The discussion revolves around a time dilation problem involving mesons, specifically focusing on their decay times and distances traveled in different reference frames. Participants explore calculations related to the decay of mesons produced in a particle accelerator, considering both non-relativistic and relativistic effects.
Participants exhibit a mix of agreement and disagreement regarding the interpretations of the problem and the calculations involved. There is no consensus on the correct approach or final answers, as various methods and interpretations are debated.
Some participants note that the problem's wording may lead to ambiguity, particularly in how it frames the assumptions about time dilation and the velocities involved. There are also discussions about the implications of using different methods for solving the problem, such as the spacetime interval versus the gamma factor.
This discussion may be useful for those interested in special relativity, time dilation effects, and particle physics, particularly in understanding the nuances of calculations involving relativistic speeds and decay processes.
And how does one find ## \Delta t'## from v and the proper time without applying at least implicitly using the Lorentz transformation?Orodruin said:You certainly do not need the Lorentz transformations in this type of problem if you apply the spacetime interval correctly. Even in the case where the meson is moving in both frames you are considering you can relate the times in the frames by using the spacetime interval through ##\Delta x = v \Delta t## and ##\Delta x' = v' \Delta t'##. In fact, I would argue that it is much easier than bringing out the Lorentz transformations. Whenever you can use geometrical arguments in relativity (which is most of the time), they tend to be much simpler than looking at coordinates transformations.
I already told you how to solve the problem of relating the times if you are given the speeds. You will have to be more specific about exactly what problem you want solved.Andrew Mason said:And how does one find ## \Delta t'## from v and the proper time without applying at least implicitly using the Lorentz transformation?
AM
Well, my point is that when you use the space-time interval in that way, you are implicitly using the Lorentz transformation:Orodruin said:I already told you how to solve the problem of relating the times if you are given the speeds. You will have to be more specific about exactly what problem you want solved.
No, this is not using the Lorentz transformation. It is using the invariance of the spacetime interval, which in my opinion is more of a fundamental concept than Lorentz transformations (which is just a coordinate transformation between some arbitrary Minkowski coordinate frames). Obviously you must get the same result, since the Lorentz transformation is based on the invariance of the spacetime interval and some assumptions regarding the frames it relates. I also disagree that it is the ”long way”. If you want to use the Lorentz transformation you must first derive it from the invariance of the spacetime interval, then you must be careful in considering what is what.Andrew Mason said:Well, my point is that when you use the space-time interval in that way, you are implicitly using the Lorentz transformation:
Using the space-time interval:
\Delta S^2 = c^2t^2 = c^2t^2 - x^2 = c^2t^2 - (vt)^2 = (c^2 - v^2)t^2
t' = t\frac{1}{\sqrt{1-v^2/c^2}} = \gamma t
Which is just a long way of applying the Lorentz transformation to determine t' from the proper time ##t' = \gamma (t - 0)##
AM
Why would you derive the Lorentz transformation from the invariance of ΔS2? The Lorentz transformations came first, after all. Lorentz derived them before Einstein developed Relativity. They follow from the constancy of the speed of light in all frames of reference. The invariance of the space-time interval derives from the Lorentz transformations. Otherwise, how does on show that the space-time interval is invariant?Orodruin said:Obviously you must get the same result, since the Lorentz transformation is based on the invariance of the spacetime interval and some assumptions regarding the frames it relates.
I also disagree that it is the ”long way”. If you want to use the Lorentz transformation you must first derive it from the invariance of the spacetime interval, then you must be careful in considering what is what.
Historically perhaps. But the historical way of developing the theory is not necessarily the most straightforward one or the most elegant to present. After all, we have had over 100 years to develop the theory since Einstein. The invariance of the spacetime interval is geometrically more fundamental al it does not rely on anything else than the geometry of Minkowski space. You can ask the same question of Euclidean geometry: Would you consider the particular form of rotations more or less fundamental than the fact that the distance function between points? I would argue that the latter is much more defining of the geometry of the space - it is after all directly related to the metric tensor whereas you can use any coordinates you wish and get away with it.Andrew Mason said:Why would you derive the Lorentz transformation from the invariance of ΔS2? The Lorentz transformations came first, after all. Lorentz derived them before Einstein developed Relativity. The invariance of the space-time interval derives from the Lorentz transformations. Otherwise, how does on show that the space-time interval is invariant?
AM
That is certainly one perspective. But I am not sure that the invariance of the space-time interval makes anything easier to understand or is is more illuminating or elegant or easier to apply than the Lorentz transformations. The beauty of Relativity, for me, lies in the way it completely changes and complicates our previously held (Newtonian) notions of time and space from a simple, elegant premise (that all the laws of physics are the same in all inertial frames of reference). The Schrödinger equation and Planck's equation are other simple mathematical statements that reveal a much more complicated world than had been previously thought.Orodruin said:Historically perhaps. But the historical way of developing the theory is not necessarily the most straightforward one or the most elegant to present. After all, we have had over 100 years to develop the theory since Einstein. The invariance of the spacetime interval is geometrically more fundamental al it does not rely on anything else than the geometry of Minkowski space. You can ask the same question of Euclidean geometry: Would you consider the particular form of rotations more or less fundamental than the fact that the distance function between points? I would argue that the latter is much more defining of the geometry of the space - it is after all directly related to the metric tensor whereas you can use any coordinates you wish and get away with it.
- Historically relativity was developed in a certain way, but that was gradually hinting at and moving toward a more geometrical view of spacetime. Relativity is ultimately a theory of the geometry of spacetime, regardless of how it developed. You can of course close your eyes to this and go on primarily using Lorentz transformations, but it will obscure the beauty and deeper understanding in my opinion.
Then you really should try to embrace the geometrical view of what relativity tells us about spacetime and not get hung up on an arbitrary set of coordinate transformations...Andrew Mason said:The beauty of Relativity, for me, lies in the way it completely changes and complicates our previously held (Newtonian) notions of time and space from a simple, elegant premise (that all the laws of physics are the same in all inertial frames of reference).
Andrew Mason said:But I am not sure that the invariance of the space-time interval makes anything easier to understand or is is more illuminating or elegant or easier to apply than the Lorentz transformations.