Heh! Or short-lived cosmic-ray breakdown particles are detected on the surface of the earth, whereas, in Newtonian physics, they couldn’t be, as they don’t live long enough to get from the upper atmosphere to sea-level.
(This, per my old cosmology professor, who mostly knew what he was talking about. It was a fun class. We had several creationists in it, and I think they drove the poor old prof to drink!)
Again, you atre trying to muddy the waters. The reactant is reacting to the rocket as it was when it was ejected. The rocket, according to Einstein, is perfectly at liberty to claim that it is stationary, as is the man throwing the snowball. In both cases, the ejectant is being ejected at a certain velocity, and will give a certain push in the opposite direction. Use the applicable equation, you will get the same push for each occurence. There are no other reference points apart from the rocket/skater.
All observed effects can be accounted for by a much simpler explanation.
For example :-
Muons undergo a deceleration - up to 20G has been mentioned. Nothing to do with SRT.
Clocks round the world - acceleration, deceleration, differing gravity potentials. Nothing whatsoever to do with SRT.
Ring laser - the speed of light is WRT the local gravitational field, not to the receiver (observer). If WRT the receiver, the two beams would arrive at the receiver at the same time. This does not happen.
. . If I travel at close to the speed of light towards Proxima Centauri, the distance between the Earth and Proxima Centauri is no longer 4 light years, it is 2 light years. This is not a visual effect, it is real.
My brother Tim set off at the same time as I did towards Proxima Centauri in a less powerful rocket, so he travels slower than I do. The distance between the Earth and proxima Centauri is no longer 4 light years, it is 3 light years. This is not a visual effect, it is real. . . .
Hamster King and Triopus both said :- So what’s the problem? You see your brother moving at a certain speed that is less than c. Your brother sees you moving at a certain speed that is less than c. We, back on earth, see both of you moving at speeds that are less than c.
My problem is that the universe has contracted by 50pc for me, and by 25pc for my brother. How does the universe know what speed I am doing to contract the correct amount? For every single person in the universe, it contracts the correct amount which of course is potentially different for every person in the universe. This is supposed to be a real occurance according to Einstein. So this effect is real, and yet a stick half in water is not really bent, it is only visual - please explain the difference. I am sure that a mathematician could come up with a formula which will show that the stick is really bent, and we cannot tell because when somebody puts their hand into the water to touch the stick, their hand undergoes the same transformation that the stick does, all molecules bending at the same rate and at the same time to render the process undetectable to our feelings.
Mendel Sachs in 1985 concluded that the time dilation of special relativity is unreal, being only an appearance, and that a travelling twin will not age differently because of it than his stay at home twin.
Mendel Sachs again in 2004 “The reason that nothing can move faster than c is that in SR, c is the maximum speed of propogation of (any type of) force. The reason that a body moves (effect) is that it was caused to do so by a force (originating in another body). If the body were to move faster than c the force could not catch up with it to make it move the way it does!”
Note the clear inference of the ability to travel faster than c.
Let’s say each snowball travels at 1000 m/s when it’s thrown and accelerates the skater by100 m/s.
So, after the first snowball is thrown, the skater is traveling 1100 m/s relative to it.
After the second snowball is thrown the skater is also traveling at 1100 m/s relative to it. But he’s traveling at 1200 m/s relative to the first snowball. The two snowballs are moving apart at 100 m/s.
The skater is always moving at 1100 m/s relative to the last snowball thrown. He’s going faster and faster relative to the first snowball thrown. And the line of snowballs is itself spreading apart with adjacent snowballs moving at 100 m/s relative to each other.
Are we in agreement so far? This is simply Newtonian mechanics.
