(1) From the length contraction equation, would [tex](L\sqrt{1-(v/c)^2} )^3[/tex] give the coordinate volume of an object? Here's the mass equation: [tex]M=\frac{m_0}{\sqrt{1-(v/c)^2}}[/tex] My second question is ultimately about the 'speed limit' of the universe. As seen by an observer, an object's mass will approach infinity as its velocity approaches c. From this we can say that, as seen by an observer, it will take a force equal to infinity to accelerate the object to c, and because it's impossible for anything to apply that force, we say that the speed limit of the universe is c. For the most part this makes perfect sense. As seen by an observer, an object approaching a velocity of c would experience the fallowing: length approaching zero, time approaching infinity, mass approaching infinity etc. This can be easily visualized, as something approaches c, it escapes more and more light, and in theory, if it reached c, the object would vanish, in turn revealing a zero length, infinite time, and a questionably visualized infinite mass. But what about the proper variables? They don't change with an increase in velocity. Mass, time, and length, by definition, don't change in the object's inertial frame. The object keeps its own rest mass. This is my main point as the 'speed limit' was initially set because the object's coordinate mass approaches infinity, but now I'm saying that an object's mass doesn't change in its inertial frame. That being said, I think it's obvious that the energy required for an object to accelerate itself to c, is in fact finite. Keep in mind, it's very important that the force be applied from within the object's inertial frame. If the force was applied from outside of the frame, it would take an infinite force for the object to reach c. The speed limit of c holds very true in special relativity. In theory, an observer will NEVER see an object traveling at or faster than c, for more than one reason. To be honest, I see no proof that the speed limit of c holds true in all cases. Obviously there is no proof that faster than light travel is possible, but even if it was very possible, we still wouldn't be able to observe that. To conclude, in theory, an object can reach the speed of light or greater with a finite force applied to itself from within its inertial frame. (2) Why wouldn't this statement be true? (whether or not we know how to initiate the above bolded phrase, doesn't change the theoretical case)
1. No, you don't take 1/gamma to the third power, since the length contraction is just in one direction. 2. I can't make sense of the bolded phrase. What would it mean to apply a force within the inertial frame? It sounds like you're going to pull your own hair or something. If you mean that you're going to do something that gives you a constant proper acceleration, that's not enough to get you to reach the speed of light.
I'm not sure what exactly what the bolded phrase would consist of, but that's not my point. I'll try to make a clear example here. We say you can't accelerate an object to c because the relativistic mass demands a force of infinity to do so. But what if the mass I wanted to accelerate wasn't relativistic at all, if I could "somehow" accelerate myself, it would only take a finite amount of energy to reach c. In this special case, I am my own observer so my rest mass is constant to me, and thus requiring finite energy. I'm just trying to show that in theory, nothing (yet), stops an object in its inertial frame from obtaining a velocity of c in this way.
This statement does not make any sense whatsoever. Sorry, but more nonsense. A rocket with a constant thrust will never reach the speed of light with respect to another object. Not even in the limit because the hyperbolic velocity addition function does in fact have no limit.
It's not just that I don't know how to accomplish what you're suggesting. It's that I don't know what you're suggesting. I don't think there's any way to make sense of it. What if I e.g. run behind you and give you a push once per second according to your clock. Even if we ignore the practical problems and how out of shape I am, you would still at best approximate the world line of an object with constant proper acceleration, and constant proper acceleration is definitely not enough to reach c.
That's not even close to a proper example of the special case I'm suggesting, and neither is the rocket example. There is no clear example that we could grasp, I already admitted that. Trying to find an example may or may not be hopeless. If applied a constant force per time on a rock, its acceleration would drop as its velocity approached c. That's an example of a force from outside of the inertial frame of an object, and becuase it's from outside of the inertial frame, we must use relativistic mass transformation to calculate the mass. If we are bounded to using the relativistic mass tansformation, then we no object with rest mass greater than 0 will ever travel at c or faster. So ask yourself, when are we not bounded to using the relativistic mass transformation? The answer is, when you are the object, when you are in the inertial frame, when you are observing yourself. Being the object in the inertial frame, we don't use the relativistic mass transformation, we use a simple m_0 = M. This is where you are getting confused. Because you are the object in the inertial frame, and you are supposed to be applying a force, you need to "somehow" apply a force to yourself. (what ever that means, I could be very possibly using the wrong words to describe; there is no example of this thus far) So, as you "apply a force to yourself" per time, you gain acceleration, and because your mass is not relative to your velocity, the amount of energy needed to accelerate you to c or more is finite. [tex]E=M_c*c^2[/tex] M_c signifies that the mass is constant for all v.
