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Does Special Relativity contain contradictions?

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Tamminen
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Re: Does Special Relativity contain contradictions?

Post by Tamminen » October 8th, 2018, 4:49 pm

So can we say that the relative time dilation between Alice and Bob is one and the same function f(a,h,t) when moving in flat or curved spacetime, where a = acceleration relative to an inertial frame, h = the distance betweeen the clocks and t = travel time? Is this what the equivalence principle means in this case?

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Halc
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Re: Does Special Relativity contain contradictions?

Post by Halc » October 8th, 2018, 6:12 pm

Tamminen wrote:
October 8th, 2018, 4:49 pm
So can we say that the relative time dilation between Alice and Bob is one and the same function f(a,h,t) when moving in flat or curved spacetime, where a = acceleration relative to an inertial frame, h = the distance betweeen the clocks and t = travel time? Is this what the equivalence principle means in this case?
Those are all locally measurable things, so yes, but a=acceleration or strength of the gravitational field. Alice and Bob cannot tell the difference between those. In the gravity field, they're not really accelerating.

Tamminen
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Re: Does Special Relativity contain contradictions?

Post by Tamminen » October 9th, 2018, 4:18 am

Halc wrote:
October 8th, 2018, 6:12 pm
Tamminen wrote:
October 8th, 2018, 4:49 pm
So can we say that the relative time dilation between Alice and Bob is one and the same function f(a,h,t) when moving in flat or curved spacetime, where a = acceleration relative to an inertial frame, h = the distance betweeen the clocks and t = travel time? Is this what the equivalence principle means in this case?
Those are all locally measurable things, so yes, but a=acceleration or strength of the gravitational field. Alice and Bob cannot tell the difference between those. In the gravity field, they're not really accelerating.
But if we look at this from the point of view of the inertial frame: in the case of the flat spacetime it is moving at a constant speed and in the case of the curved spacetime it is free fall. In the latter case a = acceleration against gravitational potential, so that for instance when the rocket is standing on the ground in a gravitational field of 1 g, it really accelerates at a = 1 g. So it does not matter if the spacetime is flat or curved or how curved it is, the relative time dilation is the same if a is the same. Right?

Steve3007
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Re: Does Special Relativity contain contradictions?

Post by Steve3007 » October 9th, 2018, 4:29 am

David Cooper wrote:
Steve3007 wrote:Which particular part(s) or that post of mine is/are, in your view, "not good enough"?
Where does it address the issue?
For reference, this is the post we're talking about here:
viewtopic.php?p=321015#p321015

It illustrates in detail, by way of an example, what is directly observed by A and B and what they calculate, based on both those observations and the definition of the word "simultaneous" (it means "at the same time"). I think a lot of accounts of such things as the "twin paradox" don't point out sufficiently clearly the difference between these two. They sometimes appear to blur the distinction by not going into sufficient detail and not using precise enough language, presumably because their aim is to provide a very brief synopsis.

In the example I used, in terms of direct observations: A and B both see each other's clocks ticking 3.73 times as slow as their own during leg 1 and both see each other's clocks ticking 3.73 times as fast as their own during leg 2. See the first table in the post for the precise numerical details of how this works, and how it results in the clock readings at the end of the experiment.

Based on the definition of simultaneity ("at the same time") they each calculate each other's clocks to be running slower than their own during both legs. See the Minkowski diagram and the second table in the post to see the numerical details of how this works. Obviously if "at the same time" meant "at the same single, universally/mutually agreed, 'Newtonian' time" then this would result in a contradiction. But it doesn't mean that. Somewhere in your article on your website I vaguely remember you claiming (without appearing to say why) that Special Relativity keeps this concept of a single, universal 'Newtonian' concept of time and that it is therefore self-contradictory. I disagree that it does this and I therefore disagree that it is self-contradictory.

Steve3007
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Re: Does Special Relativity contain contradictions?

Post by Steve3007 » October 9th, 2018, 5:57 am

Output01_Earth.jpg
Earth
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Steve3007
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Re: Does Special Relativity contain contradictions?

Post by Steve3007 » October 9th, 2018, 5:59 am

Output02_Jupiter.jpg
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Steve3007
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Re: Does Special Relativity contain contradictions?

Post by Steve3007 » October 9th, 2018, 6:00 am

Later, I'm hoping to deal with some of the recent comments of Tamminen and Halc. But first I'd like to give a preview of the Windows application that I mentioned in this post:

viewtopic.php?p=321323#p321323

Two outputs from that application are shown in the above two posts.

