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## Does Special Relativity contain contradictions?

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Steve3007
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### Re: Does Special Relativity contain contradictions?

In this post:

viewtopic.php?p=321278#p321278

I showed the outline of the mathematical argument showing this same link by analytical integration - Integral Calculus - rather than numerical integration.

In a subsequent reply you said this:
Halc wrote:The point seems to be that the acceleration itself causes no dilation.
I disagree and have demonstrated the equivalence between time dilation when considering acceleration (SR) and time dilation when considering the radial gravitational fields of massive bodies (GR).

Steve3007
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### Re: Does Special Relativity contain contradictions?

It computes g from r for instance in the GravitationalTimeDilation function
No, it computes 'g' from 'r' using the standard definition:
g = GM/r2
Halc wrote:I thought I just said that. I don't understand the 'No,' part.
OK, yes, I misunderstood you there. The function of that name does indeed contain that calculation. But g = GM/r2 is not itself anything to do with gravitational time dilation. It's simply the Newtonian function that relates acceleration due to gravity, a.k.a. strength of a gravitational field, to distance from the centre of the gravitating mass.

Steve3007
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### Re: Does Special Relativity contain contradictions?

It's simply a combination of Newton's Law of Universal Gravitation:

F = GM1M2 / r2

and Newton's second law of motion:

F = ma

where a = g

Halc
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### Re: Does Special Relativity contain contradictions?

Steve3007 wrote:
October 9th, 2018, 10:04 am
No. One is the gravity case, in the sense that it uses the equation from General Relativity.
I'm not talking about the sense of which equation is used. I can see it being done in two ways.
I'm talking about the sense that you're computing dilation for a planet both times, and never for an accelerating rocket.

Halc
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### Re: Does Special Relativity contain contradictions?

Steve3007 wrote:
October 9th, 2018, 9:51 am
So that's why the numbers are the same. They're both the gravity case, and neither is the acceleration case. You're not computing the difference between Alice and Bob in the rocket, just Alice and Bob in an improbably tall building using two different methods.
No i'm not because I'm not using the Alice/Bob SR-based equation for the other calculation. For that calculation, as I've said, I'm using the equations of General Relativity. I am thereby demonstrating how the equations of GR derive from those of SR.
I know you're using two sets of equations (although you didn't post the code for the GR case). But your simulation in both cases was one of a planet with gravity, not of an accelerating rocket. They got the same numbers because they're both correctly simulating the same thing, using different methods. Yes, I agree that GR equations derive from SR.
To try to make it easier to understand, I drew pictures of the relevant equations on the appropriate parts of the GUI of the application that I wrote. Did you see them?
Yes, they're fine. The fact that the number came out the same is evidence of this.

Halc
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### Re: Does Special Relativity contain contradictions?

Steve3007 wrote:
October 9th, 2018, 10:17 am
Halc wrote:The point seems to be that the acceleration itself causes no dilation.
I disagree and have demonstrated the equivalence between time dilation when considering acceleration (SR) and time dilation when considering the radial gravitational fields of massive bodies (GR).
I was talking about actual dilation (relative to something not in the gravity well, or not accelerating) vs you talking about relative dilation (what Alice and Bob locally observe in either scenario). The actual dilation is not a function of acceleration. This is why the centrifuge clock at the pole doesn't get out of sync with the clock sitting on the equator.
The relative dilation is a function of acceleration, which is why Alice and Bob cannot tell if they're accelerating or just parked on a planet.

So we're just talking about different things. We're both right.

Steve3007
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### Re: Does Special Relativity contain contradictions?

