On Sat, Dec 7, 2024 at 6:49 AM Alan Grayson <[email protected]> wrote:
> > > On Saturday, December 7, 2024 at 3:51:30 AM UTC-7 Alan Grayson wrote: > > On Wednesday, December 4, 2024 at 2:41:25 PM UTC-7 Jesse Mazer wrote: > > On Wed, Dec 4, 2024 at 4:06 PM Alan Grayson <[email protected]> wrote: > > In the case of a car whose rest length is greater than the length of the > garage, from pov of the garage, the car *will fit inside* if its speed is > sufficient fast due to length contraction of the car. But from the pov of > the moving car, the length of garage will contract, as close to zero as one > desires as its velocity approaches c, so the car *will NOT fit* *inside* > the garage. Someone posted a link to an article which claimed, without > proof, that this apparent contradiction can be resolved by the fact that > simultaneity is frame dependent. I don't see how disagreements of > simultaneity between frames solves this apparent paradox. AG > > > Can you think of any way to define the meaning of the phrase "fit inside" > other than by saying that the back end of the car is at a position inside > the garage past the entrance "at the same time" as the front end of the car > is at a position inside the garage but hasn't hit the back wall? (or hasn't > passed through the back opening of the garage, if we imagine the garage as > something like a covered bridge that's open on both ends) This way of > defining it obviously depends on simultaneity, so different frames can > disagree about whether there is any moment where such an event on the > worldline of the back of the car is simultaneous with such an event on the > worldline of the front of the car. > > Jesse > > > Are you claiming that the apparent paradox can be resolved by accepting > the fact that the car and garage frames have different conclusions about > whether the car fits in the garage? AG > > Yes, there are many facts in physics that are not truly "objective" in the sense of being independent of the choice of reference frame, like "which of these two objects has a higher velocity" (even in classical physics there is the notion of 'Galilean relativity' where the laws of physics work the same in different classical inertial frames related by the Galilei transformation--Isaac Newton postulated the idea of absolute space and therefore absolute velocity, but this was an untestable philosophical idea even in classical physics). Any facts about whether the car fits in the garage are similarly non-objective from a physical point of view, depending irreducibly on the choice of frame. > > Let me restate the problem where the car length is assumed to be longer > than the garage length in the rest frame. So, the car can never fit in the > garage when moving, since the garage length, which is initially smaller > than the car's length, contracts. So there's no possibility of a perfect > fit within the garage where simultaneity would apply. However, the car > *can* fit in the garage, from the garage frame, since the car's length > contracts. So, we have a situation where the car fits in the garage, but > only from the garage frame. This seems paradoxical insofar as there's no > objective reality of whether the car fits in the garage or not. AG > Right, as I said above there are many questions in physics which have no unique "objective reality" in that sense, so there's nothing especially unusual about the "does the car fit in the garage" question. The word "paradox" can refer either to a genuine logical paradox or just to something that runs counter to everyday "common sense" intuitions but involves no genuine contradiction, see definition 2a at https://www.merriam-webster.com/dictionary/paradox -- the car/garage paradox fits this latter definition. Also note that in relativity, the idea of the car's position and physical state at a given instant in time can be thought of as a 3-dimensional cross-section of its 4-dimensional "world-tube", so it's possible to come up with analogous "paradoxes" in a purely geometric scenario that doesn't involve relativity or time. For example suppose we have two cylinders in 3D space, arranged in an X-shaped intersecting pattern so there is some overlap in their interior regions. And suppose we have a Cartesian coordinate system with x-y-z coordinate axes, and we define a "level plane" to be a 2D surface with constant z-coordinate. A given level plane will then slice through the two cylinders and contain oval or circular cross-sections of each one, so we can ask the question "is there any level plane where the cross section of cylinder #1 lies wholly inside the cross section of cylinder #2"? In this case the answer depends on the orientation of our x-y-z coordinate axes and the angle that the level planes slice through the cylinders--for some choice of coordinates the answer might be yes, for others it might be no, there is no coordinate-independent answer to the question. Jesse -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion visit https://groups.google.com/d/msgid/everything-list/CAPCWU3LSgADf7c1Lz5-xXGrV1x0ExPHpWxuzSFmYaSqPYEzF9A%40mail.gmail.com.
