The concept of the Planck length can be fully understood on the basis of only a bit of dimensional analysis applied to the apparent fundamental constants of c, h, and G (whose existences had all been identified long before black holes and even photons were considered to be “real”). It’s implications are by no means yet fully understood, but it is obvious that they will have something to do with the relationship between quantum and gravitational phenomena – perhaps (but not necessarily) as a limit on the possible “sizes” of black holes and/or the wavelengths of “photons”.
Author: alan
Simultaneity and Synchronization
Identical clocks in relative motion can only be synchronized from the point of view of an observer relative to whom they both have the same speed. To anyone else they seem to be progressing at different rates and so are always unsynchronized and there’s no “becoming” about it. There is a sense in which the relativity of simultaneity forces the clocks to be unsynchronized, but it’s generally easier to understand the other way around so I’ll look at that first.
Imagine that, when you pass by me at 86% of the speed of light, we set our clocks to both read, say, t=0. Then they will both read the same time at that event but will not be synchronized because each will see the other as running at just half speed.
and that I also use light signals to set a clock that is fixed in my frame one light year away (in the direction towards which you are headed) to also read t=0 at what seems to me to be the same time.
Then when you reach that clock it will read t=1/.86 years but your own clock will read just 0.5/.86 years (so your clock and my remote clock are not synchronized). I will attribute that difference to your clock running slow but from your point of view the time on your clock is correct because the distance which I saw as 1 light year appears to you to be only half of that.
But from your point of view it has seemed that my clocks were running slow. So according to you, the clock you see as reading 1/.86 years must have started at a time 2/.86 years ago which is long before the time when we passed one another. Or in other words the clock-setting event I thought was simultaneous with our meeting was not simultaneous according to you.
Now what I have shown here is just that relativity of simultaneity is a mathematical consequence of the symmetric nature of the asynchronization of the clocks. But despite the lack of “becoming” there is also a logical connection the other way.
Imagine that I send a message to reach you when I think you are 1 year away from reaching me (ie at a distance of 1/.86 light years when travelling at 0.86c) and ask you to set your clock at t=-1 at that time and that I set my own clock at t=-1 at the moment I think that message reaches you. But relativity of simultaneity says that you will think I made a mistake and that those two clock settings did not happen at the same time. You will actually think that the time when I asked you to set your clock was after I had set my own and this will cause you to expect that when you reach me my own clock would have got to t=1 if running at the same speed as yours but yours will just be at t=-0.5. And since my clock will actually be reading t=0 at our meeting this will lead you to conclude that it must be running slowly (and so that they cannot be synchronized).
Black Hole Singularities
A Quoran asks: Do black holes have a singularity? (A point inside the event horizon which is infinitely dense)?
YES, black holes are theoretical objects in the theory of General Relativity which are usually associated with singularities of space-time. But NO such a singularity is NOT “a point inside the event horizon”. As a point in space-time the black hole’s singularity is actually an event rather than a particular point or location in space; and in fact for every point inside the event horizon, the singularity event is in the future of any particle that is ever at that point. In terms of the way it looks to an observer inside the event horizon, there is no longer any sense of any “centre” but oneself, and the theoretically predicted experience is not of falling inwards but rather of getting crushed by the collapse of everything around you. ….or something like that!
Length Contraction Inherent in Maxwell Equations?
There are two senses in which length contraction is “inherent” in the Maxwell/Heaviside equations.
One sense is that in some cases length contraction follows from Maxwell’s equations. FitzGerald and Lorentz showed that if the structure of matter is determined by electromagnetic forces then the equilibrium spacing of stationary particles due to purely electrostatic forces would become contracted when they are in motion due to the presence of additional magnetic forces. [And one effect of this contraction, together with a related slowing and desynchronization of clocks, would be that moving observers (using their own contracted measuring rods and slowed clocks) would not notice any effect of their own motion on the apparent speed of light (or in fact on any electromagnetic phenomena).]
Another sense is that length contraction is necessary to preserve the form of Maxwell’s equations. Poincare had already noted that if the coordinates used by a moving observer were related to those of one who is stationary by a standard Galilean transformation (ie by just a progressive shift of position with no change of length and time scales) and if Maxwell’s equations applied to (say) the stationary one then they would have to be modified in order to make correct predictions for the other. And he showed that the only kinds of coordinate transformation that leave the form of Maxwell’s equations unchanged are those involving length contraction and time dilation.