It calculates the time dilation for an observer in a gravitational well using two different methods. The first method uses General Relativity. It uses the Schwarzschild solution to Einstein's GR field equations to calculate the time dilation between an observer who is low down in the gravitational well ("lower clock"), an observer who is high up in the gravitational well ("upper clock") and an observer who is not in the gravitational well (i.e. is an infinite distance from the centre of the the gravitating mass).

In doing this, it also calculates the Schwarzschild radius for the gravitating body (the radius that a body of that mass would have to have in order for it to become a black hole). As you can see, it correctly shows the Schwarzschild radii of the Earth and Jupiter (you can look those up to confirm), demonstrating that it is working correctly.

The second method uses the results of Special Relativity to calculate the time dilation between two clocks at either end of an accelerating room/box/elevator/rocket which is not in a gravitational field. It uses the equation that was derived mathematically ( as equation 8 ) in this article:

https://thecuriousastronomer.wordpress. ... elativity/

...and iterates over a very large number of small boxes. If such a small box were in the presence of a radial gravitational field, the field within the small volume of each individual box could be regarded as uniform and therefore the equivalence principle applies. So iterating over a large number of boxes, each with a slightly different uniform acceleration/g field yields a value for the time dilation in a radial gravitational field where the field strength is a function of radius.

Making use of the power of computers to do this numerical integration over nearly a billion such boxes (see "Number of steps" in the application) in a couple of minutes of processing time yields a very accurate result. As you can see, the result using SR matches very accurately the result using GR (Schwarzschild solution). So it demonstrates very clearly how gravitational time dilation, and therefore the other aspects of GR, such as the concept of a curved spacetime, emerge naturally from the combination of SR and the equivalence of inertial and gravitational mass.

For me, it really drives home the reason why General Relativity, more than Special Relativity, is often regarded as one of the most profound insights of the human mind, springing as it does entirely from a combination of pre-existing empirical knowledge and very careful thought. The whole concept of non-Euclidean spacetime, and gravity as geometry, didn't spring from nowhere, or from an overactive imagination. It is a direct result of empirical evidence, going back as far as Galileo's experiments dropping objects from the Leaning Tower of Pisa, and careful reasoning.

Steve3007
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Re: Does Special Relativity contain contradictions?

Post by Steve3007 » October 9th, 2018, 6:53 am

Halc wrote:Gravitational force also does not cause dilation. I have a clock here on Earth, and one on my lab on Uranus where I weigh about 8/9th of Earth. But the Uranus clock is dilated more, despite the weaker force of gravity. It feels like less acceleration, yet the dilation effect is more. Clearly the acceleration, or a gravitational field that feels like acceleration, plays no direct role in the dilation. It is caused by the negative gravitational potential, which is far greater on Uranus (about 3x Earth), and nonexistent in a centrifuge.
Out of interest, I added Uranus to the application that I wrote and which I discussed in my previous post. The result is shown below. As you can see, the gravitational time dilation calculated between a point just above the Uranus cloud tops and a point much further away is the same when calculated using General Relativity and when calculated using numerical integration of a long series of small vertically stacked boxes within which the gravitational field can be taken, to a high enough level of accuracy, to be uniform.

To answer your above point: you're right that the value of 'g' just above the Uranus cloud tops is slightly less than the value of 'g' on the surface of the Earth. (The application calculates it and displays is.) But that value drops off less rapidly with increasing 'r' than it does on Earth. That's why the dilation effect is calculated to be more using both GR and SR. The SR numerical integration method is affected by this slower reduction in 'g' with increasing 'r' because it integrates over a long series of small volumes from a small 'r' to a large 'r'.
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Steve3007
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Re: Does Special Relativity contain contradictions?

Post by Steve3007 » October 9th, 2018, 6:58 am

From the calculations in the application, if we have 3 clocks, one on Earth, one at the cloud tops of Uranus and one far away from any gravitating bodies (and neither Earth nor Uranus are rotating), then after 100 years the Earth clock lags the infinity clock by about 2.2 seconds. For the Uranus clock it's about 8 second.

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Halc
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Re: Does Special Relativity contain contradictions?

Post by Halc » October 9th, 2018, 7:34 am

Tamminen wrote:
October 9th, 2018, 4:18 am
But if we look at this from the point of view of the inertial frame: in the case of the flat spacetime it is moving at a constant speed and in the case of the curved spacetime it is free fall.
Our examples correspond to neither of these cases. From the POV of flat spacetime, we're in a rocket accelerating, so no constant speed. The observer feels a 1g force. From the POV of the curved spacetime (on Earth), the observer is not in free fall, but has a force of 1g applied on him from the ground underneath him, which is indistinguishable from the thrust of the rocket.
From an inertial frame POV: The rocket guy is accelerating and the planet guy is not.
So it does not matter if the spacetime is flat or curved or how curved it is, the relative time dilation is the same if a is the same. Right?
Yes. It takes multiple measurements to observe the the relative dilation: two clocks that we can compare to each other.

