Alternative way to calculate the special relativistic time dilation factor

[Alexroma]

I propose an explanation of the special relativistic time dilation and calculation of its factor in terms of the difference in observational times for incoming and receding objects, based on the following thought experiment:

Imagine a spaceship coming from planet X to Earth with velocity V. The time during which the spaceship can be observed from Earth equals t – t_c, where t is the travel time of the spaceship (the time it takes for the spaceship to come to Earth from planet X), and t_c is the time it takes light to come to Earth from planet X; this expression of the observational time is explained by the fact that the spaceship is following behind its own light.

Alternatively, imagine a spaceship going from Earth to planet X with the same velocity. In this case, the time during which the spaceship can be observed from Earth equals t + t_c, as it takes t for the spaceship to go to planet X, plus it takes t_c for the light to convey the image of its landing at that planet to observer on Earth.

So, we have two different observational times for incoming and receding objects traveling the same distance with the same velocity. In order to reconcile the difference between these two observational times, we determine the mean observational time (t_m) as the geometric mean of them:

tm = √ (t – t_c)(t + t_c) = √ t² – t_c²

To find out the factor for time dilation, we divide the mean observational time by travel time:

t_m/t = √ 1 – t_c²/t² = √ 1 – V²/C²

Therefore, in result we have the same factor for the time dilation as the factor used in the special theory of relativity, i.e. the reciprocal of the Lorentz factor (√ 1 – V²/C²).

 

[DaveC426913]

You seem to be confusing two things here. Maybe it's just your terminology.

The approach and recession of an object at relativistic speeds has nothing to do with time dilation.

You would see the same effect if you examined your craft's flight using classical Newtonian mechanics.

 

[Alexroma]

You seem to be confusing two things here. Maybe it's just your terminology.

Thank you for the tip! It looks like I have to reformulate it.

 

[Alexroma]

The approach and recession of an object at relativistic speeds has nothing to do with time dilation. You would see the same effect if you examined your craft's flight using classical Newtonian mechanics.

I thought about it, and I can not agree. The notions of approach and recession are relativistic in their nature, as there is no special direction in space, and we talk about approach and recession as they relate to the observer. There is no notion of "observer" in classical Newtonian mechanics, so why should we have the same effect?

 

[DaveC426913]

I thought about it, and I can not agree. The notions of approach and recession are relativistic in their nature, as there is no special direction in space, and we talk about approach and recession as they relate to the observer. There is no notion of "observer" in classical Newtonian mechanics, so why should we have the same effect?

:shrug:

Perhaps I just don't understand what problem you're trying to solve. You've stated the solution, but you haven't stated the problem.

 

[Alexroma]

:shrug: Perhaps I just don't understand what problem you're trying to solve.

The problem is rather conceptual: I am trying to show that time dilation can arise not only from the relative motion between two observers, but also from the difference in direction of motion relative to the observer (approaching versus receding), as every motion is either approaching or receding, except the expansion of the Universe: it's only receding because there is no observer outside it. I think that in observational terms there must be some fundamental difference in approaching and receding motion, because light always approaches the observer and never recedes him, as the observer as such does not emit light, he just perceives it. I know, it's overly speculative, so I'd better keep it quiet...

 

[DrGreg]

...every motion is either approaching or receding...

You get time dilation when one observer circles around another; that's neither approach nor recession.

 

[Alexroma]

You get time dilation when one observer circles around another; that's neither approach nor recession.

In reality, such a perfect circling is not possible thanks to gravitation, and the planets periodically approach their stars and recede from them on elliptical orbits.

 

[DaveC426913]

In reality, such a perfect circling is not possible thanks to gravitation, and the planets periodically approach their stars and recede from them on elliptical orbits.

But it doesn't matter if the phenomenon is uncommon, or if it doesn't occur in circumstances of your choice. The fact that it does happen. That invalidates your claim that every movement requires an advancement or recession.

Heck, we could look down upon a star system from above its pole, and watch a planet rotate around it. Regardless of how eccentric its orbit, the planet is neither advancing nor receding from us, yet it will experience time dilation.The point is simply that time dilation indeed occurs independent of direction of motion wrt the observer. That should tell you you're barking up the wrong tree.