But as the skater throws more and more snowballs, relativistic effects start to take over. From the skater’s perspective the line of snowballs starts to experience Lorentz contraction, particularly the part of the line that farthest away and has the highest velocity relative to him. He’s still moving at 1100 m/s relative to the last snowball thrown, but his velocity increase relative to the first snowball is no longer linear. Instead of each new snowball increasing his speed by 100 m/s relative to the first snowball, it only increases it by 99 m/s, or 98 m/s, or 70 m/s, or 20 m/s, or 1 m/s. Eventually he reaches the point where he has to throw thousands of snowballs to increase his velocity relative to the first snowball by 1 m/s. Even though he still sees his velocity relative to closer snowballs increase by 100 mph just as it did when he started out.
Yes. That’s why its called relativity. Time and distance are different in different reference frames. It’s not that the universe magically shrinks to accommodate a moving observer. It’s that what distance is depends on where you’re measuring from.
Your argument seems to boil down to: Relativity makes no sense if you assume a universal reference frame for measure distance. I agree. You need to stop assuming that there’s a universal reference frame that you can measure things relative to.
What do you mean by ‘real’?
Mendel Sachs is a crackpot.
How are these ‘explanations’ supposed to work? The muons, being decelerated, would decay long before they arrive at the ground.
Or are you saying that these are due to general relativistic effects? Do you accept those?
As is anybody who disagrees with you/ Einstein?
Here is a list of some more crackpots according to the above definition :-
Don Lincoln (Fermilab) 2005. “…I know of no practitioner that takes relativistic mass [increase] seriously…”
Max Born 1962. “The contraction is only a consequence of our way of regarding things and is not a change of a physical reality.”
H A Lorentz. “But I never thought this had anything to do with real time.”
H Poincare. Same as Lorentz.
W G V Rosser 1964. “It must be stressed that the theory of special relativity does not say that one cannot have velocities exceeding the velocity of light in vacuo…”
Now here is a possible new definition of a crackpot. Anybody who believes that for every single person in the universe, it contracts the correct amount which of course is potentially different for every single person in the universe. This is supposed to be a real occurance according to Einstein.
HMHW took objection to this statement. “Muons undergo a deceleration - up to 20G has been mentioned. Nothing to do with SRT.”
and said “The muons, being decelerated, would decay long before they arrive at the ground.”
A clock in a gravitational field runs more slowly (according to GRT) by :-
T = To / sqrt( 1 - ( 2GM / Rc^2 )) . As (according to GRT) gravity and acceleration (which includes deceleration) are equivalent, the muons experience (and please don’t nitpick that word) gravity of 20G as above. They therefore last longer, and reach the ground.
I accept some of GRT, not all of it.
THK quote. “Yes. That’s why its called relativity. Time and distance are different in different reference frames. It’s not that the universe magically shrinks to accommodate a moving observer. It’s that what distance is depends on where you’re measuring from.”
So the universe does not magically shrink. Yes, you are correct there. But “the distance depends on where you are measuring from”. How does a distance shrink if the universe does not? This might be a mute point, but we are not discussing where the observer is, we are discussing how fast he is travelling. “What the distance is depends on where you are measuring from” is just another way of saying that the universe shrinks from that perspective.
Now let’s go back to the rocket. The rocket is propelled forward by the reaction of the thrust. With the analogy of the snowballing skater, you referenced that thrust back to an imaginary/arbitrary IFR and stated that as seen from that IFR, he cannot increase his velocity past lightspeed. But this is logical nonsense. The snowballs are thrown from the skater’s hand, and the reaction motor is in the rocket, not in your arbitrary IFR, so the thrust and the skater/rocket accelerate by the same amount for each snowball thrown or for each second of thrust. The observer in your example, in the arbitrary IFR, is just that - an observer. He takes no part in the acceleration of the rocket, which is done internally.
It’s just simple geometry. Nothing any more complicated than that, say, the x and y coordinates of something change under a rotation. Your statement is akin to the disbelief that the x coordinate is different for different observers, different frames of reference. Space and time are like x and y coordinates, and Lorentz transformations are the ‘rotations’ that mix them – there’s nothing paradoxical, controversial, or even all that challenging conceptually about this.
