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Dale
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Yes, that is the length of the “hypotenuse”
The spacetime interval between the two events. One event is the emission of the light flash and the other is the reception. Note you correctly calculated the value to be zero. The events have a lightlike separation. You seem to be confusing the spacetime interval with the proper length. The value of the spacetime interval equals the proper length only for events with a spacelike separation.Grimble said:Calculate the length of what?
Nowhere can I find any suggestion that it involves a different metric (if that is the correct term?)Wikipedia said:Minkowski diagrams are two-dimensional graphs that depict events as happening in a universe consisting of one space dimension and one time dimension. Unlike a regular distance-time graph, the distance is displayed on the horizontal axis and time on the vertical axis. Additionally, the time and space units of measurement are chosen in such a way that an object moving at the speed of light is depicted as following a 45° angle to the diagram's axes.
Clearly not:Grimble said:But are we not still dealing with the same equations?
It is not the same quantity, but there are some similarities. In both cases it is an invariant measure of distance in the space. The differences are that in space it is called distance and there is only one kind of distance, while in spacetime it is called the spacetime interval and there are three different kinds of spacetime intervals (space like, time like, and null).Grimble said:YEs, but one is the aggregate length in 3 dimensions and the other in 4, so does ds represent the same quantity in each case?
It makes little sense to use the term "invariant" to refer to ##a^2+b^2+c^2## in the context of Minkowski Spacetime.Grimble said:Surely a2+b2+c2 is still the aggregate length in Minkowski Spacetime; while -ct2+a2+b2+c2 is the Spacetime interval and both are invariant intervals.
Grimble said:if it is not the same quantity then both equations are true
Grimble said:It is still the same Spacetime
Grimble said:Surely a2+b2+c2 is still the aggregate length in Minkowski Spacetime; while -ct2+a2+b2+c2 is the Spacetime interval and both are invariant intervals.
Grimble said:The Mathematics doesn't depend on how they are drawn.
Grimble said:It just seems difficult to be sure what the terms mean when the same term ds2 means two different things...
It can be very confusing
Grimble said:The main difference in the two geometries is the number of dimensions, is it not?
Grimble said:In Euclidean 3-space, a2+b2+c2 is invariant because there is no time component.
Grimble said:in Minkowski Spacetime the equivalent would be that a2+b2+c2 would be invariant at any single specific time.
Grimble said:in classical mechanics using euclidean geometry, ds2=(ct')2+a2+b2+c2 where ct' is the time axis for a moving body
The Twin Paradox is a thought experiment in which one twin travels through space at high speeds while the other remains on Earth. When the traveling twin returns, they have aged less than the twin who stayed on Earth. This paradox challenges our understanding of time and relativity.
Proper time is the time measured by an observer who is in the same frame of reference as the event being measured. In the Twin Paradox, proper time is used to calculate the time dilation experienced by the traveling twin. It is the time experienced by the twin on their journey through space.
Coordinate time is the time measured by an observer who is in a different frame of reference than the event being measured. In the Twin Paradox, coordinate time is used to calculate the time experienced by the twin who stayed on Earth. It is the time experienced by an observer who is not in the same frame of reference as the event.
Yes, the Twin Paradox can be explained by the theory of relativity. According to the theory, time is relative and can be affected by factors such as speed and gravity. In the Twin Paradox, the traveling twin experiences time dilation due to their high speed, causing them to age slower than the twin on Earth.
While the Twin Paradox is often used as a thought experiment, there have been real-life examples that demonstrate its principles. For example, astronauts who spend extended periods of time in space experience time dilation and age slower than those on Earth. This has been observed in astronauts who have spent time on the International Space Station.