Hello everybody,
Just a few questions that are on my mind.
You've discovered, I think, a significant difference between how sound and light work. This leads to some interesting conclusions.
1) If something is travelling faster than sound, is it correct to say that an observer towards whom the object is travelling will see it closing in in complete silence (barring any ambiant noise, of course)? If so, at which point will the observer actually hear the travelling object? Immediately when it reaches him or a few seconds later? I'm about certain it'd be the former, but I'd like a confirmation.
Your question is a bit unclear, in an interesting way. I'd like to clarify it a bit.
Yes, it is correct to state that an observer toward whose position an object moving faster than the speed of sound is moving, will hear nothing from that object prior to its arrival. The unclarity is in the follow-on question: "Immediately,
when it reaches him or a few seconds later?" You imply that the object must necessarily occupy the same location as the observer, and if this happens and the object is larger than a grain of sand, said observer will cease to be interested in what happens afterward; the observer will either be dead or very busy dealing with a serious injury.
It is important to understand that there is a region surrounding the moving object and moving with it, which is composed of three components: positions where, at the current position of the object, there has not been time for the sound to reach that position from the current position of the object; positions where, at the current position of the object, there has been time for the sound to reach that position from the current position of the object; and postions on the edge of these two areas. This region approximately forms a cone (appropriately distorted by differences from place to place in the medium, such as temperature, pressure, or the presence of discontinuities in the medium), with the point at the current location of the object. As time passes, and the object moves, the position of this cone moves with the object. As the edge of the cone passes the observer's position, the observer will transition from a position where the object cannot be heard, to a position where it can.
If the object passes very, very close to the observer, then, the time delay between the passage of the object through a plane normal to the object's path and coincident upon the position of the observer, and the passage of the edge of the cone, will be very short, and the observer will essentially observe the object and the sound to "arrive" simultaneously. However, if the object passes through this plane quite a distance away, then the edge of the cone will not arrive for some time afterward, and the observer will not hear the object until "after it has passed." To put it another way, the arrival of the sound of the passage of a bullet passing very close to an observer will seem coincident with the passage of the bullet, and the sound of the shot from the rifle that fired the bullet will be somewhat later; however, the sound of the passage of a jet aircraft breaking the sound barrier will be considerably after the aircraft has "passed."
Furthermore, in the aircraft's case, the sound of its engines, and the shockwave created by its passage, will unite to produce what seems to the observer to be a very loud sound at the arrival of the cone's edge at the observer's position; this is called a "sonic boom." The bullet makes one too, but since it doesn't have onboard propulsion, and isn't large enough to make such a powerful shockwave, it is considerably less prominent.
2) Let's say we find an Earth-like planet and travel to it in a spacecraft that can reach a significant speed relative to the speed of light (let's say 50% the speed of light). An observer aboard this spacecraft keeps a telescope pointed at the planet and observes it at all times. It is correct to say that he would see events happening on the planet in "fast-forward"? Or would the shift... color shift... what's the name again?... thingamabob would render such observation useless anyway? Or am I totally off-course here?
First of all, just as with the sound wave, there is a cone-shaped region surrounding every object with respect to the arrival of light from that object. One thing I didn't discuss with respect to the arrival of sound from an object is the difference in the arrival of a sound generated by a non-moving object, and from a moving object. From a motionless object, sound spreads in a sphere. However, this sphere is actually a 4-cone in four-dimensional spacetime; the three-dimensional intersection of this 4-cone with our space at any moment in time is a sphere. When the object moves, then the 4-cone becomes manifest as a 3-cone in space; when it is motionless, we see only the spherical cross-section of the 4-cone.
And this situation is also true for light.
But the situation with respect to four dimensional spacetime of an object moving a significant fraction of the speed of light is very different from that of an object moving a significant fraction of the speed of sound. All moving objects are rotated in spacetime with respect to the observer; however, the degree of rotation for an object moving a significant fraction of the speed of sound, or for that matter many times the speed of sound, is negligible. The situation is very different for an object moving a significant fraction of the speed of light; in that case, this rotation is easily measurable. Furthermore, while it is possible for an object to move faster than sound, it is not possible for an object to move faster than light. And it is because of the geometry of four-dimensional spacetime that this is true.
Spacetime, unlike ordinary space, is not circularly symmetric; if we try to use ordinary circular trigonometry to describe spacetime, our equations do not accurately represent our observations. Instead, we must use hyperbolic trigonometry; and when we do so, then what we calculate matches what we observe. Spacetime is therefore hyperbolically symmetric; and the fact that space is circularly symmetric implies that the relation of the space dimensions to one another is circularly symmetric, but the relation of each space dimension to the time dimension is hyperbolic.
The angle around a circle is unbounded; one can attain any angular orientation and it will be defined. The situation with respect to a hyperbola is very different. For hyperbolic geometry and trig, there are angles that are undefined; and there are two such types of angles. The first type are angles whose value is infinite; this type are the equivalent of division by zero. They are meaningless. No representation of such an angle can be drawn in hyperbolic geometry.
