hyperdimensional time to melt your brain
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hyperdimensional time to melt your brain
i'm hoping to get an answer from those actually studying physics regarding this, because i've come across an interesting theory extending relativity and quantum mechanics into 3 time dimensions. whether this is a credible assumption about reality, i've yet to decide one way or another for myself, leaning on agreeing with this point of view.
the theory (existics) isn't very well documented  it is the work of very few people, with one Gavin Wince seemingly doing most of it including PR, talks, and so on. point being they don't explain in enough detail the necessity for extra dimensions of time, rather the argument for it is provided but i am interested in other explanations that could lead to the same results.
what i've found is the simplified massenergy equivalence E=mc^{2}, when rewritten to define mass rather than define energy, we end up with some strange units.
Energy = Mass * (<distance traveled by light in 1 second = Distance> / <1 second = time>)^{2}
Energy * (<1 second = Time> / <distance traveled by light in 1 second = Distance>)^{2} = Mass
therefore the Kg 'unit' of mass (strictly not weight) could be said to have a proportionate value in units of Energy * Time^{2} * Distance^{2}.
to me this begs the question of what possible meaning a power of time can have in relation to mass, or if it is at all possible to define conventional notions of density using this approach to defining mass?
i could go further with examples but it can all be thrown out if this much of it doesn't work... so i'm curious if this usage of units to define mass (specifically the time component) has a physical analogy that doesn't lead one to conclude there are at least two dimensions of time.
my own interpretation of this mess is that age and the passage of time are related quantities but are distinct measurements. for instance one might think that treating the observance of time can be done the same way as observing speed / position  but thinking this through it doesn't make sense, how do you measure the rate of passage of time since the only instrument at our disposal is a chronometer (a clock), the timeequivalent of a tape measure... i'm inclined to think that you cannot directly measure the rate of passage of time if you need to rely on timederivatives to define such a relationship, it would look silly, "find the rate of change of time with respect to, er, time?". the problem comes down to us using instruments that rely on a notion of absolute / universal time  while relativity doesn't use a preferred space it appears to use a preferred 'time', so to speak.
existics goes on to eventually redefine mass as a negative volume of time, but i get ahead of myself since i haven't found a proof depicting that relationship...
the theory (existics) isn't very well documented  it is the work of very few people, with one Gavin Wince seemingly doing most of it including PR, talks, and so on. point being they don't explain in enough detail the necessity for extra dimensions of time, rather the argument for it is provided but i am interested in other explanations that could lead to the same results.
what i've found is the simplified massenergy equivalence E=mc^{2}, when rewritten to define mass rather than define energy, we end up with some strange units.
Energy = Mass * (<distance traveled by light in 1 second = Distance> / <1 second = time>)^{2}
Energy * (<1 second = Time> / <distance traveled by light in 1 second = Distance>)^{2} = Mass
therefore the Kg 'unit' of mass (strictly not weight) could be said to have a proportionate value in units of Energy * Time^{2} * Distance^{2}.
to me this begs the question of what possible meaning a power of time can have in relation to mass, or if it is at all possible to define conventional notions of density using this approach to defining mass?
i could go further with examples but it can all be thrown out if this much of it doesn't work... so i'm curious if this usage of units to define mass (specifically the time component) has a physical analogy that doesn't lead one to conclude there are at least two dimensions of time.
my own interpretation of this mess is that age and the passage of time are related quantities but are distinct measurements. for instance one might think that treating the observance of time can be done the same way as observing speed / position  but thinking this through it doesn't make sense, how do you measure the rate of passage of time since the only instrument at our disposal is a chronometer (a clock), the timeequivalent of a tape measure... i'm inclined to think that you cannot directly measure the rate of passage of time if you need to rely on timederivatives to define such a relationship, it would look silly, "find the rate of change of time with respect to, er, time?". the problem comes down to us using instruments that rely on a notion of absolute / universal time  while relativity doesn't use a preferred space it appears to use a preferred 'time', so to speak.
existics goes on to eventually redefine mass as a negative volume of time, but i get ahead of myself since i haven't found a proof depicting that relationship...
theanger Posts : 1247
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Re: hyperdimensional time to melt your brain
I think the universe is a discrete structure, best imagined as a graph.
