An Accelerated Array of Clocks in Special Relativity: A Meaningful “NOW-at-a-Distance"

__________________________________________________________________________

Abstract:

Einstein showed us how an array of unaccelerated synchronized clocks could be set up, extending throughout all space, with a fixed spacing between adjacent clocks. To accomplish that, we only need to assume that the speed of light is exactly the same in all inertial (unaccelerated) reference frames: 186,000 miles per second (which is the fundamental assumption of special relativity). That establishes a meaningful “NOW-at-a-distance”, at each instant in that particular inertial reference frame. It allows the question to be asked and answered (by the people stationary in that reference frame): “How old is that particular distant person, RIGHT NOW? That answer MUST be MEANINGFUL to them. If it ISN’T meaningful to them, then the speed of light in that inertial frame can’t be 186,000 miles per second (which is the only thing that the synchronization was based on). So if that answer ISN’T meaningful to them, then special relativity can’t be correct. Assuming that special relativity IS correct (which I certainly believe is the case), the NOW-at-a-distance given by the array of unaccelerated clocks MUST be considered to be meaningful by the people stationary in that inertial reference frame.

But how about an array of clocks which are all equally ACCELERATED? Can they establish a meaningful NOW-at-a-distance for a person who is stationary with respect to that array of accelerating clocks (and co-located with one of them)? The answer is YES! Those clocks can’t be synchronized: they run at different rates. But if they are synchronized at the instant that they all start the acceleration, then a person (he), stationary with one of the clocks, can CALCULATE the current time on each of the other clocks, at each instant of his life during the acceleration. THAT effectively establishes a NOW-at-a-distance for him: it allows him to determine the current age of any particular distant person (her), according to him. And he MUST consider that answer to be fully MEANINGFUL to himself.

__________________________________________________________________________


Section 1. Some Additional Terminology

To “flesh out” the second paragraph in the above abstract, some additional terminology is needed.

It will help if we imagine that each of the accelerated clocks has a co-located human with it. We choose one of those humans to be the human whose “perspective”, or “point of view”, about simultaneity-at-a-distance we desire. I call that human the “Interested Observer”, abbreviated as the “IO”. For example, he is “interested” in what the current age of the distant “home twin” is (because he is the traveling twin in the twin paradox).
The human co-located with each of the other clocks in the array is called a “Helper Friend”, abbreviated as an “HF” (and sometimes ending in a suffix when I need more than one helper friend, like HF1 and HF2). Suppose the IO (he) wants to know the current age of a particular distant person … say, the home twin (she) in the twin paradox scenario, when the IO is accelerating back toward her. To get that answer, all the IO needs to do is ask the PARTICULAR HF, who happens to be momentarily co-located with her then, what her age is right then. That HF can determine the answer just by looking at her, or asking her what her age is right then. Because the IO can compute the current age of each of the HF’s at each instant of his life, that effectively establishes a “NOW” instant for the IO that extends throughout all space.

It should be mentioned that each of the HF’s don’t agree with the IO about the relationship between their current age and the current age of the IO. But that’s OK … we are only interested in the IO’s “Now-at-a-distance” … no one else’s.

___________________________________________________________________________


Section 2. How Does the IO compute each of the HF’s Current Age?

The relationship between the IO’s current ageing rate and the ageing rate of each of the HF’s, according to the IO, is given in my previous paper entitled “A New Gravitational Time Dilation Equation”, on viXra at https://vixra.org/abs/2201.0015 . The IO says that the ratio of the HF’s current ageing rate, relative to the IO’s ageing rate, is:

R(A,t) = [ 1 + L A sech_sqrd{ theta(t) } ],

where “L” is the distance between the IO and the HF (positive when the HF is leading the IO, and negative when the HF is trailing the IO). “A” is the acceleration, and “theta” is the “rapidity”, which has a one-to-one relationship with the velocity v. Theta can be arbitrarily large, but the magnitude of the velocity can’t exceed or equal the speed of light. The velocity v is equal to tanh(theta). In the above equation for R(A, t), the function “sech_sqrd” is the square of the hyperbolic secant function. The hyperbolic secant is the reciprocal of the hyperbolic cosine.