[Einstein then noted that if all the laws of physics, written in terms of the natural coordinates of an observer, have a form that is independent of the state of motion of the observer, then there is in fact no way to tell which of two relatively moving observers is “truly” stationary and whose clocks are “truly” synchronized. He therefore sought to express all the laws of mechanics (eventually also including gravity) in a form which used only the actually measured coordinates of each observer rather than those of some assumed fixed “rest”(or “aether”) frame (which would have been more complicated to do if possible, but was not actually possible to do properly because no observers could actually tell whether or not they were actually in motion).]
Why is photon neutral?
If a gauge invariant quantum field theory (QFT) has a Lie group gauge invariance, the matter fields are defined in the fundamental representation of the group, and the gauge fields are defined in the adjoint representation of the group. If both representations are non trivial, the quantum fields in both representations carry the gauge charges. In the case of the U(1) group the adjoint representation is trivial, therefore the gauge field (photon) is also defined in the fundamental representation. Here only the matter field (electron) carries gauge charge (coulomb charge) and the gauge field does not carry gauge charge. In other words the photon is electrically neutral.
Why is photon electrically neutral?
If a gauge invariant quantum field theory (QFT) has a Lie group gauge invariance, the matter fields are defined in the fundamental representation of the group, and the gauge fields are defined in the adjoint representation of the group. If both representations are non trivial, the quantum fields in both representations carry the gauge charges. In the case of the U(1) group the adjoint representation is trivial, therefore the gauge field (photon) is also defined in the fundamental representation. Here only the matter field (electron) carries gauge charge (coulomb charge) and the gauge field does not carry gauge charge. In other words the photon is electrically neutral.
Superposition Independent of Space and Time
The spin observables for an elementary particle can be studied without any reference to the particle’s position. They are then modeled as operators on a finite dimensional Hilbert space of spin states that is essentially independent of the infinite dimensional Hilbert space of “wave” functions on position space (which identify those aspects of the particle’s state that are related to its position observables). And in this context for example the eigenstates for any spin component of a spin 1/2 particle are superpositions of eigenstates for other components.
Now you might (correctly) think that this talk of spin components means that we must still be thinking about directions in physical position space. But in fact the study of any two-valued observable (with values “Yes” and “No” or “True” and “False”) forces us to also consider other observables whose eigenstates are superpositions of the Yes and No eigenstates and whose relationship to the original observable are mathematically equivalent to those between spin components in different directions despite not actually having any connection with directions in physical space. And in quantum computing, although spin directions for a single particle are often used as a conceptual model, the choice of Yes/No observable might in practice be something different.
Of course, to study the progress of a quantum calculation, while disregarding space we still need to consider the time evolution of the system; but some aspects of the relationships between possible outcomes can be studied without reference to time in a similar way to how the time-independent Schrodinger equation can be used to study stationary states and energy levels of an atom or molecule. And in these types of analysis the question of whether and how some states are superpositions of other states is still relevant even though there is no reference to either space or time involved.
Black Hole Fantasy
A black hole has been spotted heading towards Earth, and we have 200 years before it arrives. Does our species have a chance of survival?
Perhaps. But not on Earth. (Assuming any kind of astronomically plausible black hole, there is no way of avoiding substantial perturbation of the Earth’s orbit followed by tidal distortion and probable destruction of the Sun.)
However, it may be possible to colonize some asteroids and by small manipulation of their orbits ensure that they get slingshotted so as to end up at sufficient distance from the accretion disc (into which the sun and planets will be converted by tidal forces) so that the body of the asteroid is sufficient to shield its residents from the radiation.
Since the colonies will of necessity be small, most humans will be stuck on Earth and not survive; but if sufficient genetic diversity is brought along in the form of germ cells and/or frozen zygotes or blastocysts then perhaps the species itself might survive and adapt to the new low gravity environment.
Less Time for Same Distance?
For the traveller it’s not the same distance; due to length contraction it’s a smaller distance, and so takes less time. For the observer who sees the distance as a full light year it appears to take more than a year, but the traveller’s clocks appear slowed down and so will advance by less than that. (The time experienced by the traveller will still be more than a year unless the travel speed exceeds , ie just over 70% of the speed of light.)
Timelike vs Spacelike
If two distinct events are such that there is any inertial frame in which they have zero spatial distance between them, then there is no frame in which they are simultaneous and so they are said to be “timelike separated”. This is because the frame in which they have zero spatial separation corresponds to an observer who sees them both happening at the same place one after the other; and for any other inertial observer, the time between them is also nonzero (since for any v<c the Lorentz contraction factor is never zero).
On the other hand, any two events which are seen as simultaneous by some inertial observer (which is different from being seen simultaneously by that observer!) are said to be “spacelike separated”. But the appearance of simultaneity is relative to the observer and only happens in one particular frame. Other inertial observers won’t see the events as simultaneous but all will agree that it would take faster than light travel to see them both at the same place – eg to actually be present at both of them.