These sets of 4 numbers make no sense. You haven't even specified what the four numbers mean. Complete the following: The timer is started at the event (x,t)=(...,...). The timer is stopped at the event (x,t)=(...,...). Or, if you prefer: The timer is started at the event (x',t')=(...,...). The timer is stopped at the event (x',t')=(...,...).
Sure, I do not know why it is useful but, 1) The timer is started at event (x,t) = (0,0) The timer is stopped at event (x,t)=(0,-k/(vγ)) 1) The timer is started at event (x',t') = (0,0) The timer is stopped at event (x',t')=(k,-k/v)
I would eliminate gamma since it is a function of v and confuses the issue if we are talking about ①the v built-in to the problem description or ②the v built-in to the choice of observer, and say just one of: \(\begin{pmatrix} x_A \\ t_A \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \\ \begin{pmatrix} x_B \\ t_B \end{pmatrix} = \begin{pmatrix} 0 \\ \frac{-k}{v} \sqrt{1 - \frac{v^2}{c^2}} \end{pmatrix} \) or \(\begin{pmatrix} x'_A \\ t'_A \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \\ \begin{pmatrix} x'_B \\ t'_B \end{pmatrix} = \begin{pmatrix} k \\ \frac{-k}{v} \end{pmatrix} \) and remember that k/v < 0 and remember that 0 < |v| < c and remember that for both events A and B \(\begin{pmatrix} x' \\ t' \end{pmatrix} = \begin{pmatrix} \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} & \frac{-v}{\sqrt{1 - \frac{v^2}{c^2}}} \\ \frac{-v}{\sqrt{c^2 - v^2}} & \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \end{pmatrix} \begin{pmatrix} x \\ t \end{pmatrix}\). Remembering all that, we could verify the trivial case for event A (not shown), or the easy case of event B, where \( \begin{pmatrix} \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} & \frac{-v}{\sqrt{1 - \frac{v^2}{c^2}}} \\ \frac{-v}{\sqrt{c^2 - v^2}} & \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \end{pmatrix} \begin{pmatrix} x_B \\ t_B \end{pmatrix} = \begin{pmatrix} \left( \frac{-v}{\sqrt{1 - \frac{v^2}{c^2}}} \right) \left(\frac{-k}{v} \sqrt{1 - \frac{v^2}{c^2}} \right) \\ \left( \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \right) \left( \frac{-k}{v} \sqrt{1 - \frac{v^2}{c^2}} \right) \end{pmatrix} = \begin{pmatrix} k \\ \frac{-k}{v} \end{pmatrix} = \begin{pmatrix} x'_B \\ t'_B \end{pmatrix} \) But I would not remember what chinglu's interest in the problem was, since it seems self-consistent in problem setup and in evaluation in light of the Lorentz transform. // Edit -- Oh, yeah. chinglu notices that \( 0 < t_B - t_A < t'_B - t'_A\) and claims that this is not time dilation. But the definition of time dilation is "(relatively) moving clocks tick (relatively) slower". Let's see what this means for this example. \(t_B - t_A\) describes the time elapsed on a clock where events A and B happen in the same place relative to a inertial observer, since \(x_A = x_B\), while \(t'_B - t'_A\) describes the time elapsed on a clock where the same events A and B happen in different places. So if A and B are two events on the worldline of a uniformly moving clock, then \(t_B - t_A\) describes the proper time between A and B -- the time that the clock that moves uniformly between A and B measures. Any clock which measures the same time as the proper time is at relative rest with the clock that moves uniformly between A and B. That's the meaning of \(x_A = x_B\). Shorter: The observer who sees the clock relatively at rest measures the proper time on his own clock. But for the other observer, \(x'_A \neq x'_B\) and therefore sees the clock that moves uniformly between A and B as not at relative rest. For this observer the clock is moving with speed \( |v| \). And since \( 0 < t_B - t_A < t'_B - t'_A\) it is clear that whatever time this observer sees, it is longer than the proper time. But the clock that moves uniformly between A and B is the clock that also measures proper time, and therefore the observer who sees that clock as moving sees the time kept by that moving clock as slower than the time of his own clock. Shorter: The observer who sees the clock as moving relative to his standard measures a time longer than the proper time on his own clock. Thus this example does not contradict that moving clocks tick slowly. Big point: Since relativity is all about relative motion, it is pointless to try and claim one clock is really at rest because all clocks are at rest with respect to themselves.