To get the current age of the HF (for the case where they are each zero years old at the instant t = 0 when the acceleration begins), R(A,t) must be integrated, from t = 0 to t = tau, where tau is the IO’s age when we want to compute the HF’s age. The result is the “age change equation”:

AC(tau) = tau + L * tanh(A tau) ,

where “tau” is the current age of the IO, and AC is the current age of the HF.

When the above results are used to determine what happens when the IO instantaneously changes his velocity, it gives exactly the same answer as given by the well known
co-moving-inertial-frames (“CMIF”) simultaneity method. That’s fortunate, because the CMIF simultaneity method is easy to compute. The value of the accelerated array of clocks results that I have given ISN’T in its ability to get the answer to twin-paradox-type scenarios … the CMIF method can do that easily, and the accelerated array of clocks definitely doesn’t give the answer easily. The value of the accelerating array of clocks is that it GUARANTEES that the CMIF results are fully MEANINGFUL and “real”, and that the CMIF simultaneity method is the ONLY correct simultaneity method.

__________________________________________________________________________


Section 3. Conclusions

Many (maybe most) physicists consider special relativity to be a finished, closed theory. But it is NOT, for several reasons. First of all, I’ve previously shown (on viXra at https://vixra.org/abs/2109.0076 ) that the special relativity version of the exponential gravitational time dilation equation is incorrect. And I’ve shown in this paper that the CMIF simultaneity method is the ONLY correct simultaneity method. So special relativity certainly wasn’t finished before I came along. And it is STILL not finished, because physicists still disagree about special relativity. Beneath the surface, there are significant differences in the way different physicists INTERPRET the various results of special relativity.

For example, consider the fact that different INERTIAL reference frames (moving relative to one another) DISAGREE about the current age of some distant person (like the home twin in the twin paradox). Given that, many physicists conclude that the whole idea of simultaneity-at-a-distance is MEANINGLESS, and should be ignored. Some recommend omitting the twin paradox from the physics curriculum entirely. The instantaneous ageing of the home twin (she), according to the traveling twin (he), when he instantaneously reverses his velocity at his turnaround, is troubling to many physicists. And her negative ageing (i.e., her getting YOUNGER), if he accelerates AWAY FROM her, is even much more troubling to them. Those physicists generally prefer to explain special relativity by talking about more abstract concepts, like path lengths through spacetime. But the results that I’ve obtained for the accelerating array of clocks REQUIRE that the CMIF results MUST be considered to be fully MEANINGFUL, and that the CMIF method is the ONLY correct simultaneity method. There really isn’t any “wiggle room”, anymore, on that issue.

(The above content is also available at https://vixra.org/abs/2302.0119 ).

 

Log in or Sign up to hide all adverts.

Hi Mike,

Assuming SR, flat spacetime with no gravity, then all accelerations are coordinate accelerations as far as SR is concerned. No need to think about gravity.

In that context, if you are correct, then the "now-at-a-distance" of your accelerated observer should be identical to the "now-at-a-distance" of an identical co-moving inertial observer, correct?

 

Log in or Sign up to hide all adverts.

Yes, that's correct. I've confirmed that the "NOW-at-a-distance" result given by the accelerated array of clocks is exactly the same as that given by the CMIF simultaneity method. The value of the accelerated array of clocks is that it REQUIRES that the CMIF answer must be considered to be fully meaningful and "real".

 

Log in or Sign up to hide all adverts.

I really don’t see how you think that the constant speed of light establishes this idea that there is some constant now at a distance. This premise actually produces the opposite effect.

For instance, you could say that an object travels a distance.

d = v t

Then you can say that a ray of light travels a distance.

d_r = c t

You can set up a right triangle using the Pythagorean Theorem comparing their measurements.