You can’t cherry-pick physical theories; they come as a whole, not in parts to be individually accepted and rejected. In particular, the equivalence principle (which you now seem to accept) and the fact that gravitational fields lead to time dilation directly imply special relativistic effects; special relativity is just general relativity in the absence of a gravitational field.
Take two points in a plane, p[sub]1[/sub] and p[sub]2[/sub], with their respective coordinates being (x[sub]1[/sub], y[sub]1[/sub]) and (x[sub]2[/sub], y[sub]2[/sub]). Their ‘x-distance’ is given by x[sub]1[/sub] - x[sub]2[/sub]; their ‘y-distance’ correspondingly by y[sub]1[/sub] - y[sub]2[/sub]. Their ‘Euclidean distance’ is given by the Pythagorean d[sup]2[/sup] = (x[sub]1[/sub] - x[sub]2[/sub])[sup]2[/sup] + (y[sub]1[/sub] - y[sub]2[/sub])[sup]2[/sup].
A rotation of the coordinate frame by some angle sends the coordinates of the points to different coordinates. This entails a change in both ‘x-distance’ and ‘y-distance’. Somebody only considering x-distance might think this paradoxical: how can the same physical situation be described by different x-distances? How can the same two points be at different x-distance for me, than they are for you?
But of course, his confusion would merely result from the fact that he considers a quantity that is ultimately only an artifact from a particular choice of coordinates to be physically meaningful; but the only physically meaningful quantity is the Euclidean distance, which indeed stays the same under all rotations.
The same is the case with special relativity. Here, we call the ‘x-distance’ length, and the ‘y-distance’ time; and again, it is not either that has to be invariant in different descriptions, but a combination thereof. This is just a reflection of the fact that space and time don’t exist on their own, but rather, are parts of what we today call spacetime. This spacetime, however, is not Euclidean, but Minkowskian, and so the proper distance measure to use is s[sup]2[/sup] = (x[sub]1[/sub] - x[sub]2[/sub])[sup]2[/sup] - (ct[sub]1[/sub] - ct[sub]2[/sub])[sup]2[/sup], where the ‘-’ I’ve highlighted essentially tells you that the t-direction is a timelike one, as opposed to a spacelike one. The transformations that keep this ‘length’ invariant are just the Lorentz transformations, so from these simple considerations, the whole of special relativity follows. It’s thus nothing more complicated than plane geometry, except for a change in sign.
As to the physical justification of this, note that this ‘Minkowski distance’ vanishes whenever the ‘x-distance’ is equal to the ‘y-distance’, which I’ve written suggestively as c (the speed of light) times t, the time. Thus, for all points (commonly called ‘events’ in this context) that are removed from each other by a spatial distance such that a light ray has enough time to traverse this distance in the time that separates both events, the ‘Minkowski distance’ is 0; for all events such that light would have more than enough time to get from one to the other, it is positive; and for all events such that light doesn’t have enough time to travel between them, it’s negative. This just encodes the empirical finding that light travels at the same speed in all frames of reference; and it leads in particular to the finding that nothing can move faster than the speed of light, anymore than there is a distance between two points in the Euclidean plane that is shorter than a straight line.
But the frame I referenced WASN’T imaginary. It’s the reference frame of the first snowball thrown. And while it was arbitrary I had to pick SOME frame to use as an example. Any other frame would have yielded the same result.
Yes. I acknowledge this. The skater speed relative to the last snowball thrown is ALWAYS the same.
The observer in my example is THE FIRST SNOWBALL thrown, so it certainly did take part in the acceleration of the skater.
We’re trying to keep this simple, right? In this universe all that exists are the skater and the snowballs. Pick any snowball as your “observer” and the skater will still never move faster than c relative to it.
The mistake you’re making is assuming that you can simply add two accelerations together to determine their cumulative effect. That’s true in Newtonian physics, but not relativity.