I'm talking about the sense that you're computing dilation for a planet both times, and never for an accelerating rocket.
In the second method I am using the result for an accelerating rocket together with the equivalence of gravitional and inertial mass.
Yes, I agree that GR equations derive from SR.
Good. I guess after all that we agree then.
(although you didn't post the code for the GR case)
Here's the code:

Code: Select all

``````        private void button1_Click(object sender, EventArgs e)
{
time_dilation_to_infinity = SchwarzschildSolution(M_planet, r_lower);
double time_dilation_upper_to_infinity =  SchwarzschildSolution(M_planet, r_upper);
time_dilation_lower_upper = time_dilation_to_infinity / time_dilation_upper_to_infinity;
time_dilation_X_years = seconds_in_1_year * years - (time_dilation_to_infinity * seconds_in_1_year * years);

textBoxTimeDilSchwarz.Text = time_dilation_to_infinity.ToString();
textBox1GRTimeDil.Text = time_dilation_lower_upper.ToString();
textBoxTimeDilSchwarzXyears.Text = time_dilation_X_years.ToString();
}

// General Relativity
// Returns the tick rate outside of a gravitational well (i.e. at infinite distance from it),
// compared to the tick rate inside the well.
// Given the mass of the gravitating body and the distance of the clock in the well from the
// centre of that body.
// Uses the equation for the Schwarzschild solution to Einstein's GR field equations.
{
return Math.Sqrt(1 - (2.0 * G * mass / (radius * c * c)));
}

// General Relativity
// Returns the Schwarzschild radius of a spherically symetrical body of mass 'mass'.
{
return 2.0 * G * mass / (c * c);
}
``````

Steve3007
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### Re: Does Special Relativity contain contradictions?

Halc wrote:I was talking about actual dilation (relative to something not in the gravity well, or not accelerating) vs you talking about relative dilation (what Alice and Bob locally observe in either scenario). The actual dilation is not a function of acceleration. This is why the centrifuge clock at the pole doesn't get out of sync with the clock sitting on the equator.
The relative dilation is a function of acceleration, which is why Alice and Bob cannot tell if they're accelerating or just parked on a planet.
Hold on a sec. I'll go back and look at that example that you gave.
So we're just talking about different things. We're both right.
That is a relief!

Halc
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### Re: Does Special Relativity contain contradictions?

Tamminen wrote:
October 9th, 2018, 10:06 am
I am describing the initial situation when the rocket is in free fall instead of standing on the ground. It accelerates relative to that inertial frame.
It accelerates if it is in a gravitational field. Either way, there is no local test to detect the difference.

An interesting non-local test that can be done in such free fall: Arrange a set of objects into a sphere at rest in a vacuum. If there is no gravity, or the field is totally uniform, the sphere will stay a sphere. If it deforms, it will get longer in one dimension, thinner in a second dimension, and essentially unchanged in the third. The dimension where it gets shorter is north/south, perpendicular to your orbit (but you can't tell which is which). The dimension unchanged is east/west (the direction you're moving), again ambiguous. The dimension stretched out is up/down, and again, you can't tell which is which. The orbital period can be determined as being 4 times the time it takes the North/South sides to meet in the middle as it compresses.

Halc
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### Re: Does Special Relativity contain contradictions?

Steve3007 wrote:
October 9th, 2018, 10:45 am
I'm talking about the sense that you're computing dilation for a planet both times, and never for an accelerating rocket.
In the second method I am using the result for an accelerating rocket together with the equivalence of gravitional and inertial mass.
In the second method, you're using the equations for an accelerating rocket to compute relative dilations of clocks at a different radius in a gravitational field. You never actually compute numbers for Alice and Bob in an accelerating rocket that's not in a gravitational field. Numeric integration would not be required to do that if you use their own accelerating frame. If you use an inertial frame, they yes, you'd need to do the integration to either keep the coordinate time steps constant, or keep the proper time steps constant.

Steve3007
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### Re: Does Special Relativity contain contradictions?

I think it's this:
viewtopic.php?p=321269#p321269
Halc wrote:I thought of a way to do the centrifuge experiment. Put the one clock on the equator (moving at 450km/sec linear speed relative to Earth) but not accelerating much. Put the centrifuge at the North Pole where linear speed is only from the centrifuge, and also 450km/sec, as tight a radius as we can build. Same speed and gravitational potential, so those should cancel out. The only difference would be the acceleration. I say the clocks stay in sync. The articles that seem to address this point say this, but there is a lot of misunderstanding in the forums, and I admit to not being an expert.
Ok so, as I understand you, you're trying to set up an experiment which compares 450km/sec linear speed with 450km/sec speed in a circle, with therefore large acceleration towards the centre of the circle; the centre of the centrifuge.