Steve3007
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Re: Does Special Relativity contain contradictions?

Post by Steve3007 » October 9th, 2018, 7:35 am

Here's the result for the Sun. Note that the Schwarzschild radius is correctly calculated as just under 3km. Also note that in this case I've used a step size of 10 metres and that, as a result, there's a very slight difference between the time dilation between upper and lower clocks measured using the GR and SR methods. This means that the non-uniformity of the gravitational field in a 10 metre tall box, near the Sun, is bad enough to show up. Interesting.
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Halc
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Re: Does Special Relativity contain contradictions?

Post by Halc » October 9th, 2018, 8:02 am

Steve3007 wrote:
October 9th, 2018, 5:57 am
Output01_Earth.jpg
So I looked at this (and the Jupiter one). Not sure what all these numbers are meant to demonstrate. It just shows dilation due to varying points of negative gravitational potential. I don't contest any of it. I didn't pick Jupiter in my example since the gravity there is so much larger than on Earth. Saturn is the closest, but I chose Uranus because you actually weigh less there.

The lower clock is at sea level, which is expected.
The upper clock is out past the moon, which is hardly an example of a uniform gravitational field.
We're just not going to be able to compare these numbers to the rocket one.

Problem is, with a short height of building/rocket (mine was 300km), you barely have enough digits in your dilation fields to notice the difference.

The 2nd dilation figure has an extra digit, which makes reading it confusing. Hard to count the 9's. I think it truncates a trailing zero.
The Jupiter numbers don't have the same separation of the two clocks since the lower one is raised (to 'sea level') but the upper one is not correspondingly raised.

There are no numbers for a rocket accelerating at 1G. For the rocket, there is no 'clock at infinity' to which a comparison like that can be made.

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Halc
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Re: Does Special Relativity contain contradictions?

Post by Halc » October 9th, 2018, 8:27 am

Steve3007 wrote:
October 9th, 2018, 6:00 am
The second method uses the results of Special Relativity to calculate the time dilation between two clocks at either end of an accelerating room/box/elevator/rocket which is not in a gravitational field. It uses the equation that was derived mathematically ( as equation 8 ) in this article:

https://thecuriousastronomer.wordpress. ... elativity/

...and iterates over a very large number of small boxes. If such a small box were in the presence of a radial gravitational field, the field within the small volume of each individual box could be regarded as uniform and therefore the equivalence principle applies. So iterating over a large number of boxes, each with a slightly different uniform acceleration/g field yields a value for the time dilation in a radial gravitational field where the field strength is a function of radius.
I see. The point of the output was to have the two numbers be the same (between upper and lower clocks).
How is it that they're the same??? In the planet case, the upper clock has 'acceleration due to gravity (g)' at about 1/30000th of the lower clock, and in the rocket case they're accelerating pretty much identically.

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Halc
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Re: Does Special Relativity contain contradictions?

Post by Halc » October 9th, 2018, 8:34 am

Steve3007 wrote:
October 9th, 2018, 6:53 am
The SR numerical integration method is affected by this slower reduction in 'g' with increasing 'r' because it integrates over a long series of small volumes from a small 'r' to a large 'r'.
The SR method should have no reduction in g at all since the formula is for a uniform gravitational field. There is no r in that formula, just like there is no h in the GR formula above it. You sure you used it in the bottom calculation?

Steve3007
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Re: Does Special Relativity contain contradictions?

Post by Steve3007 » October 9th, 2018, 8:39 am

Halc wrote:I see. The point of the output was to have the two numbers be the same (between upper and lower clocks).
No, the two numbers were calculated using the two methods, as described. The point was to see if this resulted in them being the same. It did.
Halc wrote:The SR method should have no reduction in g at all since the formula is for a uniform gravitational field. There is no r in that formula, just like there is no h in the GR formula above it. You sure you used it in the bottom calculation?
Steve3007 wrote:The second method uses the results of Special Relativity to calculate the time dilation between two clocks at either end of an accelerating room/box/elevator/rocket which is not in a gravitational field. It uses the equation that was derived mathematically ( as equation 8 ) in this article:

https://thecuriousastronomer.wordpress. ... elativity/

...and iterates over a very large number of small boxes. If such a small box were in the presence of a radial gravitational field, the field within the small volume of each individual box could be regarded as uniform and therefore the equivalence principle applies. So iterating over a large number of boxes, each with a slightly different uniform acceleration/g field yields a value for the time dilation in a radial gravitational field where the field strength is a function of radius.
I can post the code to make it clearer.

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