The mistake you are making is that the observer you are talking about does not actually do anything, he does not take part in the acceleration of the rocket ship for instance. In your snowball example he does take part in the first acceleration, but then he is out of the loop so to speak. According to SRT, that rocket ship could have been travelling at 0.5 c before the accelerations started anyway, there is no way to tell absolute motion. You are correct in one respect - from the FR of eg the first observer to throw the snowball, the skater will not be observed to reach c, bnut this is kinematical only. Only the observer who observes fom the FR with the motive force in it can be invoked. Einstein/Lorentz had two frames and only two frames in the equations. One has the motive force in it, the other has in it the object being acted upon. A particle accelerator is a perfect example. The force is generated in one frame, and is used to push a particle around a track. The particle is the other frame. The observer is in the FR of the particle accelerator. All experiments, without exeption, which “prove” time dilation et al, have been done under those circumstances. The motive force in the particle accelerator is electro magnetic, and is limited by definition to c. The particle accelerator is run up and pulls or pushes the particle. As the particle approaches c, the accelerator cannot accelerate it any more, giving the mistaken impression of a mass increase (all the energy goes to increase the mass…). No experiment to detect time dilation et al has ever been done in which the object undergoing acceleration carries its own motive force and is accelerated to an appreciable fraction of the speed of light by it. This experiment could not be done at present, as the rocket when reaching c would disappear from view, and no force from Earth could reach it to react with it to test its mass or check its clocks or measure its length for example. A light powered ship is envisaged, this being powered by light pressure on its massive sail. This ship would definitely be limited to c, as its motive force is the Sun, which is in a different FR to the ship, so its light could not possibly push the ship any faster than c relative to the Sun itself. This would be seized upon by relativists (if there are any left by then) as proof that nothing can travel faster than light.
H Margenau stated “A theorem of mathematics can be true, yet have no bearing on reality.”
I prefer this statement :- The real world can always be expressed by mathematics, but mathematics cannot always represent the real world. A good example to follow this up with is this
‘Suppose we have a cubical vessel whose volume is 8 cubic feet, and we wish to find the length of one of its edges … We let x be the required length, and all we have to do is solve the equation x3 = 8. But this equation has three solutions, viz 2, [(-3)1/2 – 1], -[(-3)1/2]+1, all having the same mathematical validity. But we know that the only one of these solutions that can possibly correspond to the reading of a measuring rod is 2 …’
Similarly, Lorentz’s equations may be good mathematically, but do not correspond to the real world.
HMHW quote. “It’s just simple geometry. Nothing any more complicated than that, say, the x and y coordinates of something change under a rotation. Your statement is akin to the disbelief that the x coordinate is different for different observers, different frames of reference. Space and time are like x and y coordinates, and Lorentz transformations are the ‘rotations’ that mix them – there’s nothing paradoxical, controversial, or even all that challenging conceptually about this.”
The interesting word there is conceptually. Einstein was talking about reality. With velocity, lengths DO shorten, time DOES dilate, mass DOES increase. All this happens differently for every single observer in the universe, and we are supposed to believe it is a real physical occurence. Just because the maths says it can happen does not mean it can actually happen. x^3 = 8 : is the solution 2 or is it [(-3)1/2 – 1] or -[(-3)1/2]+1 ?
Correct me if I’m wrong Tom, but you seem to be saying that the relativstic rocket is both at rest and (eventually) exceeding c in it’s own (accelerated) rest frame. Clearly it’s a rest in it’s rest frame, not that it would be a problem for special relativty if it’s coordinate velcoity in a non-inetial frame were to exceed c as special relativyt only says that objects with real, non-zero mass must always travel at below c in inertial frames.
Well, just as x and y-distance behave differently to every observer. Or do you have a problem with that, to? I can paint you another picture if it’s not clear, but it’s really simple: get some graph paper out, plot two points at x[sub]1[/sub] = 3, y[sub]1[/sub] = 2, x[sub]2[/sub] = 5, y[sub]2[/sub] = 7, – or whatever values you see fit! --, then rotate the axes of your coordinate system, and observe the change in both x and y-distance, corresponding to the changes in length and time. It’s just that simple!
