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Why is time dilation interesting?
The principle of relativity tells us that K’ (the person who owns the moving clock) will reverse the roles of the two clocks. K’ will say that the clock owned by K is ticking slowly.
There is no non-arbitrary way to answer the question, "Who is right?" (
There is no absolute time.
The size of the effect is the same as for Lorentz time dilation, but it is conceptually quite different.
For Lorentz there was a preferred frame (the ether), and all clocks moving through the ether ran slowly with respect to absolute time. Now there is no preferred frame. Each person says that clocks moving w.r.t him are running slowly.Just to be sure you understand ...
The reason K and K’ disagree about the clocks is that although they see the light taking different paths, they both say the light is travelling at c relative to themselves. It is NOT a matter of them being too dumb to correct for the travel time of the light- the corrections disagree because they disagree about the relative speed of the light wrt to different objects.
Are there any "real" effects of all this?
Consider cosmic ray muons which are produced in the upper atmosphere, 25-50 km above the ground. These particles only live about 2 microseconds, in which time they can only travel a few hundred meters at the speed of light. Nevertheless, a large number of them make it to the ground. How is this possible?
The solution is time dilation. Cosmic ray muons travel at a speed that is about 99% that of the speed of light, that is, v/c=.99. Hence, the time dilation factor is about 7.09. Hence, if a muon at rest decays in 2 microseconds, one traveling at v=.99c decays in 2 times 7.09 or 14.18 microseconds. In this time, muons can very nearly reach the ground.
In order for SR to be self consistent all clocks must be affected the same way. Thus, time dilation is not a property of any specific mechanism, but of time itself.
What does the muon see?
From its point of view, there is no time dilation, but the atmosphere (which is moving) is Lorentz contracted. In the 2 µs muon lifetime, the earth, travelling close to c, has time to collide with the muon because it starts close by.
Both observers agree the muon will reach the surface.
They disagree about why this is so, i.e. they disagree about the values of some particular quantities used in the calculation leading to the conclusion about whether the event occurs.
Simultaneity
(see Einstein, ch 9)Suppose a train is moving past
Fred, who is standing on the embankment. Barney is riding in the middle of the train. Two lightning strokes hit the ends of the train at times such that Barney sees the two flashes at the same time. At that instant, he is passing by Fred (i.e., they are at the same place when they see the flashes).
The question is, "Were the two lightning strokes simultaneous?"
Barney says:
The lightning hit the two ends of the train, which are the same distance from me. Since I saw the two flashes at the same time,
Fred says:
When the lightning struck (some time in the past) the front of the train was closer to me than it is now, and the rear was farther. Thus the two flashes traveled different distances. Since I am seeing them at the same time, they must not have been simultaneous.
There is no way to tell who is right, so:
Simultaneity of distant events
depends on the motion of the observer.
As with time dilation, this effect results from the invariance of the speed of light. In this case, it depends on the fact that Fred sees both flashes moving at c, even though they are moving w.r.t. each other.
This raises a subtle point. It is impossible to prove that the light moves the same speed in both directions. That is an assumption, based on the apparent isotropy of space, and the simplicity of Maxwell's equations. It is not important in this situation, because if we assume that the speed of light is different just exactly to cancel the effect, it will double the effect when the train moves to the left.
Note: We could consider a different pair of strokes, which Fred says are simultaneous. (They wouldn’t be at the ends of the moving train.) Barney will say that they aren’t simultaneous.
Being at the same place (at different times) is also relative. This was known by Galileo. Think about the meal you eat while travelling on a boat.
Notice: we assumed that simultaneity at the same place was objective. If Fred was at the same place as Barney at the time one light flash reached, and Barney says they reached simultaneously, Fred was also there when the other flash hit.
Important note: Once again relativity is NOT a trivial reminder that "it takes a while to see things because light has a finite speed." Romer knew that, and it led to no changes in anyone's picture of space-time. If relativity seems obvious to you, you have not yet got it.
Relativity is a description of the symmetry of space-time. If the "c" that appeared in the equations didn't correspond to the speed of something we use to view events, or even to the speed of anything at all, the structure of the theory would not change at all. It would just be harder to make up little stories to illustrate the theory.
(We'll soon see the equations in full form.)
The twin paradox
Suppose Alice and Beth are twins. Alice sets off in her rocket so fast that the time dilation factor becomes 10
. She travels away from Earth for 10 years, as measured by Beth, who has remained on Earth. Alice then turns around and returns to Earth at the same rapid pace.
When Alice returns home, Beth has aged 20 years. How much has Alice aged?
There appears to be a paradox. According to the Lorentz transformation, during the time Alice is travelling:
Beth says
: I measure Alice’s clock to be running slow by a factor of ten, so she has aged only two years.Alice says
: My clock is fine. I measure Beth’s clock to be running slow by a factor of ten, so she has aged only 2 years.They start and end standing right next to each other, so
a direct comparison of clocks is possible. Who is correct?