The second type is not more difficult to envision, but is far more difficult to describe. You may have forgotten or never learned this, but "cone" and "hyperbola" have more meaning than they are usually envisioned to: both have two
lobes. What we ordinarily think of as a "cone" is actually only half of a cone; a true geometric cone has two ordinary cones meeting at their tips. In the case of a circle, ellipse, or parabola, this distinction is not material to the definition of the figure; but in the case of a hyperbola, it is. These four figures, circle, ellipse, parabola, and hyperbola, are generated by the intersection of a plane at various angles to the cone. A circle is generated by the intersection of a plane normal to the axis of the cone; an ellipse, by the intersection of a plane that is not normal to the axis, but whose angle to the axis is less than the angle of the cone; a parabola by a plane parallel to the edge of the cone; and a hyperbola by a plane whose angle to the axis of the cone is greater than the angle of the cone. The first three only appear in one lobe of the cone, because such planes only intersect one lobe; however, the hyperbola appears in both lobes, because such a plane intersects them both.
The second type of angle, therefore, represents an angle in the
opposite lobe of the cone from the one we are in. And to describe these angles,
we use imaginary numbers. Now, it's well understood what it means to have an angle whose value is a real number, that is, a complex number whose imaginary component is zero; and this is true for all four geometries, circular, elliptical, parabolic, and hyperbolic. But we have no idea what it really means to have an angle whose value is a complex number with a non-zero imaginary component; and this is so because our mathematics for describing space are all composed of real numbers, since space is circularly symmetric. Various means of interpreting this are offered; my favorite is Feynman's interpretation, which is that an object with such an angle is
moving backward in time.
There are many who will object that that is not what Feynman meant; but that is not the subject here, and it doesn't really matter whether that interpretation or another is correct, the fact of the matter is that we do not commonly observe such objects, and the implications of it are therefore immaterial. In addition, they are immaterial to your question. I will therefore drop the matter at this time, though I may take it up in another thread (and there is a candidate that has been languishing for quite some time that I may resurrect to discuss this with several here who have detailed knowledge in these matters, so that I can pick their brains and see if what I'm thinking makes any sense).
Now, what does all this mean? Well, it means that first, no material object can move at the speed of light, because to do so its angle in hyperbolically symmetric spacetime must be an undefined number; and second, no material object can move faster than the speed of light, because many of its parameters would be defined by values that contain imaginary numbers, and we have no idea what this means but we've never observed it, and likely are incapable of doing so; and furthermore, to accelerate to a speed greater than that of light involves at some point going the speed of light, which is undefined.
This way of looking at spacetime is called "Special Relativity," and is the brainchild of Albert Einstein.
Now, one of the implications of SR is that the effects of very fast motion are symmetrical to all observers; that is, there is no absolute frame of reference for motion, and so two observers moving relative to one another can each say, "I am motionless, and s/he is moving," and both be right. Whether this is true in the frame of a third observer is immaterial; there is no third observer who can definitively be stated to be motionless with respect to space. All motion is therefore relative, and that's why they call it "relativity." But there is an absolute standard of motion in relativity;
acceleration is absolute. An observer whose frame is
accelerating can state, "I am accelerating, and s/he is not," and the other observer, whose frame is not accelerating, can equally state, "I am not accelerating, and s/he is," and both be right, but the opposite is not true. The situations for motion and acceleration are therefore different.
The answer to your question, therefore, is that the observer on the ship would observe the events on the planet by blue-shifted light; but would also observe that they moved more slowly than equivalent events on the ship,
in that observer's frame. An observer on the planet would observe the same thing: that the light from events on the ship was blue-shifted, but that equivalent events proceeded more slowly on the ship than on the planet. From this arises the twin paradox.
The twin paradox asks, "Suppose two events separated by an amount of time dictated by physical law, happen both on the ship and on the planet. On which does the second event happen first, the ship or the planet, since the rates of time are different for them?" And the paradox is, one cannot define a good answer to this question. The answer for an observer on the ship is, "on the ship," and the answer for an observer on the planet is, "on the planet." How can these be reconciled?
The answer is, both are correct.
IN THEIR OWN FRAME OF REFERENCE. And there is no third "more correct" frame; because there is no absolute motion.
Now, in order to get the people on the ship and the people on the planet together, so that they can compare notes, one of two things has to happen: either the planet has to accelerate, or the ship has to decelerate. But this creates an asymmetrical situation; now, one of the observers has experienced an accelerated frame of reference, and the other has not; and
ACCELERATION IS ABSOLUTE, so now they have different histories, and one or the other will be observed to have experienced "more time," and both will agree upon this observation. They will furthermore agree that the one who has NOT accelerated has the correct observation; the one who did has had hir frame of reference altered by the acceleration in such a manner that hir perception of history
in hir new frame of reference will be the same as the unaccelerated observer's.
Hopefully that helps you think about these matters.