Thankyou. As an aside, I note that I have consistently been considering the position x'=-k, with k greater than zero, so that the coordinate is negative. You have been considering x'=k, with k less than zero, so that the coordinate again is negative. So, we're discussion the same situation, but our opposite signs are a potential point of confusion. In this post I will adopt your convention that k is negative. Given the above information about where your timer starts and stops, I make the following points: 1. Your coordinates are correct for those two events, in the primed and unprimed frames. 2. In the unprimed frame, the "start" and "stop" events both occur at the same spatial coordinate. In the primed frame, they occur at different spatial coordinates. 3. Therefore, the proper time is the time difference between the events in the unprimed frame, in this example. 4. The time interval between the start and end events is shorter for the unprimed frame than for the primed frame, in your example. 5. Special relativity tells us that a proper time between any two events is always the SHORTEST possible time, and that the time measured in any other frame (i.e. not a proper time) should be longer. 6. Your example AGREES with what special relativity predicts, since the time interval in your unprimed frame is shorter than in the primed frame, and the unprimed frame in this case measures the proper time. So, since your example agrees with what special relativity predicts, it seems you can have no complaints. Do you have any other questions you wish to ask?
Excellent. From the view of the unprimed frame, the clock at k elapses more time than the clock at the unprimed origin. Therefore, we have a moving clock that is not time dilated as indicated by the OP. Further, the moving clock is time expanded. Therefore, we can not conclude in relativity that all moving clocks are time dilated.
You have not resolved anything here. In the unprimed frame, the k observer is a distance -k/γ when the origins are the same. Even the simplest mind understands Δt = Δd/v, hence Δt =-k/(vγ). Now, LT is applied with x = 0 since in the unprimed from we are at the origin or x=0. Δt' = ( Δt - vx/c² )γ =-k/(vγ)γ = -k/v Any simple mind can understand dt'/dt = γ. Therefore, the moving clock beats time expanded.
You realise that it is impossible to say whether the unprimed or primed frame is "moving" or "stationary", do you not? The theory of relativity has no preferred reference frames. If it helps you, you can regard the unprimed frame in your example as "moving" and the primed frame as "stationary". All that really matters is relative motion. That's why it's called "relativity". It really depends on how you define a "moving" clock. Do you agree that the proper time will always be the shortest time interval measured? Yes or no? If you want to defeat relativity, I suggest you try to invent a scenario in which the proper time is longer than the time in some other frame, for the same pair of events.
Here’s another take, on absolute space and the speed of light. I ran into it somewhere, and thought that it had some promise. I'm not swearing by it, but thought you might be interested. Here's my summary of it. The Lorentz derivation left absolute space behind, given the assumption that measurements are limited to 3 dimensions. Relativity provides computational capability, but does not give us access to the underlying reality, yet, there is a reason that the speed of light is the universe’s maximum speed, and why the internal change of a system approaching the speed of light slows to zero. The speed of light is not merely a universal constant, but is the linear, dimensional relationship between space and time. There can be only one such relationship, and thus there is only one speed for the propagation of energy through space. Objects moving at speeds less than ‘c’ are amalgamations of static (particle cores) and dynamic (photon/kinetic) energy. The observed time dilation in an object moving through absolute space is solely a product of the universe’s photonic speed limit. A moving object is composed of static (rest) and dynamic (propagating) energy. Its net motion through absolute space constitutes the external manifestation of its dynamic energy, herein called V. Any motion of its dynamic constituents normal to V has no measurable effect on the net motion of the entire object, this motion being called Vi. Since ‘c’ is the metric of energy’s propagation through absolute space, and since the external and internal components of an object’s dynamic energy are perpendicular to each other, their relationship to each other can be depicted as a triangle. height: Vi, length: V, diagonal: ‘c’ The speed of an object’s internal motion defines the rate of the internal changes it experiences, such as the decay period of an unstable relativistic particle. An object’s internal motion is related to its external motion by Vi^2 = c^2 - V^2. The speed of light, ‘c’, is the metric of all change; a motion of zero represent no change; a motion of ‘c’ is maximum change. The normalized rate of an object’s internal change is the ratio of its internal components to the speed of light. Ri = Vi/c. When Vi = 0 then Ri, the rate of internal change, goes to zero. The relative length of a unit time interval in this system is the magnitude of its time dilation, TD, this being the inverse of the rate of internal change, Ri. TD = T/T0 = 1/Ri = c/Vi, where T0 is the length of a time interval at rest (V=0). If internal change is reduced to half of normal, for instance, an internal event takes twice as long to occur. Substituting and solving, we get T/T0 = TD = 1/[square roof of(1-V^2/c^2)], which is the same time dilation given by the Lorentz transformation, yet, it is based solely on the real, physical limitations of a moving system, no reconfiguration of space and time being necessary. As the external motion goes to ‘c’, the rate of internal motion goes to zero. Velocity addition can also express ‘c’, by performing two successive Lorentz transformations, for a V of V1 and V2, this being the addition of ratios, not magnitudes, giving V = (V1+V2) / [ 1+ (V1*V2/c^2) ] = (c+V2 / c+V2 ) = c. Thus, the speed of light appears the same way in any direction in any moving reference frame. Velocities could not add exactly like distances unless ‘c’ = infinity, there then being no upper limit to their magnitudes. So, vector addition could not be used, due to the dimensional relationship between time and space, although it could be an approximation for adding velocities.
