On 12/12/2024 5:43 PM, Alan Grayson wrote:
On Thursday, December 12, 2024 at 6:34:01 PM UTC-7 Brent Meeker wrote:
On 12/12/2024 12:27 PM, Jesse Mazer wrote:
So what the hell is going on? It then occurred to me that
this situation is somewhat analogous to the Twin Paradox
(TP), where the two frames seem identical, yielding an age
contradiction when the twins meet. But the solution to the TP
is the recognition that the frames are NOT equivalent due to
the accelerations of only the traveling twin whose clock can
be shown, with SR or GR (although they likely give different
numerical values), that the traveling twin's clock ticks at a
SLOWER rate than the clock of the Earth-bound twin,
accounting for the age difference when they meet. So, how to
apply the lesson of the TP to the issue at hand? How is the
garage frame different from the car frame? The answer is
ACCELERATION! Specifically, in the problem at hand, these
frames can only be equivalent if they have _equivalent_
_histories._
Acceleration is only relevant to the twin paradox if it happens
between the initial and final events you are analyzing,
specifically between the moment the two twins depart from a
common location and when they reunite at a common location. Any
acceleration done by either twin *before* the departure moment
would be completely irrelevant to predicting their ages on
reuniting, if you know their ages at departure and subsequent
paths between departure and reuniting, that's sufficient to get a
prediction of their ages on reuniting that doesn't depend in any
way on what happened prior to them departing from one another.
Similarly, in the car/garage paradox we can assume some initial
conditions where the front of the car has not yet entered the
garage and both the car and the garage is moving inertially--what
happened *before* those initial conditions will be irrelevant to
the analysis of what happens after, for example it makes no
difference if the car accelerated before that moment, or if we
replace the car with space rock with the same rest length that
has been moving inertially for billions of years, while the
garage is mounted to a rocket and accelerated towards the space
rock shortly before the initial conditions.
Jesse
As is well known the twin's paradox is shown to be independent of
any acceleration by the related triplet's paradox:
This can't be correct.
It not only can be, it is. Remember I've /taught/ relativity theory.
Acceleration is only required in the sense that the path not be
geodesic, which are maximum length paths between give events. This is
the same in GR or SR. It's not a question of clocks running slow.
That's loose talk; clocks in these thought experiments are always
assumed to be perfect and to measure proper time along their world line.
Brent
For example, accelerations in GR cause clocks to slow, and absent
accelerations, the traveling twin won't return to meet the Earth-bound
twin. Further, using SR, and modeling changes in velocity to connected
straight line segments, from the pov of Earth-bound twin, the
traveling twin's clock runs slow on each segment or partition, and
allowing the partitions to decrease in length to the limit, any
acceleration can be modeled. AG
Brent
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