation as it completely does away with classical physics from the get go. And uses the idea of a "god finger" pushing an object through space, and futhermore does not consider the point of observational reference.

 

 

 

If we prefer a more complex example such as a rocket; and we use a hypothetical rocket that ejects mass at nearly "C" and its mass is 99% fuel, we arrive at the same reasoning in examples 1 and 2. And that is, relative to an observer, under the laws of classical physics it is impossible for an object to reach C.

 

 

 

 

I am not attempting to debunk anything here, I would just like to use this specific example as a way of refreshing my mind on the subject. I am interested in responses including the equations, and explanations of why "even if classical mechanics gives these results, it just doesnt cut it". Or whatever responses one feels are neccessary.

[Boof-head]

You have to remember that, in physics-logic, a force isn't a velocity. Force is in fact an illusion; we "know" that F = ma; here acceleration is what displaces an equilibrium, which is constant velocity.

 

The equation of motion here for a mass, m, is derived from the force F and the known, or derived acceleration. It depends what you know about any two things in the equation so you can calculate the third thing; mass is real and acceleration is real, so is the derivative of a, or velocity (if you have mass). Again, motion is always measured in terms of distance and time; mass is measured by "extent" and by the way it reacts to imaginary forces which are in fact accelerations.

 

SR tells us there isn't a large enough force to accelerate mass to the speed of (massless) particles of light - which travel "at the speed of light", or accelerate to it in zero time. That is, no massive body or particle with mass can be accelerated to c, because it always only reaches a v (an equilibrium) after accelerating, therefore you are required to accelerate it for a long time. This time-factor is actually infinite time.

 

So if you accept a speed-limit c, and that a mass m can only reach a finite velocity, you have to accelerate it for an infinite time, so that: "you cannot accelerate mass to c, in less than infinite time"; the algorithm cannot "halt", which is impossible because "it does" when a mass reaches some (terminal) v.

[arkain101]

Thanks for response. I think it was able to help me narrow down my question and clarify what I was trying to say.

 

 

I'm not sure how to express this mathematically without "looking into it" to demonstrate that regardless of the force used to accelerate an object, it must always depend on the of the inertia of another object to supply the velocity.

 

What I am bringing forward here is that the desired velocity for the "object sent in motion" can never reach the equivalent velocity of the source of acceleration (be that the sources max velocity is C, or 100m/s).

 

In any case of acceleration, a proportional amount velocity exchange will occur between the greater and lesser mass.

 

example: If an electron is shot from earth from a hypothetical gun floating in space that can accelerate an electron to C almost instantaniously, an equal force will act in the opposite direction, in other words on the gun. Therefore, the gun will of be observed to move in the opposite direction. The only way to give the electron 100% of the potential velocity would be to apply infinite mass to the gun. Then it would behave as an "unmovable" foundation of which to push from.

 

But here lies the impossibility of infinite mass. Therefore, no "system" as it were can deliver 100% of the potential velocity to its projectile frame.

 

 

Edit: Added:

 

Come to think of it, when it comes to the mass of a high velocity object, is what I have brought forward here in any way what relativity is actually saying when it comes to infinite mass?

 

To elaborate:

 

Relativity theory calculates that the mass of a moving object nearing the speed of light will increase, specifically, to a proportion of the calculated gamma value.

If I remember correctly:

 

 

[math] \gamma = \frac {1}{\sqrt{1 - \frac {v^2}{c^2}}}[/math]

 

Or otherwise said, that calculations demonstrate that as the velocity of a 'massive' object increases its relativistic mass nears ever closer to infinity.

 

At the same time, the classical interpretation demands a infinite mass of which to propell from for any mass to reach the velocity C, propelled by a force with a maximum velocity of C.

 

When it comes to visualizing two bodies being blasted apart at relatively high velocities, the example of an electron shooting gun is poor in a sense due to the fact that the equal and opposite reaction force will not conserve momentum strictly in the form of motion. What I mean is that, when considering an example with something as small as an electron being shot from a much more massive body such as a gun, the equal opposite force between the electron and the atoms of the body that is accelerating the electron I would expect to be tranformed into other forms of motion/energy such as heat or EMR, as opposed to strictly macroscopic motion.