The answer is that Alice, the twin who turned around, has aged less. The situation is not symmetrical, because in order to return to Earth, Alice must have
accelerated. Our descriptions of how things looked to different observers (Lorentz transformations) so far do not describe accelerated observers, so we only know how things look to Beth. Of course Alice must agree that Beth is older, when they now stand side-by side. Now we can put together a conclusion about how Beth must have looked to Alice while Alice was accelerating. While turning back (accelerating toward earth) , Alice must observe Beth's clock to be running fast, not slow.
In other words, this is not a paradox at all but just a reminder that the SR transformations only work between reference frames which are not accelerating (AT LEAST with respect to each other, leaving aside the question of absolute acceleration.) But you can also see that from SR we can draw conclusions about how things MUST look to accelerating observers.
The symmetry of special relativity
That's because G's relativity made assumptions about how some particular quantities (distance, time, mass…) changed between different frames (i.e. that they didn't change) , and those assumptions turned out to be wrong.
In Special Relativity
Lorentz transformation
Remember how Galileo related space and time measurements made by moving observers (motion is along the x direction):
Galileo-Newton Lorentz-Einstein
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Time intervals are the Same for all observers |
g is the time dilation factor:
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Time is no longer an absolute quantity.
Notice that distances (y,z) perpendicular to the relative motion of the two coordinate frames are unchanged, as we showed was necessary.
The equations are a bit simpler and more symmetrical if we use ct instead of t and define b= v/c:
Both types of transformation are INVERTIBLE. That is, if you transform to a primed frame using relative velocity v, the transform using relative velocity -v, you get back the original coordinates. That's obvious for Galilean transforms, and you can easily check it for Lorentz transforms. That's one of the properties that make the transforms part of what's called a GROUP of symmetries. More importantly, since you use the SAME TYPE OF TRANSFORM either way, you can't get a clue as to which frame is the "proper" one.
Important note: even if there were no such thing as light, or anything else that traveled at speed c relative to other stuff, all of the essential points of relativity are contained in the new rules for converting between coordinate systems. Our arguments about "what would observer A see by using light" are not essential, just convenient paths toward these transformations. The key point is NOT about practical limitations on observations, but rather what sets of variables different observers have to use to get nature to obey the same simple laws.
There is a speed limit
Galileo says
velocities add like vectors:This can’t be correct in special relativity, because everyone gets the same speed, c, for light. There is a general law of combination of velocities.
(derivable from the L.T.) If I measure b1 and b2 as above (b º v/c) (both along same direction), then you will measure the object to have
.
Note that
b = 1 means v = c. You can verify that if b1 = 1 orExample:
I measure a proton to be moving 0.75 c. You are moving 0.75 c in the opposite direction (i.e., not with the proton). You will not measure 1.5 c for the proton’s speed, but rather 0.96 c.
For small velocities (
b1 « 1 and b2 « 1) this reduces to Galileo’s expression, because b1b2 is extremely small and can be ignored. This is an example of the correspondence principle; a new theory must agree with the old one in the old one’s region of validity.You might ask,
"What happens if I keep pushing on the proton? Won’t it keep accelerating indefinitely?" We’ll address this question soon.
Notice- Relativity might have sounded like some vague "everything goes" claim initially. Now we are deriving specific new physical laws from it.
Relativity is a constraint on the physical laws. It says "No future physical law will be found which takes on different forms in different inertial frames."
The unification of electricity and magnetism
Einstein’s one simple postulate solves a lot of problems.
Consider the magnetic force on a moving charge due to the electric current in an electrically neutral wire (no electric field):
The magnetic force occurs when the charge is moving. If we look at it from the charge’s point of view
(i.e., in its own "rest frame"), there can’t be a magnetic force, but there must be some kind of force, because the charge is accelerating.So, the principle of relativity tells us that the charge must see an electric field in its rest frame.
(Why must it be an electric field?) How can that be? The answer comes from Lorentz contraction. The distances between the + and - charges in the wire are Lorentz contracted by different amounts because they have different velocities. The wire appears to have an electrical charge density. (The net charge in a current loop will still be zero, but the opposite charge is found on the distant part of the loop, where the current flows the opposite direction.)
When we change reference frames, electric fields partially become magnetic fields, and vice versa. Thus, they are merely different manifestations of the same phenomenon, called electromagnetism.
The first oddity of Maxwell's equations was that the magnetic force existed between MOVING charges. But now we say that there's no absolute definition of MOVING.
The resolution is that whether the force between two objects is called electric or magnetic is also not invariant.
(Neither is the exact magnitude of the force, but that's another problem.)