Here are Seth Lloyd's insights: https://www.youtube.com/watch?v=t9zcBKoFrME
Here are Seth Lloyd's insights: https://www.youtube.com/watch?v=t9zcBKoFrME
Abstractness Posts : 470
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Re: hyperdimensional time to melt your brain
haven't had time yet to watch, can't while at work
but... quick check on Seth Lloyd on wiki and i think i know what he would say.
for reference i've been looking at theories like loop quantum gravity re discrete structure, so i'm visualizing spin networks / spinfoam / etc at this scale. until coming across existics (and even now) im in the LQG bandwagon for the time being.
from what i can tell having more than one dimension of time is not incompatible with a discrete universe, and for all we know time is quite likely to also be discrete at that scale (by extension of its deep connection to spacial dimensions via spacetime).
i suppose i should clarify the nature of the existics viewpoint  it is an interactional model; for things to exist they must interact with other distinct, noncomplementary things at least once. this applied to the concept of differing rates of time results in a structure of infinite relative frames of reference between any two objects.
again, not a whole deal of information available so i've formed my own rationale for infinite reference frames...
if you have two observers, A and B. A's perception of time is sped up, and B's perception of time is slower. from an arbitrary 'now' moment the age of both objects will grow with respect to their passage of time. classically and from what i can tell in relativity as well, rate of passage of time between two observers has always been treated as a ratio. ie, if A and B check their clocks at the start and at the end of some 'event', if A's clock is twice as far ahead as B's since the start, we would say A experienced twice the rate of time as B during that event.
by 'event' i do mean strictly an interaction of some sort between A and B. consider this: A and B will both experience something different, A will see and interact with (from A's point of view) a slowerreacting B, and viceversa for B interacting with A.
inspecting this situation i realized that A will interact with B in A's reference frame, relative to A's experience. what isn't immediately apparent is A's version of B will interact with B's version of A  A, as seen by B in B's rate of time, as seen by A in A's rate of time, and so on. this alludes to an infinite series (depthwise) of reference frames as well as a divergence of reality (A and B experience potentially two different results, at different rates, and possibly in a different order on larger scales). existics offers a much more involved analysis regarding this but i feel this is the easier way to explain the situation.
establishing that A and B's interaction event requires infinite reference frames, the picture starts to look like a repeated fraction. i'm still working through the math presented but existics comes up with this...
old approach to time:
2 units observed by A, vs 1 unit observed by B > A/B = 2 (A's passage of time is twice as fast).
A/B = N
B/A = U
N . U = 1
existics:
the equation A/B = 1 cannot be possible unless A is B (reasoning behind unique identity), and by definition two objects cannot be exactly identical, opposite or complimentary. instead it is replaced by equations 
A/B = N*
B/A = U*
N* . U* = I
I ends up being >=1
N* and U* are, due to the infinite reference frames, repeated fractions of:
N* = (A + ( A + (A + A / B) / B) / B) / B ... = (A + N*) / B = A / (B  1)
U* = (B  (B  (B  B / A) / A) / A) / A ... = (B  U*) / A = B / (A + 1)
B needs to be above 1, otherwise intermediate calculations must be used to first raise B above 1 and then normalize it back afterwards.
but from these figures you can see that the actual ratio of rate of passage of time is (if the theory is correct) nonlinear for two interacting objects. nonlinearity to me implies a necessity for more than one time dimension to describe this complex relation between age and rate of time for two interacting objects.
existics goes on to (poorly) explain a third time dimension, an axis of simultaneity, as well as making reference to 3 dimensional timevolume transformations.
i forget who but one physicist has explained this another way  we are always moving through time at the speed of light, when we move through space, we in a sense rotate our spacetime momentum from timelike to spacelike. in fact existics goes on to redefine acceleration as the ratio of two different rates of time, which to me makes a lot more sense in the context of gravitation...
so yeah, am i going mad or does this make any sense.
but... quick check on Seth Lloyd on wiki and i think i know what he would say.
for reference i've been looking at theories like loop quantum gravity re discrete structure, so i'm visualizing spin networks / spinfoam / etc at this scale. until coming across existics (and even now) im in the LQG bandwagon for the time being.
from what i can tell having more than one dimension of time is not incompatible with a discrete universe, and for all we know time is quite likely to also be discrete at that scale (by extension of its deep connection to spacial dimensions via spacetime).