What I am saying is that there is no way of telling if the rocket is at rest or not. The converse of this is then automatically true - there is no way of detecting its speed. Of course its speed can be measured WRT any other FR, but what is the speed of that one? etc etc.
I see that we agree that objects can exceed light speed. As the Earth is a non IFR, and as is a rocket accelerating at 1G, that rocket can then exceed light speed WRT the Earth.
To HMHW. I don’t have a problem at all with the maths. Read what Margenau has to say. Read what I said. My problem is with reality. I am travelling towards Proxima Centauri, and the distance from me in my rocket to Proxima Centauri cannot be 2LY for me, and 1LY for another observer who is at that moment alongside me but travelling faster in the same direction, despite what the maths says.
Of course I have a problem with distances being different for every observer. That is what this whole discussion has been about. Of course it is simple mathematically or graphically, but space cannot be rotated at a whim like a piece of paper, and moreover rotated differently for every observer. The effects of SRT are not real, they are kinematical. Even Einstein himself realised this in 1921, and despite what was said on this forum a while ago, I have found no evidence that he did not hold this view for the rest of his life. No experiment which shows a relativistic effect has ever been done where the motive force is in with the FR being accelerated. They have all been done in particle accelerators, where the motive force is in one FR (the particle accelerator) while the object (particle) being accelerated is [in] another.
What difference does “taking part in the acceleration” make? If I’m standing beside a train track I can measure the speed of a passing train. I don’t have to “take part in the acceleration” of the train to observe it.
Right. Which is why we’re discussing the skater’s velocity only in RELATION TO THE FIRST SNOWBALL. It doesn’t matter what velocity the skater has relative to some outside observer.
What are you talking about? The skater is the one throwing the snowballs. And what do you mean by “kinematical”? If one doesn’t reach c “kinematically”, then how does one reach c?
Acceleration isn’t required to produce time dilation. GPS satellites experience two different forms of time dilation. They’re moving relative to ground observers, so their clocks tick slower (relative to us) and they’re higher in the gravity well, so their clocks tick faster (relative to us). The two time displacements predicted by special and general relativity match the actual displacement of the GPS clocks.
And exactly that realization, plus the realization that light moves at c in every frame of reference (as predicted by Maxwell’s equations), leads to special relativity.
But the relative speed of reference frames is all we ever have access to, anyway, so that data determines the full physical situation. Of course, you could imagine the whole set of frames moving relatively to some ‘absolute background’, and at arbitrary speed, but that motion would be wholly unobservable and without effects, so it can be set to zero and the background abolished, as Occam would tell us.
This isn’t right. Even relatively to non-inertial frames, nothing can ever exceed the speed of light.
Not despite, but because of! That length contraction is exactly what the math says. Your problem is just that you don’t like what the math says, and for some reason think that reality should behave according to your preconceptions, rather than according to logical conclusions drawn from experimental data.
But that’s exactly what the notion of a reference frame is about! Every observer, due to having his own viewpoint, uses a different set of coordinates to place things in space and time. These may be rotated to one another, displaced by a constant amount, or moving with a constant velocity, and with respect to the physical content of the universe, these rotations and displacements can’t make any difference – that’s Galilean relativity. Your complaint, in a Galilean universe, would simply be that it can’t be right that two different observers have two different notions of x-distance. But of course, x-distance is not a physically meaningful quantity, if viewed on its own – the Galilean universe is a manifold, a continuum with x, y, and z coordinates that can be rotated into one another.
Now, our universe is not Galilean, but Einsteinian relativistic, and the appropriate means to ‘translate’ between two different frames of reference, two different coordinate systems are not Galileo’s, but Lorentz’ transformations. This Einsteinian universe, in which you complain that x-distance is different for every observer, is, however, a manifold, a continuum with x, y, z, and t coordinates that can be ‘rotated’ into one another. This isn’t any more complicated than the Galilean case, and not any more paradoxical.
