Posted on 02/21/2008 7:47:47 PM PST by NormsRevenge
So what is the calculation of twin 1 in regards to twin 2. If moving in the same direction, no clock difference. When moving in opposite directions - I imagine both would calculate the other has a serious time dilation problem.
In the first instance, they are in the same inertial reference frame. In the second instance, they are not in the same frame. A couple of transformations could place them in the same frame however, by splitting the relative velocities between them.
Note that one can not determine absolute clock rates from relative velocities for either case here, because only relative velocity info is known. The clock rates for both cases are indeterminate. That indeterminacy is the basis for the "twin paradox". One must have info regarding prior accelerations of the pair to get a handle on absolute clock rates. That info regarding acceleration(s) is what breaks the paradox.
In the cases presented in prior posts, Earth was the reference point to determine the acceleration histories. Note that even there, the velocity of the Earth is not known, so the absolute velocities of all the other movers aren't known either.
The reference to determine absolute velocities and clock tick rates, is the inertial frame comoving with the expansion of the universe. Any inertial frame is equivalent to that. The Big Bang should not be thought of as an explosion where 4d stuff flies out from some point. It can be seen as a phase transition in the underlying stuff, or a change in the dimensional structure of space, that happened everywhere at once, and resulted in the appearance of the the 4d stuff and the new 4d spacial structure of space it's embedded in.
The simple picture... The big bang began with an inflationary period, where the radial "expansion" of space was faster than the speed of light. The radius varied with time as:
R(t)=e-E*t,
where E ~ Tc2/Mp. Mp is the energy equivalent of the mass of the proton, Tc is the critical temperature when inflation begins, and the dimensional units are h=c=1. When h=c=1, energy and time are reciprocals.
The inflation period lasted ~10-33 secs and resulted in an expansion on the order of 1030. So the universe essentially came to size very early and happened everywhere. That is why the universe looks flat, because it is so large. Only a small part of the universe can be seen from here. Thereafter, the expansion is ~t2/3.
Both inflation and expansion can be seen as a change in dimensional structure. Once that change in dimensional structure was substantially established at the end of inflation, then particles began to drop out, or appear in the new 4d space and interact. That is the point where "stuff" began to acquire velocities different from the rest frame of space itself and clocks which tick at lower rates than the rest frame of the expanding space.
To estimate an absolute velocity one looks at the motions of the centers of gravitationally bound structures like galaxies. The expansion of space can be obtained by redshift data and by removing the extraneous velocities of those objects acquired through interactions from the inflationary period. Those extraneous velocities are those that show up as deviations in redshift data.
Should be ∝t2/3...
Actually it's a term used instead of admitting that current cosmological models don't work out mathematicly in certain situations. (ie. why do the edges of galaxies rotate just as fast as the center?)
And how come acceleration is not a term in the time dilation formula? (I've heard your explanation about relative accelerations before, I cannot really comprehend it yet). I've read that it surprised a lot of people that: while clock rates have to be adjusted in GPS satellites WRT earth, they do not have to be adjusted with respect to each other, even though they have non-zero relative velocities - multiple disparate adjustment likely a mathematical impossibility - so a good thing.
An interesting paradox.
Can any object have a rotational component about the nominal center of the universe, if such exists?
LOL. I don't know boss. The details aren't known, only a general overview. What is known is that the universe appeared as a phase change of part of the energy contained in the vacuum. That's an event which just requires some nucleating event as a trigger. That phase change in the energy also accompanied by a change in dimensional geometry that corresponds to the new phase.
This universe has 4 extended dimensions and 6/7 curled up dimensions. Prior to inflation, all 10/11 dimensions were curled up, where the radius of curvature is on the order of the Planck length. Inflation is the interval during which those 4 dimensions we observe expanded from a Planck radius to the point where R(t2/3) at ~10-36secs. Up to that point, the forces of nature were indistinguishable/unified. Inflation took the Planck length radius to ~1μ. That's about the point where the universe still looks like a hot vacuum, and 4d hasn't quite emerged. After that is when massive particles such as quarks started to drop out.
"I guess that the inflation solves the problem of the universe escaping an instant black hole death?"
There was no black hole in the beginning. A black hole is a collection of matter and radiation, and black holes aren't all the same. In the beginning, the composition included an equivalent amount of antimatter. The vacuum is composed of both. The final composition of this universe contains ~10-9 more matter than antimatter. That was established after the expansion phase began. Some high energy antimatter particles decay at a faster rate than their matter equivalents by the weak force. The full story isn't known, but that's part of the reason for the excess matter.
"how come acceleration is not a term in the time dilation formula?"
That time dilation formula is from special relativity and applies only to velocity. Time dilation is a function of velocity, not acceleration. That applies also in general relativity also. Acceleration is velocity as a function of time, or a change in the clock rate as a function of time. The clock rate itself depends only on the instantaneous velocity and an acceleration just steps the clock rate down. Info regarding which frame is really moving, or has accelerated just allows IDing the slower clock.