I don't know for sure either, but I suspect the clocks wouldn't stay in sync, and here's my reasoning:

The Alice/Bob experiment shows that two clocks at either end of an accelerating box will get out of sync with each other. The Bob-clock is dilated compared to the Alice-clock. So this would be true of two clocks positioned at different points on the radial axis of the centrifuge.

They'd effectively be at either end of a box that's accelerating towards the centre of the centrifuge. In terms of the equivalence principle I think they'd be in the equivalent of a gravitational field whose force vectors point radially outwards and whose magnitude is proportional to v2/r. So I think there would be time dilation between different clocks at different values of 'r' in the centrifuge. And there would therefore be time dilation between any of those clocks and a clock that is not rotating.

---

Incidentally, straight after that you said this:
The article you quoted misstates the principle of equivalence: "Einstein’s principle of equivalence tells us that whatever is true for acceleration is true for a gravitational field."

That is just wrong, so I question the rest of the article. The principle says there is no local test to determine which situation you're in.
For the reasons I've been discussing recently, I don't agree with you here. It's not wrong as such. It just doesn't make it clear that acceleration is in every measurable way equivalent to a uniform gravitational field.

Steve3007
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### Re: Does Special Relativity contain contradictions?

Halc wrote:In the second method, you're using the equations for an accelerating rocket to compute relative dilations of clocks at a different radius in a gravitational field.
Yes, by using the complete equivalence between an accelerating reference frame and a reference frame that is stationary in a uniform gravitational field.
You never actually compute numbers for Alice and Bob in an accelerating rocket that's not in a gravitational field.
I use numbers that are entirely equivalent between acceleration and a uniform gravitational field. That's why I can use an equation that was calculated for acceleration, not gravity, as in the article that I cited.
Numeric integration would not be required to do that if you use their own accelerating frame. If you use an inertial frame, they yes, you'd need to do the integration to either keep the coordinate time steps constant, or keep the proper time steps constant.
I'm not sure what you're saying in this part. The reason why numerical integration is required is because the gravitational field is not uniform; it is radial. So the acceleration to which it is equivalent is a whole series of boxes with different accelerations.

This is why, in your centrifuge example, I think clocks at different distances from the centre of the centrifuge would experience dilation relative to each other. Because they are in the equivalent of an inverted radial gravitational field where the force is directly proportional to r rather than being inversely proportional to its square.

Steve3007
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### Re: Does Special Relativity contain contradictions?

Steve3007 wrote:I don't know for sure either, but I suspect the clocks wouldn't stay in sync, and here's my reasoning:

The Alice/Bob experiment shows that two clocks at either end of an accelerating box will get out of sync with each other. The Bob-clock is dilated compared to the Alice-clock. So this would be true of two clocks positioned at different points on the radial axis of the centrifuge.

They'd effectively be at either end of a box that's accelerating towards the centre of the centrifuge. In terms of the equivalence principle I think they'd be in the equivalent of a gravitational field whose force vectors point radially outwards and whose magnitude is proportional to v2/r. So I think there would be time dilation between different clocks at different values of 'r' in the centrifuge. And there would therefore be time dilation between any of those clocks and a clock that is not rotating.
A quick rethink of this: I still think that there would be time dilation between clocks at different radii in the centrifuge, and that (in GR terms) it acts like a kind of inverted gravitational field. But I suspect that this can also be explained in SR terms and that therefore you are right to say that a clock that is moving at linear speed v would stay in sync with a clock that is positioned in the centrifuge such that its tangential speed is also v.

So, after all this discussion, I suspect that, as you've said, we're basically in agreement.

Halc
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### Re: Does Special Relativity contain contradictions?