If you are going to assume a mass increase with velocity, and that follows from the Lorentz equations, that mass increase has to be WRT a FR. The only FR that makes sense to use is the one with the force in it. All experiments to show mass increase have been done under those circumstances.
The Hamster King Quote:
Originally Posted by tomh4040
The mistake you are making is that the observer you are talking about does not actually do anything, he does not take part in the acceleration of the rocket ship for instance. In your snowball example he does take part in the first acceleration, but then he is out of the loop so to speak.
What difference does “taking part in the acceleration” make? If I’m standing beside a train track I can measure the speed of a passing train. I don’t have to “take part in the acceleration” of the train to observe it.
I agree on that point, but we are talking here about the Lorentz equations which are all linked, and the one about mass increase is definitely referred back to the FR with the force in it; so are the others.
Quote:
According to SRT, that rocket ship could have been travelling at 0.5 c before the accelerations started anyway, there is no way to tell absolute motion.
Right. Which is why we’re discussing the skater’s velocity only in RELATION TO THE FIRST SNOWBALL. It doesn’t matter what velocity the skater has relative to some outside observer.
It doesn’t matter about the first snowball. The acceleration from each and every snowball is the same as the mass of the skater does not increase relative to it.
Quote:
You are correct in one respect - from the FR of eg the first observer to throw the snowball, the skater will not be observed to reach c, bnut this is kinematical only.
What are you talking about? The skater is the one throwing the snowballs. And what do you mean by “kinematical”? If one doesn’t reach c “kinematically”, then how does one reach c?
I have used the word kinematically exactly as Einstein used it. In this context it means in appearance only.
Quote:
No experiment to detect time dilation et al has ever been done in which the object undergoing acceleration carries its own motive force and is accelerated to an appreciable fraction of the speed of light by it.
Acceleration isn’t required to produce time dilation. GPS satellites experience two different forms of time dilation. They’re moving relative to ground observers, so their clocks tick slower (relative to us) and they’re higher in the gravity well, so their clocks tick faster (relative to us). The two time displacements predicted by special and general relativity match the actual displacement of the GPS clocks.
I used acceleration because that is how a particle in an accelerator is moved. I stand by my statement above. Can you contradict it? Acceleration produces speed, and speed produces time dilation, as does gravity. The GPS clocks were synchronised to an ECI, and stay synchronised despite their being a continual acceleration between them. They were synchronised using LET not SRT. The speed of light is invariant in the ECI, but not in the FR of each satellite. See one of my previous postings quoting TVF.
Since it’s the simplest hypothesis to account for the experimental data, I’m sure Occam would be delighted.
Then tell me: why should the notion of distance be the same for all observers? It’s perfectly consistent for it not to be. You’re still stuck with the idea that space and time ought to be separate entities, but they just aren’t – like x-distance and y-distance in the Galilean universe, they’re just two coordinates, and can be rotated into one another.
Try to accelerate your car on a frictionless plane, or in space. What, you can’t? But then the source of the acceleration can’t be contained within the car alone, can it? Because indeed, it’s the friction between the tires and the road that makes it possible to impart an equal but opposite momentum to the road, assuring conservation of momentum, and moving your car. There is no system such that it can accelerate itself out of itself. But anyway, even if there were, it would not pose a problem for special relativity, since that’s merely about relative velocities, and the details of the acceleration or force just don’t enter. But the point seems to continue being lost on you…
No, any (inertial) frame can be used – the Lorentz transformations connect all inertial frames, and give the experience of an observer within any of these frames. And wrt all those frames, there is no motion faster than c. And again, the notion of a frame of reference ‘with the force in it’ may make sense in George Lucas movies, but not in physics. Forces act between objects, and lead to a change in motion, such that both objects will no longer be in the same frame of reference; since the force between those objects acts on both, it can’t be ‘in’ either frame.