"I had said this before, "Clocks also run slow in the curved space of gravitational fields, even if the particle isn't moving in translation." That was wrong. Clocks do not change tick rate at all by being hung in a gravitational field, or curved spacetime. The clock rate is only a function of velocity. I had been misled by what said In those cases, the signal simly looses momentum as it escapes the field, or gains it as it enters. Stationary clocks in the field run at the same rate.
"while clock rates have to be adjusted in GPS satellites WRT earth, they do not have to be adjusted with respect to each other, even though they have non-zero relative velocities - multiple disparate adjustment likely a mathematical impossibility"
The satellite's send their position data and the device on the ground does the calcs. It's the precision and accuracy of the calculations and info considered by the device on the ground that matters. Once the clocks are sychronized, they're in synch everywhere, with good calcs. The big errors in GPS are distortion of the signal by the ionospere, then the atmosphere. More expensive equipment is calibrated at local surveyed points on the Earth to measure those effects "at the moment" and that data is used to make corrections for measurements made around that time.
"Can any object have a rotational component about the nominal center of the universe, if such exists?"
The net spin of the universe is zero. Conceputually a particle pair produced early on could have moved in a partial orbit of the iniverse. The other particle of the pair would cancel out that component, because of conservation of angular momentum.
This universe has 4 extended dimensions and 6/7 curled up dimensions.
Are dimensions things of physical reality, or mathematical constructs to simplify calculations (like states)? I've always been fascinated by scale, which appears to me to be a somewhat comprehensible property of all 4 dimensional measures. (Or perhaps scale is another dimension?) Scale, tiny to gigantic, seconds to eons, at least we have some sense of it, meager though it is. For other then x,y,z,time dimensions: not so much. But X,Y,Z or R,theta, phi? Now, past, present, future? But in time, clearly we exist in a instant, so are insubstantial in space-time, a projection on some momentary screen, instantly of no weight, volume, or consequence in 4-d. Even the screen perhaps being part of the production. Little surprise that formula can fail.
There was no black hole in the beginning. A black hole is a collection of matter and radiation, and black holes aren't all the same. In the beginning, the composition included an equivalent amount of antimatter. The vacuum is composed of both.
So anti-matter does not have normal gravity? Is it repulsive? Does it interact with matter gravitationally? Does anti-matter fall into a normal matter black hole? Feynman believed anti-matter to be time reversed matter. Does his theory still hold?
Time dilation is a function of velocity, not acceleration.
But you couple the real-world effect of acceleration as a factor in determining time differences between different frames. How does one account for it in calculations? I cannot see how the paradox is resolved!
Points, waves, strings, dimensions, all all other similar things in math are representations of things used to know and understand reality. Dimensions are like tracks the energy follows, they're used to note where the energy is located and to describe the location.
Scale isn't a dimension. It's a feature of a dimension, other measure, or a region defined by dimensions. 4d space allows for an arbitrarily large scale expanse. The curled dimensions don't, but they do have real effects that show up in 4d. The Large Hadron Collider could provide evidence for them after it starts generating data. Due to symmetry, it could be that instead of the 4d dimensions being extended to large radius, compared to the original r, all the dimensions started with large r and the 4ds shrunk. Some calcs are easier looking at it that way.
"Feynman believed anti-matter to be time reversed matter. Does his theory still hold?"
Antimatter is a negative energy solution in quantum physics. Dirac is the one that recognized that negative energy flowing backwards in time, looks like positive energy flowing forward in time, if the sign of the charge is reversed. Schrodinger was hung up on that and chose his equation to work with QM, because of that conceptual problem. So, antimatter has normal mass and gravitates. The first antimatter pair looked at was the electron-positron pair.
Only when the pair is combined does it not gravitate. In that case, the pair can't be seen at all, except for fleeting moments, shorter than t=h/2πEpair. The dielectric constant of the vacuum and cosmological constant are due to that. As is the evaporation of black holes after they suck up antiparticles and the matter partner escapes, or fine structure in spectroscopy.
"But you couple the real-world effect of acceleration as a factor in determining time differences between different frames. How does one account for it in calculations?"
Velocity is v=dx/dt. Acceleration is a=dv/dt.
a dt=dv
∫a dt = ∫dv = v
If a is constant, v=a*t
The paradox can not be resolved, unless one has extra info. One must know which one was moving faster. That means one must simply know which one accelerated, or which one moved, or moved faster. The numerical value of the acceleration is completely irrelevant. Only the velocity matters. It doesn't even matter if the particle turns, which requires an acceleration. Only the tangential velocity matters and if it's a quick turn compared to hte whole trip, ignore it.
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