 

The same principles should still apply even at the atomic scale. That is, that the acceleration of any projectile will be hindered by a loss of potentially useful energy in the opposite direction as it were.

 

The conclusion of the classical interpretation to my understanding is that a body will not acquire infinite mass on its journey toward light speed. On the contrary, it will in fact be required to lose ALL of its mass in order to do so, and furthermore, in order to satisfy the conservation of energy, this lost mass of the projectile must be utilized in some fashion, where it could be transformed into some form of energy, and/or applied as mass to the body of which supplied the acceleration. This process exists, this is precisely what light is, a massless form of energy of a velocity C.

 

[math]E = MC^2[/math] : In some way, the [math]C^2[/math] can be explained as the equal opposite force acting on both the massive and non-massive bodies. In this classical model.

 

[math]C^2 = \frac {E}{M}[/math]

 

[math]C = \sqrt {\frac {E}{M}}[/math]

 

 

 

Is there not some kind of controversy occurring here between classical and relativistic physics?

[Boof-head]

No, you seem to have reached a conclusion which does stand up, acceleration and forces are conserved.

As you state, to prevent a recoil by any "gun" that can propel a particle with mass, the gun needs to have a lot of mass - actually only an infinite mass will do it.

 

Since infinite mass will collapse, or a particle will have zero possibility of escaping it, this completes the logical circle.

[arkain101]

Post #3 edited.

[Boof-head]

If you consider the Lorentz shift, given by the formula you posted, which for a massive body relates its velocity to "the rest of the universe", viz:

 

[math] \gamma = \frac {1}{\sqrt{1 - \frac {v^2}{c^2}}}[/math]; where [math] \gamma [/math] is a factor - the Lorentz one, determining a shift in perspective for velocity, against a constant v (like the planet we live on, say); which is then the global parameter. This speed constant, c is the limit for velocity of massive particles.

 

The linear velocity form is [math] x'\, =\, \gamma (x\, -\, vt); t'\, =\, a(t\, -\, bx) \;\;\;[/math]; where a and b are constants determined by Galilean transformation.

[actually a G transform will always yield: a = 1, b = 0, so how are they preserved in a Lorentz frame..?]

 

Gamma relates something about removing a value - the squared particle v / constant 'universal' v ratio from the "selection" it makes in space and time. This is deterministic - a massive particle is always in "the same spatial extent" whereas quantum ones can be in more than one spatial extent - or at least this is what we see...:huh:

[dkv]

relativity is all wrong because violates the principle of one frame one clock. From the realtive frame the relative time appears heterogeneous. I have discussed this in detail.

[Boof-head]

relativity is all wrong because violates the principle of one frame one clock.

"Relativity is all wrong" I assume means "Albert was up the creek"; your principle of one frame, one-clock is not meaningful; there is no such thing as "one frame", since GR/SR has 2 frames, the L-frame and C-frame, like any formalism that deals with "motion" between them (which means: relative motion).

 

How do you determine relative motion from your "one" frame?

[CraigD]

We have Two objects (A and B) positioned with zero distance between them, they have equal masses. Also, we have an observation frame that is at rest relative to the intial state of A and B. Then, we apply this force between the two objects. They each take 50% of the velocity and fly apart 1/2 of C.

 

(this is just a simple example sharing the obvious behavior with the given forces)

 

The next question is, what is the fastest possible speed we can obtain for "one" of the objects if we adjust the mass of each object using the same hypothetical force?

For a problem in classical mechanics such as this one, it helps to begin by identifing which quantities are conserved and which are not. In this case, based on Arkain’s description, it’s sensible to assume mass and momentum are conserved, and energy is not.

 

From that, we can calculate the change in energy of the system. Initially, A and B are at rest relative, so the system’s energy is 0. After, the energy is

 

[math]\frac12 M \left( \frac{C}2 \right)^2 +\frac12 M \left( \frac{C}2 \right)^2 = \frac14 M C^2[/math]

 

where [math]M[/math] is the mass of either body A or body B, which are give to be equal. Note that the quantity [math]\fracC2[/math] in this example is just the speed given in the problem’s description, not a special value related to the speed of light, because in classical mechanics, the speed of light isn’t anything special, and speeds can have any non-negative value.

 

Now we can consider the question of the greatest speed possible for one of the bodies by writing their equations, allowing the masses of the two bodies to be different and requiring the change in system energy to be the same as in the first example.