Sorry, but it's nonsense. c is a speed. For any of this to be consistent, V and Vi must have dimensions of velocity as well. They cannot be energies. So V is supposed to be some kind of absolute speed. How can we measure V? Answer: We can't, because measuring absolute motion is impossible. What is this "internal motion" supposed to be? Vague terms like "internal change" are meaningless unless defined. Ri is not a rate of change. In the previous paragraph it was defined as the ratio of two speeds. While this is all a very clever way to "reverse engineer" the standard time-dilation formula, it doesn't work. It doesn't work because here "V" is supposed to be an absolute speed, whereas in relativity the "v" in the time dilation formula is a relative speed. The time dilation formula given here predicts time dilation in only one "direction", whereas relativity predicts a reciprocity. In other words, the theory of relativity says that if you're moving relative to me then I'll see your clocks running slower than mine AND you'll see my clocks running slower than yours. The "theory" being discussed in this post has an absolute speed V, so the time dilation will only work "one-way". This is contrary to real-world experimental evidence and observation, and so the entire alternative theory can safely be discarded on this basis alone.
This is a little off-topic, and doesn't imply that I necessarily agree with Sciwriter, but I think think isn't strictly correct, James. You can make a model that exactly matches SR and includes an absolute rest state. It just wouldn't be as parsimonious, and the absolute rest state wouldn't be measurable: If there is an ether with an absolute rest state, and clock are absolute dilated and rods are absolutely contracted according to their motion relative to the ether, then you'll still find SR-predicted mutual length contraction and time dilation when you set up inertial reference frames, because in setting up a reference frame, you can't synchronize the clocks in the ether rest frame. Using einstein sychronization to set up an inertial reference frame guarantees that the clocks are unsynched in such a way that they'll measure clocks and rods in relative motion to be dilated and contracted exactly as predicted by SR. The consequence of this, of course, is that there's no way of distinguishing the ether rest frame from any inertial frame, which naturally leads to SR. Afterthoughts I think it's quite difficult for people who are familiar with SR to go back to that kind of unmeasureable absolute-rest paradigm, because it feels redundant and arbitrary or something, but I also think that it's a potentially stepping stone for people who aren't familiar with SR. Anyone can understand the idea of absolute rest state. Anyone can understand the idea of absolutely moving clocks ticking slowly. Anyone can understand the idea of absolutely moving rulers being contracted. But, people have trouble wrapping their head around relative motion producing mutual time dilation and length contraction, which is where I think carefully exploring the consequences of an absolute ether model might help.
Hi Sciwriter, What you say sounds to me like a slightly garbled and misinterpreted version of a geometrical description of special relativity with the premise of an unmeasurable (arbitrary) absolute rest frame thrown in. Many of the concepts described in your post seem to correspond to some degree to four-vectors. The problem is that the suggested 'real, physical limitations' are not specifically measureable, ie there is no way to measure the actual motion of a specific system, or whether it is in fact at rest. So while the idea of absolute motion is consistent, it seems useless, which is why it is discarded in special relativity. Without the idea of absolute motion, we can get the same results with a simpler model.