i suppose i should clarify the nature of the existics viewpoint  it is an interactional model; for things to exist they must interact with other distinct, noncomplementary things at least once. this applied to the concept of differing rates of time results in a structure of infinite relative frames of reference between any two objects.
again, not a whole deal of information available so i've formed my own rationale for infinite reference frames...
if you have two observers, A and B. A's perception of time is sped up, and B's perception of time is slower. from an arbitrary 'now' moment the age of both objects will grow with respect to their passage of time. classically and from what i can tell in relativity as well, rate of passage of time between two observers has always been treated as a ratio. ie, if A and B check their clocks at the start and at the end of some 'event', if A's clock is twice as far ahead as B's since the start, we would say A experienced twice the rate of time as B during that event.
by 'event' i do mean strictly an interaction of some sort between A and B. consider this: A and B will both experience something different, A will see and interact with (from A's point of view) a slowerreacting B, and viceversa for B interacting with A.
inspecting this situation i realized that A will interact with B in A's reference frame, relative to A's experience. what isn't immediately apparent is A's version of B will interact with B's version of A  A, as seen by B in B's rate of time, as seen by A in A's rate of time, and so on. this alludes to an infinite series (depthwise) of reference frames as well as a divergence of reality (A and B experience potentially two different results, at different rates, and possibly in a different order on larger scales). existics offers a much more involved analysis regarding this but i feel this is the easier way to explain the situation.
establishing that A and B's interaction event requires infinite reference frames, the picture starts to look like a repeated fraction. i'm still working through the math presented but existics comes up with this...
old approach to time:
2 units observed by A, vs 1 unit observed by B > A/B = 2 (A's passage of time is twice as fast).
A/B = N
B/A = U
N . U = 1
existics:
the equation A/B = 1 cannot be possible unless A is B (reasoning behind unique identity), and by definition two objects cannot be exactly identical, opposite or complimentary. instead it is replaced by equations 
A/B = N*
B/A = U*
N* . U* = I
I ends up being >=1
N* and U* are, due to the infinite reference frames, repeated fractions of:
N* = (A + ( A + (A + A / B) / B) / B) / B ... = (A + N*) / B = A / (B  1)
U* = (B  (B  (B  B / A) / A) / A) / A ... = (B  U*) / A = B / (A + 1)
B needs to be above 1, otherwise intermediate calculations must be used to first raise B above 1 and then normalize it back afterwards.
but from these figures you can see that the actual ratio of rate of passage of time is (if the theory is correct) nonlinear for two interacting objects. nonlinearity to me implies a necessity for more than one time dimension to describe this complex relation between age and rate of time for two interacting objects.
existics goes on to (poorly) explain a third time dimension, an axis of simultaneity, as well as making reference to 3 dimensional timevolume transformations.
i forget who but one physicist has explained this another way  we are always moving through time at the speed of light, when we move through space, we in a sense rotate our spacetime momentum from timelike to spacelike. in fact existics goes on to redefine acceleration as the ratio of two different rates of time, which to me makes a lot more sense in the context of gravitation...
so yeah, am i going mad or does this make any sense.
theanger Posts : 1247
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Re: hyperdimensional time to melt your brain
At the moment I don't have time to think about 3 dimensional time, remind me in a month or so.
Abstractness Posts : 470
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Age : 27
Location : Switzerland
Re: hyperdimensional time to melt your brain
Jeezus Cripes! The title for this thread is so appropriate!
I'd imagine that any outside observers reading these threads would probably think that we're either insane, or nuclear physicists...
Same thing, in the end, I suppose...
I'd imagine that any outside observers reading these threads would probably think that we're either insane, or nuclear physicists...
Same thing, in the end, I suppose...
Re: hyperdimensional time to melt your brain
Those rads raddle your brains
Brodur Posts : 143
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Re: hyperdimensional time to melt your brain
a month later, and it makes much more sense to me. somehow.
pondering about this i came across something interesting which may have something interesting to say about present and period axises... light. well, photons at least.
this never made sense to me, hoping someone can answer it...
1. a photon takes time to travel between two spatial points, or another way to put it, a photon travels through space and time at a set rate.
2. anything moving close to the speed of light experiences an inversely proportional time dilation effect relative to everything else, via lorentz transform.