( v t )^2 + ( c t )^2 = ( c t )^2

Notice side b and c are congruent. No right triangle can have it’s hypotenuse and side measure be congruent. The solution is the amount of time that has passed is zero. Zero is the only value that allows the equation to work out correctly. It would mean we are describing a truly dimensionless object.

What relativity does to solve this problem is assigning a different time variable to the other frame of reference, usually denoted as time prime or tau. Now, that they no longer use the same variables, those variables are no longer mathematically forced into being the same value. That has to be MEANINGFUL!

Instead, the equation becomes;

( v t )^2 + ( c tau )^2 = ( c t )^2

where t’ = tau. This allows one to create a relationship between the two observers time by simply saying they are not equal or the same as one another. This produces an equivalent equation of the proper time. That is the amount of time experienced by a distant observers watch in constant relative motion.

That observer has to measure time more slowly in order to measure a light ray to travel a shorter distance straight up and down, instead of at a diagonal.

 

Mike;

Time.
The purpose of a clock is to produce a uniform sequence of artificial events (ticks) used as a standard for recording events of interest. An additional requirement for a clock is maintaining a total of ticks. A typical clock has a 24 hr cycle to match one Earth rotation, with an auxiliary calendar for days, yrs. etc. A complete clock would have memory for larger units of accumulated time.
Age is accumulated time. A clock rate is the rate of aging. Since time is continuously increasing, so is age.

The 'twin' problem asks, is there any difference in elapsed time for two clocks that separate and reunite after a random time interval?
The simplest case involves Ann moving with constant velocity (speed and direction) and Bob departing and reuniting with Ann. They have identical clocks.
In fig.1, A and B are in a rest frame. To get facts, you need measurements.
B sends a light signal that causes A to emit a time encoded signal t1 to B.
If B maintains constant velocity he receives it at t2. A and B are synchronized.
If B accelerates toward A, he receives it earlier and assigns (green) it slightly before t1.
If B accelerates away from A, he receives it later and assigns it slightly after t1.
Ann's aging rate decreases/increases depending on Bob's motion.
Both clock rates are constant until someone changes velocity. It's Bob's perception of Ann's clock rate that changes.
As B accelerates, his ticks will stretch meaning he is losing time.

In fig.2, the popular 'twin triangle'. We can ignore the impossible instant reversal since it requires zero time, and analyze the two discontinuous paths. Both clocks emit uniform signals at a rate of 1 per minute. Since B is moving at .6c. his minute is 1.25* that of A.
A receives 8 ticks while B receives 10 ticks thus B lost time compared to A.

Please Register or Log in to view the hidden image!

Please Register or Log in to view the hidden image!

Let's simplify. If a helper is playing the part of an answering machine, he is redundant.
In fig.3, in ref. frame U, moving frames Ann and Bob are at the 'now' state. All have a light cone containing historical events they are aware of.
Using 'smart' clocks, while Bob is moving, he asks (e1), sends a signal to her clock that causes it to respond with a time encoded return signal At. It should be obvious he doesn't receive the signal until e3 in the future, when he assigns it to his clock event e2.
Bob only knows what Ann's clock reading WAS at e2. Bob at his new 'now' still doesn't know what Ann's clock reads 'now'.

misc.
Simultaneity is only relative or simulated for an inertial frame. Both clocks function independently of each other. The instantaneous change of clock rate is solely a result of the instantaneous reversal of direction. No object could perform that motion without violating physical laws. If a smooth transition was added (for political correctness), the reversal would have no significant effect.

You are attempting to restore 'Universal time' which was based on instantaneous light propagation, as accepted before Newton's era. That idea died about 1750 based on astronomical observations.

A reminder, 'there is no instant knowledge'.

 

I can check the time in my own geographic time zone, and then instantly know the time in any other time zone. I don't have to wait for distant clocks to send me a light signal that tells me their current time. That is the whole purpose of synchronised clocks.