What is lost on you is – . No, I will not let this get personal, perhaps you could try that aproach as well.
That whole paragraph above is totally wrong, and designed to be misleading. A car has friction to the road through its tyres, of course it does, but it carries its own engine. The rocket ship in space also carries its own engine, and is in a frictionless medium, yet it works, and does not violate the conservation of momentum.
Again, you guys should sing from the same hymn sheet.
[Quote= These are my own pants ]
…not that it would be a problem for special relativty if it’s coordinate velcoity in a non-inetial frame were to exceed c as special relativyt only says that objects with real, non-zero mass must always travel at below c in inertial frames.
[/quote]
I aked a question earlier, I got no reply. Here it is again. Do you believe that if something is correct mathematically, it must be correct in the real world? I can’t remember who it was I asked, so I am now asking you (and anybody else who wants to answer).
It was assumed a few posts ago, that I now embraced the equivalence principle. I do not, and have had another look at the problem. Here it is again in another form.
A man in a train travelling along an embankment watches the embankment apparently move backward. He then experiences a jerk forward (WRT the train), and the speed of the embankment passing by reduces. He assumes it was caused by the brakes being applied. This is not bound to be the case however. It could be that the train was at rest, with the embankment moving backward, and then a gravitating body appeared in front of the train. The gravity from this body then caused the embankment to behave so that its backward velocity is reduced.
So the gravity from this body effects the man, the embankment, but not the train?
Why not the train?
Good idea…
Well, that was a short-lived New Year’s Resolution!
That’s because it ejects part of itself: the reaction mass of the expelled exhaust. The exhaust matter is the road; the rocket motor is the tires.
The fact that we are all in agreement might be some kind of nasty conspiracy…or we might simply happen to be right here. We beseech you, by the bowels of Christ, consider that you might be wrong.
Of course not. The square root of negative one is well-defined, mathematically; it is the basis of the Complex Number plane. As far as anyone knows, it doesn’t mean anything in the real world.
But…the square root of four is two…and that doesn’t really have any meaning in the real world. There aren’t any “squares” in nature; there isn’t any “square root function” out there.
There are many patterns that do exist in both nature and math. The Fibonacci expression of whorls in pine-cones and sunflowers. Also, there are real-world patterns that are well described by square functions; the drop-off of the force of gravity with respect to distance is one, and the distance fallen by an object dropped from a cliff is another.
Mathematics describes reality, but it isn’t the same thing as reality itself.
Because of quantum effects, 1 + 1 = 2 doesn’t work if you’re talking about protons. 1 proton + 1 proton = 1.999 proton and a bit of energy. That’s why fusion power works…
All very valid. No absolute frame of reference exists, so you can add or subtract velocities, using the Lorentz equations, any way you want. The results will not be contradictory. They also won’t be the results you would expect from ordinary arithmetic addition.
If I’m going at half the speed of light (from your point of view) and I fire a gun with muzzle velocity of half the speed of light, the sum total is not the speed of light. 1/2 + 1/2 does not equal 1. (It actually works out to about .71 c.)
The hymn book works; it is borne out in particle accelerators, GPS satellites, interplanetary probes, and other high-speed experiments. We aren’t just making stuff up here.
You know, if you’d actually thought better of it, as opposed to just wanting to make a show out of yourself, you’d just deleted the half-sentence instead of leaving it dangling there as a testament to your superior self-restraint.
But the force that propels it acts between the tires and the road; and it’s necessary for it to do so, as otherwise, it would not move. The force that propels the rocket acts between the rocket and the reactant. Nothing is able to move itself acting just on itself. This is just Newton’s laws, the first and the third (everything stays in a state of constant motion unless acted upon by a force, and to every force acting on a body, there exists an equal and opposite one, to wit), to be precise. Really, these are the basics we should clear up before moving on – if you’re planning to take on Newton as well as Einstein, I fear I won’t have the patience to go along for the journey…