3. anything moving at the speed of light in theory would have zero time passing (in theory because only things that are massless can go exactly that fast, and they always moved that fast and always will for the duration of their existence).
to my knowledge, photons are dualnatured (wave and particle, not either and not both), correlated, propagating disturbances in the electric and magnetic fields. because the disturbance experiences time evolution itself (photon phase), does that mean that the photon undergoes changes of phase angle in its waveform as it travels?
if a photon undergoes a change, and moves at the speed of light, does that mean time evolution occurs even at light speed? otherwise it suggests that photons are 'timefrozen' simple packets of energy in the aforementioned quantum fields and don't have any internal dynamics at all...
pondering about this i came across something interesting which may have something interesting to say about present and period axises... light. well, photons at least.
this never made sense to me, hoping someone can answer it...
1. a photon takes time to travel between two spatial points, or another way to put it, a photon travels through space and time at a set rate.
2. anything moving close to the speed of light experiences an inversely proportional time dilation effect relative to everything else, via lorentz transform.
3. anything moving at the speed of light in theory would have zero time passing (in theory because only things that are massless can go exactly that fast, and they always moved that fast and always will for the duration of their existence).
to my knowledge, photons are dualnatured (wave and particle, not either and not both), correlated, propagating disturbances in the electric and magnetic fields. because the disturbance experiences time evolution itself (photon phase), does that mean that the photon undergoes changes of phase angle in its waveform as it travels?
if a photon undergoes a change, and moves at the speed of light, does that mean time evolution occurs even at light speed? otherwise it suggests that photons are 'timefrozen' simple packets of energy in the aforementioned quantum fields and don't have any internal dynamics at all...
theanger Posts : 1247
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Re: hyperdimensional time to melt your brain
ok so, thinking about this some more, found an interesting viewpoint with strange consequences.
a photon should not experience time. ok
a photon therefore, from its perspective, must not be moving either!?
that depends on your definition of 'moving'.
we define motion as a ratio of change in position relative to a change in elapsed time in a given reference fame (RF). in other words, v = dx / dt
when dt = 0 we get a singularity, which means our choice of coordinates is not suitable to describe the motion of a photon. it does not conclude that the photon is everywhere at once (or other silly things) simply because it gets from A to B in 0 time.
this is simple to prove actually... a spacetime diagram in a rest RF would show ordinary objects moving mostly vertical (parallel to time axis) and the closer dt gets to dx (dx / dt tending to 1 from 0 in natural units) the longer it takes for the same measure of time to be experienced for such objects. a photon in such diagrams is depicted moving at 45 degree lines. dx / dt = 1; that means that from the perspective (relative frame of reference) of a photon, however absurd, would show the photon's own motion entirely along the horizontal, having no vertical component!
this proves easily enough that as far as a photon is concerned, it exists only for a single instance of time. so far so good.
but what of this horizontal 'movement' that happens in an instant... remember that if the photon disappeared and reappeared, we would see only two points on the photon's spacetime diagram illustrating such a discontinuity of events (as well as on any regular object's spacetime diagram). but instead we have a line joining the two locations, and we know that this line exists in regular objects' spacetime diagrams sloped at 45 degrees. so the motion must not only be apparent, but is also physically real for all observers NOT moving at the speed of light.
this means that the motion of a photon is in fact motion in some sense in all frames of reference. what makes it special from the photon's frame of reference, is that it's momentum is entirely timelike.
this is where that third time dimension comes in, that makes it necessary to distinguish between distinct frames / slices of events (moments) and the amount of apparent time experienced as the duration of those moments. this means that all our traditional treatment of time conflates the concept of moment and instant... fascinating stuff...
edit  clarify:
regular objects have many instants of the same moment when they are not 'moving' and merely aging  standing still in space is, in a sense, drawing out the one moment indefinitely. the opposite happens with a photon, where it stands still in a single instant, but travels through many moments.
a photon should not experience time. ok
a photon therefore, from its perspective, must not be moving either!?
that depends on your definition of 'moving'.
we define motion as a ratio of change in position relative to a change in elapsed time in a given reference fame (RF). in other words, v = dx / dt
when dt = 0 we get a singularity, which means our choice of coordinates is not suitable to describe the motion of a photon. it does not conclude that the photon is everywhere at once (or other silly things) simply because it gets from A to B in 0 time.