 

I can look across the room and check the time!
For 'local' time there is no problem. How do you know the time on a lander on Mars?
How long will it take to synchronize a clock 10 ly distant?

 

My example was for time zones all around the world, so it was not really just local time. Even if the planet were a light year across, we could still synch all the clocks on it eventually, and from then on we could instantly know the time any place.

I could send the lander from earth to Mars at a constant velocity, and I would always know the time on the lander would lag behind my own time by a factor of 1/gamma.

It would take 10 years for a one-way signal, or 20 years for a round-trip signal. But the point is that once that is done, you theoretically know the time there instantly. You don't have to wait another 10 years every time you want to know.

 

You can speculate and calculate, but that does not guarantee success. A distant clock may move into the gravitational influence of a large mass,
You wouldn't be aware of that, and it would take 20 yrs to find out! Knowledge requires facts.
A large array of synchronized clocks is only possible as an isolated experiment.

 

Signs that indicate you might be doing SR wrong when it comes to thought experiments:
1. If you think the gravitational influence of a large mass might randomly appear someplace. (SR does not deal with gravity at all.)
2. If you think an observer has to wait until the light from a distant clock reaches their eyes before they can know what the time is in that location. (SR allows for sets of pre-synchronised clocks and therefore "instant knowledge" of the time shown on all clocks in that set.)
3. Etc.

 

Neddy;

You are living in the past when people thought light propagation was instantaneous, which implies universal time. They thought what they saw was happening as they viewed it regardless of distance. Finite light speed was discovered by astronomical observations about 1670.
That means a person cannot be aware of an event until it occurs, which requires a transit time for the images of the event to the viewer.
A large (astronomical) scale system of synchronized clocks cannot be maintained with any degree of efficiency, due to distance and variation of conditions. Constant velocity is an ideal concept achievable for small intervals of space and time. What purpose is there for a synchronized clock 10 ly distant?

The real world must include gravity, thus SR is not applicable in a GR world.
Speculation including calculation consists of conditional 'if' statements.
Counterexamples include the Titanic, 2 space shuttles, numerous aviation losses, etc. Incidents that involve extreme amounts of calculations, safety regulations, etc. The proposed system would work in an ideal world of perfect humans where nothing new occurs or nothing fails.
Both Lorentz and Einstein realized there was no common/universal time for local and distant events. Thus they derived coordinate transformations, where the distant coordinates could be calculated based on the local coordinates on the basis of constant velocity. Notice, the local coordinates are already known facts based on measurements. There is no magic or telepathy involved.

Mike is using the equivalence principle of GR to supposedly solve the 'twin' problem in SR, without considering the difference in aging and rate of aging.
Seems confusing to me.

 

In an SR thought experiment, the purpose of a synchronised clock 10 light years distant is to be able to define the time coordinate of any event that occurs at the location of the clock. That time coordinate is only valid in the inertial frame in which that array of synchronised clocks is stationary. This way, the time of an event can be defined when the event occurs, even though it would take 10 years for the event to be seen by an observer 10 light years distant.

For example, in the standard twin scenario, imagine that the stay-home twin has an array of synchronised clocks that the traveling twin travels past. When the traveler turns around to head back toward the stay-home twin, there is a clock there and the time on that clock marks the time of the turnaround event in the stay-home twin's inertial frame. So, for example, if the traveler is 10 light years away from the stay-home twin at that time, the stay-home twin will not know the time of the turnaround in her reference frame for another 10 years, but the time is already defined as soon as it happens. This allows us to draw a space time diagram without any light signals at all. All of your space time diagrams seem to have light signals bouncing back and forth, but it seems to me that those are not necessary.

You seem to be hung up on the idea that some unknown variable could affect the outcome, but that is missing the point. For example, in the in the standard twin scenario, the stay home twin can always assume the traveling twin is younger than themselves by a factor of 1/gamma. It does not matter if the travelling twin had actually died, unbeknownst to the stay-home twin. The point is not the traveling twin's actual age. The point is about what the time would be in the inertial reference frame of the traveling twin, not whether that twin has actually lived to that age or not.