 

The simplest case is when the masses of the two bodies remain equal, but different than the initial give [math]M[/math]. In this case, we can write the after energy equation for the different mass [math]M'[/math] and speed [math]S[/math] and set it equal to the original energy,

 

[math]\frac14 M C^2 = \frac14 M' S^2[/math]

 

and solve for [math]S[/math]

 

[math]S = \sqrt{\frac{M}{M'}} \,\cdot C[/math]

 

The answer to the “what is the fastest possible speed” question, then, depends on what is the smallest possible mass [math]M'[/math]. As [math]M'[/math] approaches zero, [math]V[/math] approaches infinity, so, if there’s no limit on how small [math]M'[/math] may be, there’s no limit on how great [math]S[/math] will be.

 

If a hunch, however, that Arkain meant to consider cases where the mass of one of the bodies remains the same, and the two bodies no longer have the same mass - a “big gun, little bullet” scenario. In this case, assuming the mass of A is unchanged, the after energy equation is

 

[math]\frac14 M C^2 = \frac12 \frac{M}2 S_A^2 +\frac12 M_B S_B^2[/math]

 

Where [math]M_B[/math]is the mass of B, and [math]S_A[/math] and [math]S_A[/math] are the after speeds of A and B. To solve this, we need another equation, the equation for the system’s momentum

 

[math]0 = \frac{M}2 V_A + M_N V_B[/math]

 

[math]V_A[/math] and [math]V_B[/math] are the after velocities of A and B, vector quantities, but by choosing our coordinate system specially, we can relate them simply to the bodies speeds, and rewrite the momentum equation

 

[math]\frac{M}2 S_A = M_B S_B[/math]

 

or

 

[math]S_A = \frac{2M_B}{M} S_B[/math]

 

and from this, rewrite the after energy equation

 

[math]\frac14 M C^2 = \frac12 \frac{M}2 \left(\frac{2M_B}{M} S_B\right)^2 +\frac12 M_B S_B^2[/math]

 

and solve for [math]S_B[/math]

 

[math]S_B = \sqrt{\frac{M^2 C^2}{4M_B^2 +2M_BM}} [/math]

 

Again, the answer to the “what is the fastest possible speed” question is “unlimited, depending on the smallest possible [math]M_B[/math]”.

 

Note that it wasn’t necessary in any of the above to actually calculate a force or an acceleration. The event described could have involves large forces and accelerations that occurred over short time intervals, or small ones that occurred over long intervals, and the forces and accelerations could have varied over time in complicated ways. This is fortunate, because calculating the actual forces involved would have required more information, and more calculations. Because we know that in classical mechanics, changes in velocity can occur only due to acceleration, and acceleration only due to force, we know that some forces must have been involved, but need not know the details.

 

In short, classical mechanics don’t impose any maximum speeds. So the conclusion

… no matter how much energy is put between two objects, an object can never reach C because the force will be devided between the two objects. This is exactly the same conclusions that special relativity makes, however here we have reached it using a different process of reasoning than special relativity requires.

is incorrect, and the answer to the question

Is it not a fact that prior to even using the theory of special relativity, classical physics will demonstrate the impossiblity of C?

is No.

 

It’s important to note, that, despite its usefulness for calculating speeds smaler than a small fraction of the speed of light, it’s a near certainty that classical mechanics doesn't accurately describe reality for speeds greater than a small fraction of the speed of light.

[sanctus]

Actually what your classical reasoning proves is just that if you have a force able to ideally accelerate an object to a given speed v, then since you need two masses the force will always be split and hence you never reach the given velocity v but always less.

 

You do not prove c as a limiting speed, because in classical mechanics you can also imagine a force accelerating a given mass to eg.2c, then your 2 masses reasoning proves that no mass will ever reach 2c...

[Boof-head]

There isn't any requirement to establish a constant c in General Relativity; Einstein assumed this was the case in all reference frames, which was the Special theory's foundation and led to the deductive step of GR; we know we can do this in reverse, or uncover c by starting with GR, but we don't need to.

 

The speed of light exists independently of relative motion, is all GR needs. It appears as a ratio of the elastic constant in electric charge/magnetic charge, or permittivity/permeability -> permittance/permeance which is determinable without starting with c.