this is simple to prove actually... a spacetime diagram in a rest RF would show ordinary objects moving mostly vertical (parallel to time axis) and the closer dt gets to dx (dx / dt tending to 1 from 0 in natural units) the longer it takes for the same measure of time to be experienced for such objects. a photon in such diagrams is depicted moving at 45 degree lines. dx / dt = 1; that means that from the perspective (relative frame of reference) of a photon, however absurd, would show the photon's own motion entirely along the horizontal, having no vertical component!
this proves easily enough that as far as a photon is concerned, it exists only for a single instance of time. so far so good.
but what of this horizontal 'movement' that happens in an instant... remember that if the photon disappeared and reappeared, we would see only two points on the photon's spacetime diagram illustrating such a discontinuity of events (as well as on any regular object's spacetime diagram). but instead we have a line joining the two locations, and we know that this line exists in regular objects' spacetime diagrams sloped at 45 degrees. so the motion must not only be apparent, but is also physically real for all observers NOT moving at the speed of light.
this means that the motion of a photon is in fact motion in some sense in all frames of reference. what makes it special from the photon's frame of reference, is that it's momentum is entirely timelike.
this is where that third time dimension comes in, that makes it necessary to distinguish between distinct frames / slices of events (moments) and the amount of apparent time experienced as the duration of those moments. this means that all our traditional treatment of time conflates the concept of moment and instant... fascinating stuff...
edit  clarify:
regular objects have many instants of the same moment when they are not 'moving' and merely aging  standing still in space is, in a sense, drawing out the one moment indefinitely. the opposite happens with a photon, where it stands still in a single instant, but travels through many moments.
theanger Posts : 1247
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Re: hyperdimensional time to melt your brain
finally have a bit of clarity on this whole thing, took a few months to sit down and work through the concepts...
the best hole to poke in relativity is the event horizon concept, so lets start there...
RF = relativistic frame; ill get tired of writing it so many times
as you know (or don't) if you're in a uniformly accelerating RF (a RF that undergoes constant acceleration relative to everything else in the universe), then your trajectory in flat spacetime is more or less an ideal hyperbola. since all motion is relative, the same applies for anyone looking at you from a stationary RF  you appear to be accelerating to a stationary RF and the stationary RF relative to you appears to be accelerating in the same way but opposite direction.
suppose you construct a trajectory like this in 1+1 dimensions (t,x), it takes the form:
1 = x^{2}  t^{2}
conveniently the asymptotes represent the light cones of an event at (0,0), assuming a natural coordinate system where c = 1.
the hyperbola here has the special property that if we wanted to perform a lorentz transform for an object on the hyperbolic path, relative to the origin, the point would simply move to a different location on the hyperbola, therefore the path does not change but specific values of x(t) would change. details aside...
suppose you have two people A and B, in a spaceship moving within this accelerated RF, and shortly before turning away from the origin (as t approaches 1 from below, for example) A ejects and follows a constantvelocity RF (a line path) away from B.
A and B will both see eachother accelerating away from one another. however something, concerning, happens when A's path crosses the x=t asymptote  A can still see B afterwards, but the reverse is not true, B cannot see A age or move beyond the point where A crosses the light cone (asymptote). the asymptote is said to be an event horizon or acceleration horizon depending on who you ask.
Why is this a problem?
suppose A separates from B at t = 1, x = sqrt(2)
A will cross B's event horizon at x = t = 0.41421 and will hit x = 0 at t = 1
if we examine where everything is at t = 1, we can see that A has reached the center of our plot / graph at (0, 1) and B is still accelerating away and is now at (2^{0.5}, 1). clearly A is on the side of the horizon that B is unable to observe.
now, objectively, this is not a problem in itself. we know in our absolute space coordinate system (the graphs / plots) where everything is and when. the problem is, if our RF was centered on B OR on A, we could not. another way to say the same thing is:
"all the above observations were taken in C's RF, which is stationary relative to these trajectories".
from here on let's assume the identity of a third observer / person called C. for argument's sake lets say C was, in its RF, at x = 1 at all times. C could start off moving in positive x, it doesn't really matter, the point is the reality of B and A exist equally for C, but A does not exist for B anymore beyond some 'age' of A as seen by B & C. this easily proves that the reality of A depends on where you are and how you're moving; there are many RF's where A cannot exist, and many RF's which contradict those observations.
