force
vvvvvvv
-> ==rod== ->relativistic motion
-table- slot -table-
As we all know, the speed of light in vacuum is the same for all inertial frames of reference, and because of that, movement in inertial frames of reference is subject to Lorentz transformations.
For this reason if there are two spaceships travelling to opposite directions at speed s (from an external observer's point of view), if ship A measures the speed of ship B as it perceives it, it's not 2*s, but something less (and always less than c). It happens because ship B appears contracted in the direction of the movement.
Let's assume that ship A measures the speed of ship B by starting a clock when the bows of both ships coincide, and stops the clock when the bow of ship A coincides with the stern of ship B. [snip]
In all pictures, time is on the vertical. The pictures are roughly to the scale of the problem (A views B as having about half its length). There are Minkowski diagrams, by the way.
On the center is the situation according to the external reference frame in which ships A and B have the same length and move at the same speed; ship A is red, ship B is blue.
On the left is the situation as viewed by ship A plotted on the external reference frame; ship A is orange, ship B is light blue. In gray are the simultaneity lines of A (the more horizontal gray lines) and the lines of constant position of A (the more vertical gray lines). In English, this means 'the points which A considers to be simultaneous' and 'the points which A considers to be stationary'.
On the right is the situation as viewed by ship B plotted on the external reference frame; ship A is purple, ship B is green. In gray are the simultaneity lines of B (the more horizontal gray lines) and the lines of constant position of B (the more vertical gray lines). In English, this means 'the points which B considers to be simultaneous' and 'the points which B considers to be stationary'.
Note that there is some overlap between A and B in the 3 images in the middle lines; these are where the stern of ship B coincides with the bow of ship A according to ship A (left), the external observer (center) and B (right).
In the left and right pictures, you have to explicitly include the Lorentz factor on any distances and times measured along the gray lines (due to Minkowski geometry).
As you can see, the situation is quite asymmetrical as viewed from within the ships (not so much for the external observer).
If anything is unclear, please say so and I will attempt to explain it better.Since it took 1 second for the bow of ship A to go from the bow of ship B to its stern, and since A measured B to be exactly half the length of A, it would mean by reciprocity that it would take 2 seconds for the bow of ship B to go from bow to stern of ship A.
However, by using the clock of ship B, it takes only a half second for the bow of ship A to go from the bow to the stern of the ship B. Since this is from the time reference of ship B, it would seem that ship B is measuring a half second for this
If anything is unclear, please say so and I will attempt to explain it better.
This picture is the combination of the 3 images above, plus 3 more lines. The 3 new lines show, in the 3 different frames, the time that the bow of ship B takes to move from bow to stern of ship A. The yellow line and the pink line show the times as measured at the bow of A and B, respectively, and are on the same scale (as the ships have the same speed in this diagram); the cyan line is in the external reference frame, and is in another scale. If the yellow line is 1 second, then the cyan line is sqrt(3/2) (as HHS stated) and the pink line is 2 seconds (it is roughly twice as long in the image, and doing the math will show this to be the case). If the time interval in question is measured at another point in the ships, you can just slide the respective line keeping the endpoints in the same simultaneity lines (for the cyan line, this means "horizontally").
As you can see, these time intervals measure quite different things. And since they only share one event (the origin), it doesn't really make sense to compare them.
Now we go to the clocks. If we are going to assume that the folks at ship A are smart enough to check the clock of B at both relevant events and measure the delta, this question can be answered objectively. Otherwise, it will depend on where the clock is located in ship B.
Assuming smart folks, we can place the clock somewhere convenient -- say, at the stern of ship B. Then, we have a Minkowski triangle with two known sides (the yellow line and the lowest light blue line) and one unknown side (the gray line between them). This unknown side is the time interval elapsed in the bow of ship B between the two relevant events as seen by A.
Now, the side can be calculated if we have the rest length of the ships. I will represent this length by L (this is the length of the orange and purple lines). We know that whatever this rest length is, the light blue lines (and the green lines too) will be half of this: L/2. The yellow line has "length" equal to c * 1 s. Putting in the proper time formula (which is a special case of the space-time version of Pythagoras theorem), we have that the desired time interval dt is
dt = sqrt[(1 s)^2 - (L/2c)^2]
This is how much time will elapse in a clock on ship B as seen by ship A in the problem at hand.
I hope this is clear enough; this kind of thing is much easier to explain in person, having instantaneous feedback from the subject if something is unclear...If ship A had a measurement device at the half point of the ship which looks what the clock at B's bow is showing (at the moment when B's bow is exactly at this half point of ship A), would it see a different value than the measurement device located at A's bow? If yes, what would the two readings be?
S = (c * delta-t)^2 - (delta-x)^2 - (delta-y)^2 - (delta-z)^2And so on. The trickiest thing to calculate are the speeds of A and B according to the external reference frame, as well as the gamma factor for this speed; this is because you need to calculate the hyperbolic angle and use inverse hyperbolic tangents and so on
2 = gamma = 1/sqrt(1 - (L/(2s*c))^2)
4 - (L/(1s*c))^2 = 1
L = sqrt(3) * 1 s * cdt = sqrt((1 s)^2 - (-sqrt(3)/2 s)^2) = sqrt(1 - 3/4) s = 0.5 sActually, no calculation of angles is necessary, just a plain old quadratic equation:
I understood your explanation, and it makes it clear what A is measuring using its own clock. However, I still have hard time deducing the answer to the original problem: Let's assume there's a visible clock on the bow of ship B, which A can see during the entire experiment. What does someone located at the bow of ship A (or at some other point in ship A) see as the value of this clock at different times, and how does this relate to what someone on ship B sees (when looking at his own clock)?
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Drama, too long, didn't read, lol.
The General Theory of Relativity predicts that if a particle enters the event horizon of a black hole, it would require an infinite amount of energy to stop it from falling into the singularity at the center. On the other hand, the Pauli exclusion principle states, basically, that you would need an infinite amount of energy to make two particles be at the same place at the same time (which is what is happening in a singularity). How are these two opposite forces consolidated?
The General Theory of Relativity predicts that if a particle enters the event horizon of a black hole, it would require an infinite amount of energy to stop it from falling into the singularity at the center.
First of all, no scientist, when being precise, would ever talk about "an infinite amount of energy" or some such. For example, you've probably heard laypeople say things like "if you had an infinite amount of energy you could accelerate a spaceship to the speed of light." That's not a meaningful statement because there is no such thing as an infinite amount of energy, and it certainly doesn't follow from relativity. What relativity tells us is that no finite amount of energy is sufficient to accelerate a massive object to the speed of light. That certainly doesn't imply there is such a thing as "infinite energy."
When physicists talk of singularities and such, it doesn't actually mean they believe there are precise physical manifestations corresponding to the ways mathematical models fail.
To a layman (like me) there is little difference between the concepts of "you would need an infinite amount of energy to" and "no finite amount of energy is enough to", as they sound like two ways of expressing the same thing.
