# Alternative way to calculate the special relativistic time dilation factor

• Alexroma
In summary: Regardless... Heck, we could look down upon a star system from above its pole, and watch a planet rotate around it. Regardless...

#### 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 – tc, where t is the travel time of the spaceship (the time it takes for the spaceship to come to Earth from planet X), and tc 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 + tc, as it takes t for the spaceship to go to planet X, plus it takes tc 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 (tm) as the geometric mean of them:

tm = √ (t – tc)(t + tc) = √ t2 – tc2

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

tm/t = √ 1 – tc2/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).

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.

DaveC426913 said:
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.

DaveC426913 said:
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?

Alexroma said:
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.

DaveC426913 said:
: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...

Alexroma said:
...every motion is either approaching or receding...
You get time dilation when one observer circles around another; that's neither approach nor recession.

DrGreg said:
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.

Alexroma said:
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.

DaveC426913 said:
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.

Alexroma said:
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.

DaveC426913 said:
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.

DaveC426913 said:
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.

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 * tc 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?

DaleSpam said:
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?

DaleSpam said:
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.

dacruick said:
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 * tc 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".

dacruick said:
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.

DaveC426913 said:
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.

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 = tm/tar = √ (tar – tc/2)(tar + tc/2)/tar2

In this formula:

ε is time dilation factor

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

tar 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; tar = L/2V

tc is time that takes light to cover distance L; tc = 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.

Alexroma said:
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.

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

Should I call it "quasi-observational"?

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

DaveC426913 said:
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.

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

Alexroma said:
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".

DaleSpam said:
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".

Alexroma said:
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.

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DaveC426913 said:
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!

## 1. What is the special relativistic time dilation factor?

The special relativistic time dilation factor is a concept in physics that describes the difference in the passage of time between two objects moving at different velocities. It is a consequence of Einstein's theory of special relativity, which states that time is not absolute and can be affected by factors such as velocity and gravity.

## 2. How is the special relativistic time dilation factor calculated?

The special relativistic time dilation factor is calculated using the equation γ = 1/√(1 - v²/c²), where γ represents the time dilation factor, v is the velocity of the object, and c is the speed of light. This equation takes into account the fact that time appears to pass slower for objects moving at high velocities.

## 3. What is an alternative way to calculate the special relativistic time dilation factor?

An alternative way to calculate the special relativistic time dilation factor is to use the Lorentz factor, which is represented by the symbol β. The Lorentz factor is calculated using the equation β = 1/√(1 - (v/c)²). This equation is essentially the same as the one used for γ, but it uses the ratio of the object's velocity to the speed of light instead of the square of the velocity.

## 4. How is the special relativistic time dilation factor used in practical applications?

The special relativistic time dilation factor is used in practical applications such as satellite navigation systems, where precise time measurements are crucial. The accuracy of these systems is affected by the difference in the passage of time between the satellites and the receivers on Earth, which is taken into account by the time dilation factor.

## 5. Does the special relativistic time dilation factor have any limitations?

Yes, the special relativistic time dilation factor is only applicable in situations where objects are moving at speeds close to the speed of light. It also does not take into account other factors such as acceleration and gravitational effects, which can also affect the passage of time. Additionally, it is a theoretical concept and has not yet been fully proven by experiments.