the problem then becomes consistency  physics MUST be consistent for all observers everywhere regardless of the consequences, unless you subscribe to the theory that physical constants and laws are locally constant only (in some far away space Pi = 3 for instance and gravity is half as strong; i disagree but i can't disprove it either). and very clearly, physics is not being consistent, since depending on your choice of coordinates A either exists or it does not.
to say that A disappears would violate conservation laws. to say it doesnt requires a description for A after it crossed the event horizon in B's RF for t > 0.41. to say it exists in one frame and not another does not necessarily violate conservation laws however you will not find a comfortable answer without resorting to additional dimensions / degrees of freedom.
i think this is a very simple example but it demonstrates the point  that relativity is bogged down by poor choice of coordinates, or lacks an additional freedom (dimension) to express the full nature of A as seen by B and C.
earlier i mentioned the third time dimension is hard to explain, because we mostly deal with rate and age. i was wrong:
T1 = period, clock time.
T2 = present, moments that can have infinitesmal duration. ratio of moments to clock time IS lorentztransformderived time dilation.
T3 = passage, rate of time per time; how quickly does time itself move / propagate / evolve / whatever.
from the first post i explained how A and B interact through an infinite number of RF's in between them, in other words with a virtual / residual 'image' of the distant object and vice versa. that means that the image itself may and provably is distinct from the original object. therefore, in this example, 3D time explains the two viewpoints as such:
A in B's RF stops moving along T1, and the lost 'speed' along T1 shifts into T2  from B's perspective it is moving through present moments without aging. the A that B can 'see' and 'interact' with, is not the same A for which we have plotted a linear trajectory!
to reconsile this the theory goes on to suggest that varying T3 implies parallel universes that differ only by the rate of time. since B is in an accelerated frame, it is no longer dealing with A from its original T3 but with an image of A in the lower T3 (since rate of time lowers with velocity), and since the two A's are, for our intents and purposes, distinct realities, we are allowed to have conflicting reference frames. to the point, 3D time allows one to recover A's position and momentum after it has crossed the event horizon by using temporal coordinates from B's frame.
i'll leave it there for now, but the theory goes on still to suggest that events lower in T3 the longer they drag out in T1. to put it simply, light, which in its frame is instantaneous, travels through T3 on its journey as well, which has ramifications about its frequency and wavelength measurements no longer agreeing, and even proposing the mechanism for light to redshift over distance, negating the necessity for a big bang. getting far too ahead of myself so ill stop there...
the best hole to poke in relativity is the event horizon concept, so lets start there...
RF = relativistic frame; ill get tired of writing it so many times
as you know (or don't) if you're in a uniformly accelerating RF (a RF that undergoes constant acceleration relative to everything else in the universe), then your trajectory in flat spacetime is more or less an ideal hyperbola. since all motion is relative, the same applies for anyone looking at you from a stationary RF  you appear to be accelerating to a stationary RF and the stationary RF relative to you appears to be accelerating in the same way but opposite direction.
suppose you construct a trajectory like this in 1+1 dimensions (t,x), it takes the form:
1 = x^{2}  t^{2}
conveniently the asymptotes represent the light cones of an event at (0,0), assuming a natural coordinate system where c = 1.
the hyperbola here has the special property that if we wanted to perform a lorentz transform for an object on the hyperbolic path, relative to the origin, the point would simply move to a different location on the hyperbola, therefore the path does not change but specific values of x(t) would change. details aside...
suppose you have two people A and B, in a spaceship moving within this accelerated RF, and shortly before turning away from the origin (as t approaches 1 from below, for example) A ejects and follows a constantvelocity RF (a line path) away from B.
A and B will both see eachother accelerating away from one another. however something, concerning, happens when A's path crosses the x=t asymptote  A can still see B afterwards, but the reverse is not true, B cannot see A age or move beyond the point where A crosses the light cone (asymptote). the asymptote is said to be an event horizon or acceleration horizon depending on who you ask.