 

[Alexroma]

But it doesn't matter if the phenomenon is uncommon, or if it doesn't occur in circumstances of your choice. The fact that it does happen. That invalidates your claim that every movement requires an advancement or recession.

Every observation, in principle, needs time, as you can not do it faster than the speed of light. By the time you get the result of your observation, the circling object would move away from the point of your observation. That means, it's receding.

 

[DaveC426913]

the circling object would move away from the point of your observation. That means, it's receding.

No it doesn't. Look up the definition of receding. It means 'distance increases'.

An object has quite a bit of freedom of movement without ever having to change its distance from the observer.

 

[Dale]

Heck, we could look down upon a star system from above its pole, and watch a planet rotate around it. Regardless of how eccentric its orbit, the planet is neither advancing nor receding from us, yet it will experience time dilation.

Hi Dave, the part in bold is not correct. Although your overall point is.

The point is simply that time dilation indeed occurs independent of direction of motion wrt the observer. That should tell you you're barking up the wrong tree.

Alexroma, please pay attention to Daves statement here, it is very important. This is a key prediction of SR called transverse Doppler shift. It is considered one of the most important predictions of SR because is is not just quantitatively different from Newtonian mechanics, but qualitatively different.

Transverse Doppler has been experimentally confirmed, if your formula cannot replicate it then your formula is contradicted by experiment.

 

[dacruick]

Would someone mind explaining this to me? I recognize that I may be a bit out of my league here but, If the observer sees the spaceship land c * t_c after it actually did land, how does that affect time dilation?

My understanding of time dilation is that one relativistic second is longer than one classical second. So how would the time it takes the information (the light) to get to the observer change anything about the information?

 

[DaveC426913]

Hi Dave, the part in bold is not correct. Although your overall point is.

Yeah, you're right. With an eccentric orbit you'd see a tiny bit of distance change.

But he's totally barking up the wrong tree. Imagine observing a planet 5 light years away with an eccentric orbit. We're looking down on the system from above so its orbital planet is normal to us. He thinks he's going to see recessional motion that will cause time dilation?

 

[Alexroma]

Transverse Doppler has been experimentally confirmed, if your formula cannot replicate it then your formula is contradicted by experiment.

Thank you for pointing it out. That's a serious argument. I'll look at it closer.

 

[Alexroma]

Would someone mind explaining this to me? I recognize that I may be a bit out of my league here but, If the observer sees the spaceship land c * t_c after it actually did land, how does that affect time dilation?

It really doesn't affect it in terms of the special theory of relativity. So, Dave was right in his first reply: I chose a wrong terminology for the title of my post. To call it "Alternative way to calculate the special relativistic time dilation factor" is indeed confusing. I should have called it just "Alternative way to calculate the time dilation factor".

My understanding of time dilation is that one relativistic second is longer than one classical second. So how would the time it takes the information (the light) to get to the observer change anything about the information?

Information and light are not the same. In a sense, information is the message, and light is the medium.

 

[Alexroma]

Yeah, you're right. With an eccentric orbit you'd see a tiny bit of distance change. But he's totally barking up the wrong tree. Imagine observing a planet 5 light years away with an eccentric orbit. We're looking down on the system from above so its orbital planet is normal to us. He thinks he's going to see recessional motion that will cause time dilation?

I got your point. I don't have the ready answers at this stage, because my idea is only three days old, and it's still an idea, it's not even a concept. I really appreciate your comments. Still, I believe they don't kill my idea, because gravitation, which makes planets go in circles, causes time dilation too. I am looking forward to save my idea with gravitation, as it makes the objects to approach each other, but it's a long way to figure it out in details.

I know, it looks like it does not make sense "to reinvent the bicycle" as soon as we already have STR and GTR, but I still like to approach things differently. That's just my nature.

 

[Dale]

Thank you for pointing it out. That's a serious argument. I'll look at it closer.

You are welcome. Here is a good place to start:
http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html#Tests_of_time_dilation

 

[Alexroma]

I finally figured out the application of my method of calculation of time dilation to the objects moving in circles and all other non-rectilinear trajectories. First of all, I did not properly present my method. Starting from the title, instead of calling it “Alternative way to calculate the special relativistic time dilation factor”, I should have called it “Non-relativistic explanation and calculation of the time dilation”, as I don’t employ the frames of reference for my calculation.