Yes, exactly. The coordinates of events can be defined in a scenario, and readily transformed from one reference frame to another, without any observer waiting for any light signal to allow them to actually see the event with their eyes to be sure it happens.

I find Mike's idea confusing also, but you and I are discussing a side issue. You always seem to want to look at how long it takes for light signals to travel some distance so that an observer can "know" some event happened by seeing it with their eyes. That is fine, but it is an extra step that we can usually avoid by simply transforming coordinates of events from one reference frame to another, without using any light signals.

 

Neddy;

Mike states:

1. true. The equivalence principle was used to account for the increasing clock rates at various altitudes above the earth surface, based on the gravitational gradient from a large radial distance to the surface. The system of satellite clocks must be corrected daily for accuracy.

2. false.
In the SR environment, using a pair of clocks A and B, as the pair accelerates (with out gravity), the green Ax and Bx axes (aka axis of simultaneity) rotate relative to the U x axis.
If A sends a light signal at At=0, at At=3.85, A assigns Bt=2.30 to At=2.10. The ratio of B-time to A-time is 2.30/2.10=1.10.
If A sends a light signal at At=1, at At=5.07, A assigns Bt=3.63 to At=3.00. The ratio of B-time to A-time is 3.63/3.00=1.21.
While A is calculating the time ratio, it is changing!
Measurement is the verification tool for science. To know (have evidence or facts) requires measurement. Anything else is assumption/speculation/wishful thinking/hoping...etc.

3. How does he ask?
________________

In SR the time jump (instant reversal) relies on the behavior of the Bx axis of simultaneity. If a curved path connects the 2 discontinuous segments, allowing a single B observer to follow the entire route, the Bx axis sweeps across the entire A path, and the jump disappears.

Please Register or Log in to view the hidden image!

 

Hmm. I'm not sure if you are claiming some disagreement with Mike's idea, or not. But you still included some light signals sent back and forth. Those are the things I was wondering about. Do we need those to understand it?

 

Neddy;

The basic problem is the meaning of 'know'.
a. hold information in the mind: to have information firmly in the mind or committed to memory [how does the info get into the mind?]
b. be certain about something: to believe firmly in the truth or certainty of something [how do you establish certainty of a distant event?]
c. comprehend something: to have a thorough understanding of something through experience or study [where is the experience?]

1. You acknowledge the delay. The time is recorded if and when it happens.

2. It does matter to Mike who states "it allows him to determine the current age of any particular distant person". There is no provision in his calculations to detect the death of the other person.

3. Science relies on measurements to verify experimental results. When Einstein predicted starlight bending around the eclipsed sun, observers measured the displacement relative to the same area without the sun.
A dispute over property lines can be resolved with a survey.

Please Register or Log in to view the hidden image!

Measurement is involved in most human activities.
Light is today's universal measuring rod for astronomical distances.
------------------------------
Fig.1 is A's description of events.
A and B are both anauts weightless in space. B departs from A, selects Bt=15 to perform his reversal. The green lines are his x' axis (aka axis of simultaneity). They do not magically appear, but result from a (blue) light measurement made by B. As B reverses direction the Bx' axis rotates cw until Bt=20 where his motion is parallel to A. The inbound segment is a mirror image of the outbound segment. The red line indicates td of .50 for the constant velocity portion. The purpose of fig.1 is to correct for the impossible instantaneous reversal and 'time jump'.

Fig.2 is B's description of events.
B assumes a pseudo rest frame with constant velocity. For the Bt interval 15 to 25, B experiences a g-field (via the equivalence principle). That alters his perception of A's motion. He interprets her falling in the g-field and returning to him. He perceives her path as flattened due to a constant round trip transit time for light during the reversal.