Why is this a problem?
 some quick math:
position of separation:
1 = x^{2}  t^{2}
x^{2} = 1 + t^{2} = 2
t = 1
x = 2^{0.5}
slope of A's path afterwards:
t = (x^{2}  1)^{0.5}
dt/dx = x(x^{2}  1)^{0.5}
m = t'(2^{0.5}) = 2^{0.5}(2  1)^{0.5} = + 2^{0.5}
since t<0, m = 2^{0.5}
linear locus of A's trajectory:
t_{A}  (1) = 2^{0.5}(x_{A}  2^{0.5})
t_{A} + 1 = 2^{0.5}(2^{0.5}  x_{A})
t_{A} = 1  2^{0.5}x_{A}
intercept with x = t:
let x = t;
t_{A} = 1  2^{0.5}t_{A}
t_{A} + 2^{0.5}t_{A} = 1
t_{A}(1 + 2^{0.5}) = 1
t_{A} = 1 / (1 + 2^{0.5}) ~= 0.41421
position of A and B at t = 1:
x_{A}(1) = 0
x_{B}(1) = 2^{0.5}
suppose A separates from B at t = 1, x = sqrt(2)
A will cross B's event horizon at x = t = 0.41421 and will hit x = 0 at t = 1
if we examine where everything is at t = 1, we can see that A has reached the center of our plot / graph at (0, 1) and B is still accelerating away and is now at (2^{0.5}, 1). clearly A is on the side of the horizon that B is unable to observe.
now, objectively, this is not a problem in itself. we know in our absolute space coordinate system (the graphs / plots) where everything is and when. the problem is, if our RF was centered on B OR on A, we could not. another way to say the same thing is:
"all the above observations were taken in C's RF, which is stationary relative to these trajectories".
from here on let's assume the identity of a third observer / person called C. for argument's sake lets say C was, in its RF, at x = 1 at all times. C could start off moving in positive x, it doesn't really matter, the point is the reality of B and A exist equally for C, but A does not exist for B anymore beyond some 'age' of A as seen by B & C. this easily proves that the reality of A depends on where you are and how you're moving; there are many RF's where A cannot exist, and many RF's which contradict those observations.
the problem then becomes consistency  physics MUST be consistent for all observers everywhere regardless of the consequences, unless you subscribe to the theory that physical constants and laws are locally constant only (in some far away space Pi = 3 for instance and gravity is half as strong; i disagree but i can't disprove it either). and very clearly, physics is not being consistent, since depending on your choice of coordinates A either exists or it does not.
to say that A disappears would violate conservation laws. to say it doesnt requires a description for A after it crossed the event horizon in B's RF for t > 0.41. to say it exists in one frame and not another does not necessarily violate conservation laws however you will not find a comfortable answer without resorting to additional dimensions / degrees of freedom.
i think this is a very simple example but it demonstrates the point  that relativity is bogged down by poor choice of coordinates, or lacks an additional freedom (dimension) to express the full nature of A as seen by B and C.
earlier i mentioned the third time dimension is hard to explain, because we mostly deal with rate and age. i was wrong:
T1 = period, clock time.
T2 = present, moments that can have infinitesmal duration. ratio of moments to clock time IS lorentztransformderived time dilation.
T3 = passage, rate of time per time; how quickly does time itself move / propagate / evolve / whatever.
from the first post i explained how A and B interact through an infinite number of RF's in between them, in other words with a virtual / residual 'image' of the distant object and vice versa. that means that the image itself may and provably is distinct from the original object. therefore, in this example, 3D time explains the two viewpoints as such:
A in B's RF stops moving along T1, and the lost 'speed' along T1 shifts into T2  from B's perspective it is moving through present moments without aging. the A that B can 'see' and 'interact' with, is not the same A for which we have plotted a linear trajectory!
to reconsile this the theory goes on to suggest that varying T3 implies parallel universes that differ only by the rate of time. since B is in an accelerated frame, it is no longer dealing with A from its original T3 but with an image of A in the lower T3 (since rate of time lowers with velocity), and since the two A's are, for our intents and purposes, distinct realities, we are allowed to have conflicting reference frames. to the point, 3D time allows one to recover A's position and momentum after it has crossed the event horizon by using temporal coordinates from B's frame.
i'll leave it there for now, but the theory goes on still to suggest that events lower in T3 the longer they drag out in T1. to put it simply, light, which in its frame is instantaneous, travels through T3 on its journey as well, which has ramifications about its frequency and wavelength measurements no longer agreeing, and even proposing the mechanism for light to redshift over distance, negating the necessity for a big bang. getting far too ahead of myself so ill stop there...
theanger Posts : 1247
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