Thanks, guys, as I was wondering if it was too good to be true, and I do remember that absolute space was an objective for this analysis, as well as the 4Dness of space-time, which may or may not have anything to do with 'four-vectors'. I may have caught the gist, but could still have garbled some of the details. I'll have to go look for the thing someday in my Library of Babel, but now I'm living 'free' in Hawaii on a mountain top and working out of a tent.
Are you talking about the primed clock at x'=k? How do you know how much time elapsed on that clock, since you only have a single time reading for it? You have a the reading on the primed clock at x'=0 when the timer began, and a reading on the primed clock at x'=k when the timer stopped. Two different primed clocks. You don't have an elapsed time for either one, just a single reading for each. To get the elapsed time on the primed clock at x'=k, you need a reading on that clock when t=0. Do you know how to get the reading on the primed clock at x'=k when t=0? Try this. Lorentz transform: \(t = (t' + \frac{vx'}{c^2})\gamma\) \(t' = \frac{t}{\gamma} - \frac{vx'}{c^2}\) Substitute in x'=-k, t=0: \(t' = -\frac{vk}{c^2}\) So, in the unprimed frame, the primed clock at x'=k reads t' = -vk/c^2 when t=0, and t'=-k/v when t=-k/(vγ), right? Now you can get the elapsed time: \(\Delta t' = \frac{vk}{c^2} - \frac{k}{v}\) \(\Delta t' = \frac{k}{v} \times \frac{v^2 - c^2}{c^2}\) \(\Delta t' = \frac{-k}{v\gamma^2}\) \(\Delta t' = \frac{\Delta t}{\gamma}\)
This is all so nice. Let's see what the primed frame says. The observer or clock at k sees the unprimed origin coming toward it. That origin is a distance k when the two frame origins are the same. The primed frame has no choice but to conclude with speed v and k distance implies the unprimed origin will meet the observer at k based on \(\Delta t' = \frac{-k}{v}\) Yet you have \(\Delta t' = \frac{-k}{v\gamma^2}\) Which is a contradiction.
[*]then you'll still find SR-predicted mutual length contraction and time dilation when you set up inertial reference frames, They will not be the same values as with SR. Assume, you have frame A at rest with the ether by chance and B moving v relative to the ether. A would have absolute 0 time dilation and length contraction. Yet, B would be length contacted and time dilated relative to the ether. So, the length contraction and time dilation are absolute and not reciprocal. Therefore, your view is wrong.
On the subject of numbers that have seemingly no meaning, anyone attached to the science community have an answer to what these ones represent? 1 4 6 10 11 26 (And no there isn't a right or wrong answer, it's a set of numbers someone implied about something some time back, but I could never find any references from alternative sources)
No, Pete summed your views. You run into a contradiction. You must represent your views with math. Pete did. And, I will admit, I was waiting on this contradiction. This means the OP is absolutely correct
chinglu: Right. I thought you wouldn't be able to respond to #267, just as you have failed to respond to my arguments over and over again. No, he gave his views. Where? Go through posts #117, #118, #165, #177 line by line and point out any errors or contradictions you can find there. I have invited you over 20 times to do this, and you have never yet started to do it. See posts #117, #118, #165, #177, where the math is posted in full. What contradiction? It is proven wrong in posts #117, #118, #165 and #177, which remain totally unrebutted.
So you agree that in the unprimed frame, the primed clock at x'=k elapses \(\frac{-k}{v\gamma^2}\) between t=0 and \(t=\frac{-k}{v\gamma}\), yes? Not a contradiction, chinglu, but relative measurements. Measurements in different frames don't necessarily agree with each other - you know that, right? In this case, the relative difference means that you're comparing different intervals. Both intervals end when x'=k coincides with x=0. This event is not relative, because that's where the clock of interest is located - everyone agrees that the clock at x'=k reads t'=-k/v as it passes the x=0 marker. But your starting event is relative: You say the timer starts when the origins coincide. But this is relative, because the clock of interest is at a different location. In the primed frame, time is defined by t'. The x and x' origins meet when t'=0, so the origins meet when t'=0 at x'=k. But in the unprimed frame, time is defined by t, not t'. The x and x' origins meet when t=0, so the origins meet when t=0 at x'=k. You can't start the timer twice, so you must choose one. Which is it? Does the timer start when t=0 at x'=k, or when t'=0 at x'=k?