As my method is non-relativistic, an observer can be located at any point in space, so for the purposes of calculation of the time dilation of a moving object, we substitute the trajectory of its movement, no matter how complex it is, for a straight line (in a physical sense, it’s possible because time dilation does not depend on the trajectory) and place the observer in the middle of the distance L = Vt, where V is velocity of the object and t is time during which the observer would observe the object going distance L if he saw it immediately, without delay caused by the finite speed of light.

The effect of time dilation results from the difference of times of observation of an object approaching the observer who is centrally located on the straightened trajectory of the object's movement and receding from him:

ε = t_m/t_ar = √ (t_ar – t_c/2)(t_ar + t_c/2)/t_ar²

In this formula:

ε is time dilation factor

t_m is geometric mean of times of observation of an object approaching the centrally located observer and receding from him; t_m = √ (t_ar – t_c/2)(t_ar + t_c/2)

t_ar is time during which the centrally located observer would observe an object either approaching him or receding from him (going distance L/2), if he saw it immediately, without delay caused by the finite speed of light; t_ar = L/2V

t_c is time that takes light to cover distance L; t_c = L/C

Although the present formula is equivalent to the formula of time dilation factor used in the special theory of relativity, i.e. the reciprocal of the Lorentz factor (√ 1 – V²/C²), it shows that time dilation of a moving object may be explained by an observational effect, rather than by the actual velocity of the object.

 

[Dale]

it shows that time dilation of a moving object may be explained by an observational effect

Except that once you straighten the trajectory it has nothing to do with any actual observational effect.

 

[Alexroma]

Except that once you straighten the trajectory it has nothing to do with any actual observational effect.

Should I call it "quasi-observational"?

 

[Dale]

The relativistic Doppler shift is the geometric mean of the Doppler shift for the classical Doppler shift with the speed of light fixed relative to the emitter and with the speed of light fixed relative to the detector. I think this is related to your idea. For more details see section 6.5 at:
http://books.google.com/books?id=bU..."transverse doppler" "geometric mean"&f=false

 

[Alexroma]

Thank you, DaleSpam! It really looks related, I'll be examining it further in more detail. Still long way to go.

 

[Passionflower]

No it doesn't. Look up the definition of receding. It means 'distance increases'.

An object has quite a bit of freedom of movement without ever having to change its distance from the observer.

Quite right, for instance an object accelerating away from another object might increase or decrease the distance between it or leave the distance the same.

 

[Alexroma]

Well, I don't see a problem with interpretation of "receding" in my new explanation. There is no point to discuss it anymore.

 

[Dale]

Well, I don't see a problem with interpretation of "receding" in my new explanation.

No problem other than the fact that it has nothing whatsoever to do with the usual definition of "receding".

 

[Alexroma]

No problem other than the fact that it has nothing whatsoever to do with the usual definition of "receding".

If it still looks like a problem, I agree that I have to change it for a more vague term, as soon as it does not make any difference for calculations:

"The effect of time dilation results from the difference of times of observation of an object that is alternatively inward- and outward-bound relative to the observer who is centrally located on the straightened trajectory of the object's movement".

 

[DaveC426913]

If it still looks like a problem, I agree that I have to change it for a more vague term, as soon as it does not make any difference for calculations:

"The effect of time dilation results from the difference of times of observation of an object that is alternatively inward- and outward-bound relative to the observer who is centrally located on the straightened trajectory of the object's movement".

I'm sorry Alex. Fun's over. PF is not the place to invent fanciful new terms to help develop fanciful new theories.

PF rules:

One of the main goals of PF is to help students learn the current status of physics as practiced by the scientific community; accordingly, Physicsforums.com strives to maintain high standards of academic integrity. There are many open questions in physics, and we welcome discussion on those subjects provided the discussion remains intellectually sound. It is against our Posting Guidelines to discuss, in the PF forums or in blogs, new or non-mainstream theories or ideas that have not been published in professional peer-reviewed journals or are not part of current professional mainstream scientific discussion. Non-mainstream or personal theories will be deleted. Unfounded challenges of mainstream science and overt crackpottery will not be tolerated anywhere on the site.

 

[Alexroma]

I'm sorry Alex. Fun's over. PF is not the place to invent fanciful new terms to help develop fanciful new theories.

Fine, I am receding . Thank you for your help in figuring it out, I really appreciate it!