Please note: while the acceleration can be viewed from your frame only, velocity only exists when viewed between two frames. You can't escape the fact that you'll always measure your velocity to be below C.
The big problem here is much more fundamental than the speed limit of c. If I understand you correctly your idea expressed here violates the conservation of momentum, one of the most fundamental laws of the universe. If you want to invent a magical universe where momentum is not conserved then you are certainly free to decide that in your fantasy-land c won't be a limiting speed either.
As I said, the problem isn't to find an example, it's to properly define what you would like to find an example of, and your attempts to do that don't make sense. We seem to be back to pulling our own hair. (But internal forces cancel according to Newton's 3rd, as I'm sure you know).
It sounds like the OP is thinking about "proper velocity", not relative velocity. Proper velocity is momentum/mass, so it it not limited to c. There's plenty of info on the net about proper velocity, but for most purposes, it's not very useful. Al
No. It's momentum/mass, or proper acceleration times proper time. Or, If I travel to a star 10 ly away at 0.8c rel. velocity, I stop at star and divide rest distance by elapsed time on my clock to get (average) proper velocity. In this case, 1.33c is my (average) proper velocity. It's only useful in some cases if it isn't misused. Another way to look at it is, I left earth 7.5 yrs ago, now I'm 10 ly from earth, at rest with earth, 10 ly/7.5 yr is 1.33c. There is plenty of info about it on the net, but it's normally not very relevant. Al
True, but you're forgetting that the only time an object is observable is when its traveling at less than c. So the time that I was traveling at c, I wouldn't have been able to be measured, and so yes, at every point at which I could have been observed, it would read less than c. Very true, but as I said before, it's not an example of what I'm talking about, as I am not focused on making examples at this point. A few ideas could possibly involve light, but let's not get into personal theories. At this point it is just a theoretical case that's completely true. Given that we can "apply a constant force to our self", it would only take a finite amount of energy to accelerate oneself to c. As far as violating the conservation of momentum, I haven't even suggested an example yet, so you can't really assume so. One possible idea to get around this, off the top of my head, is some collision with light in a way where energy is transferred to you, but because the light's rest mass is zero, the negative 'action' that is demanded by the conservation of momentum would cancel out, therefore holding the conservation of momentum. (please do not quote this example) Bolded phrase is my main point of this topic.
There's no theoretical limit to how much or how long you can accelerate. You could accelerate at 500 G for 10,000,000 yrs. No problem. Your velocity relative to any other mass in the universe will still be measured to be < c. Al
Sure. As I said above, if you are going to throw one law out the window you may as well throw the rest out too. Just don't kid yourself that you are discussing anything other than fiction. It is not an assumption, if you apply a force to yourself which causes you to accelerate then momentum is not conserved. That is the nice thing about conservation laws: the specific example doesn't matter.
Interesting. You are right, I looked up the wikipedia Proper Velocity page. That is a really unfortunate name for this concept, it seems to have nothing to do with the usual things associated with the term "proper". Specifically, it is not invariant, and it is not a property measurable in an object's rest-frame. Thanks Al, it is nice to know that I can learn something even in the silliest threads!
I already stated that when velocity is able to be measured, it will always be less than c. You are not thinking of this in the correct way. I'll use a relativistic example in attempt to show you. First of all, we assume that the object is able to apply a force to onself, therefore 'dodging' the relativistic mass equation. Observer's frame [tex]t_1[/tex]:An observer witnesses an object increasing its velocity in an unknown manner. [tex]t_2[/tex]:The object's velocity is approaching c. The object's mass, as viewed by the observer, is seemingly approaching infinity, as well as the other transformations. [tex]t_3[/tex]:The observer clocks the object at a velocity slightly less than c, and witnesses it decelerate. Object's inertial frame [tex]t_1[/tex]:The object starts to accelerate, somehow harnessing a force from with its inertial frame, while still upholding the conservation of momentum. [tex]t_2[/tex]:The object's velocity approaches c, and upon a finite amount of time, the object reaches c, and possibly continues to accelerate. And remember, there is nothing wrong with this, its mass does not approach infinity to the force that was applied to the object, because the force was part of the inertial frame. [tex]t_{2 + vt}[/tex]:The object maintains its velocity for a period t, then decelerates to a speed under c. [tex]t_3[/tex]:The object is now traveling at a speed less than c. To conclude, yes you are correct, the observer never observed the object reaching c.
Given the circumstances, which by no means has been disproven thus far, it is true. Whether or not a clear example exists is the question, I've already made suggestions towards one. The concept itself is really not hard to understand. The proper properties of an object i.e. length, time, mass etc., does not change with velocity. From this we can say that it takes a finite amount of energy for the object to reach c, given that the force is 'proper' in and of it self, which is the hardest part to conceptualize.