Steve3007 wrote:
October 9th, 2018, 11:19 am
Ok so, as I understand you, you're trying to set up an experiment which compares 450km/sec linear speed with 450km/sec speed in a circle, with therefore large acceleration towards the centre of the circle; the centre of the centrifuge.
Yes, except 450m/sec, not km per sec. I said it wrong the first time. Bob at the equator is moving at that speed, and is also going in a circle, but it takes a day to do a lap, so much lower acceleration. We should run the experiment for at least a day if we were to do this for real, so Bob's net velocity at the end is zero just like that of Alice.
I don't know for sure either, but I suspect the clocks wouldn't stay in sync, and here's my reasoning:

The Alice/Bob experiment shows that two clocks at either end of an accelerating box will get out of sync with each other.
Bob is not in the box here, and the environment is very different where Alice is. Each knows which is which. I'm putting at least 1000 g's on poor Alice.
The Bob-clock is dilated compared to the Alice-clock. So this would be true of two clocks positioned at different points on the radial axis of the centrifuge.
Right, but we're not doing that. Only Alice, at the exact point where she has the same linear speed as Bob, and the same gravitational potential as well. <-- Pun mildly intended. There is no third clock somewhere else on the centrifuge.
They'd effectively be at either end of a box that's accelerating towards the centre of the centrifuge. In terms of the equivalence principle I think they'd be in the equivalent of a gravitational field whose force vectors point radially outwards and whose magnitude is proportional to v2/r. So I think there would be time dilation between different clocks at different values of 'r' in the centrifuge.
Big time, yes. But we're not doing that. Just Alice in there.
And there would therefore be time dilation between any of those clocks and a clock that is not rotating.
I think it would be the same at exactly one point in the box: the one that moves at 450m/sec. Clearly it runs faster at the center (not moving at all), and slower further out (more V and more acceleration), so somewhere in between they must balance. If acceleration is not a factor, it balances at whatever RPM gives us 450m/sec. If acceleration matters, it balances at a smaller radius than that.
If we want 1000g, we need to pick an r that is about 22m I think, 290000x smaller than the r of Bob who is accelerating at about 1/290th of a g.

Your GR equation is useless since r is identical in both instances. It is computing gravitational dilation of something stationary, not of something moving. The SR equation you use in your program only computes a relative dilation for a clocks separated by h, and we have no h here. The appropriate SR equation is the simple Lorentz factor for 450m/sec in both cases. Acceleration (g) does not play into that equation.

Incidentally, straight after that you said this:
The article you quoted misstates the principle of equivalence: "Einstein’s principle of equivalence tells us that whatever is true for acceleration is true for a gravitational field."

That is just wrong, so I question the rest of the article. The principle says there is no local test to determine which situation you're in.
For the reasons I've been discussing recently, I don't agree with you here. It's not wrong as such. It just doesn't make it clear that acceleration is in every measurable way equivalent to a uniform gravitational field.
I still stand by my assessment of that. Every locally measurable way, yes, but not every measurable way. My Uranus/Earth thing bears that out. Both locally detect a similar g, but the one with the slightly higher g has considerably less actual dilation. There is no local test for actual dilation or for r (the r is what differs between Earth and Uranus). David would be ecstatic if there was such a test, since it would prove that his is the only correct view.

Halc
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### Re: Does Special Relativity contain contradictions?

Steve3007 wrote:
October 9th, 2018, 11:31 am
You never actually compute numbers for Alice and Bob in an accelerating rocket that's not in a gravitational field.
I use numbers that are entirely equivalent between acceleration and a uniform gravitational field. That's why I can use an equation that was calculated for acceleration, not gravity, as in the article that I cited.
The rocket has no decreasing gravity as it moves. Your calculation compute a new different g at every point. The rocket has a fixed g. Alice at the top measures the same g as Bob at the bottom, not 30000 times less like in the Earth numbers. You seem not to realize that you're not simulating that.
I'm not sure what you're saying in this part. The reason why numerical integration is required is because the gravitational field is not uniform;
It is in an accelerating rocket. Both Alice and Bob measure 1g (9.8m/sec2) for all of the 100 (subjective?) years. The actual dilation would be much more than the dilation on Earth because the rocket would really be moving after that much time, but it is the relative dilation between Alice and Bob we're trying to compute here. You're not doing that.
This is why, in your centrifuge example, I think clocks at different distances from the centre of the centrifuge would experience dilation relative to each other.
There are no multiple clocks in my centrifuge. There's just the one. The second clock is on the equator.