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Does Special Relativity contain contradictions?

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Steve3007
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Re: Does Special Relativity contain contradictions?

Post by Steve3007 » October 9th, 2018, 8:52 am

This is the code that is called when the "Calculate" button is pressed for the SR method (bottom of the GUI). As described in the previous post, it combines the individual contributions from lots of small heights, within each of which the gravitational field can be regarded as uniform. i.e. numerical integration.

Code: Select all

        private void buttonCalcSR_Click(object sender, EventArgs e)
        {
            double t = 1.0;
            double r = r_lower;
            for (int i = 0; i < steps; i++)
            {
                t = t * GravitationalTimeDilation(M_planet, r, H);
                r += H;
            }
            textBox1SRTimeDil.Text = t.ToString();
        }


        // As above, but we now calculate the time dilation for a small (local) room sitting
        // at distance 'radius' from the centre of a gravitating mass (planet) of mass 'mass'.
        double GravitationalTimeDilation(double mass, double radius, double h)
        {
            double g = G * mass / (radius * radius);
            return AccelerationalTimeDilation(g, h);
        }

        
        // Special Relativity
        // For an accelerating room/rocket of height 'h' this function returns the difference
        // in the measured tick rate of a clock at the bottom of the room compared to a clock
        // at the top of the room.
        double AccelerationalTimeDilation(double g, double h)
        {
            return 1 - g * h / (c * c);
        }        
        

Tamminen
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Re: Does Special Relativity contain contradictions?

Post by Tamminen » October 9th, 2018, 9:27 am

Halc wrote:
October 9th, 2018, 7:34 am
From an inertial frame POV: The rocket guy is accelerating and the planet guy is not.
I have understood that being in free fall in a curved spacetime is being in an inertial frame. Light propagates in the same way as in an inertial frame in flat spacetime, so the relative time dilation between the top and bottom of the rocket is 0. Accelerating in relation to each of these inertial frames at a = 1 g makes the same effect on the relative time dilation, and one example of this acceleration is the rocket standing on the ground in a gravitational field of 1 g. Have I misunderstood the whole situation or is there a terminological confusion?

Steve3007
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Re: Does Special Relativity contain contradictions?

Post by Steve3007 » October 9th, 2018, 9:27 am

To try to make the above snippets of code clearer, and to try again to explain the process of iterating (numerically integrating) over a long series of small heights, within each of which the gravitational field can be regarded as uniform, here's a simple flowchart which schematically represents that code:
Flowchart.jpg
Flowchart.jpg (11.01 KiB) Viewed 1002 times
As I've said, 'h' is a very small constant value. The smaller it is, the closer each section comes to having a uniform gravitational field within it, and the closer the approximation to the GR result. But each of these small sections is calculated at a different value of 'g' because they are each at a different value of 'r'. This is what happens in a process of numerical integration. That's what numerical integration means.

We must surely all remember, at school, finding the area under a graph of a curve by dividing the graph into a series of thin rectangles? That's numerical integration. The area of each rectangle is not exactly equal to the area of the section of the graph that it represents. It's an approximation. But the thinner the rectangles (and therefore the more of them there are) the more accurate the approximation becomes.

See?

Steve3007
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Re: Does Special Relativity contain contradictions?

Post by Steve3007 » October 9th, 2018, 9:37 am

Tamminen wrote:I have understood that being in free fall in a curved spacetime is being in an inertial frame. Light propagates in the same way as in an inertial frame in flat spacetime, so the relative time dilation between the top and bottom of the rocket is 0. Accelerating in relation to each of these inertial frames at a = 1 g makes the same effect on the relative time dilation, and one example of this acceleration is the rocket standing on the ground in a gravitational field of 1 g. Have I misunderstood the whole situation or is there a terminological confusion?
I don't think you have misunderstood the situation. I think you've understood it. I think in an earlier comment you mentioned that in an accelerating reference frame, we can think of spacetime as being flat but angled i.e. uniform. You were exactly right. The same slope everywhere. The same arrows everywhere. Acceleration is, in every measurable way, equivalent to being in a uniform gravitational field. But real gravitation fields are not uniform. They are radial. They spread out from a point and get weaker with the square of the distance from that point. That's why, when we go from the SR consideration of an accelerating box to GR time dilation, we have to use a series of small boxes. Because within each of those small boxes the gravitational field is approximately uniform. The smaller the box, the better the approximation.

That's what I've been trying to explain to Halc in my last few posts.

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Halc
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Re: Does Special Relativity contain contradictions?

Post by Halc » October 9th, 2018, 9:41 am

Steve3007 wrote:
October 9th, 2018, 8:52 am
This is the code that is called when the "Calculate" button is pressed for the SR method (bottom of the GUI). As described in the previous post, it combines the individual contributions from lots of small heights, within each of which the gravitational field can be regarded as uniform. i.e. numerical integration.
OK, you seem to be using two different methods to compute the dilation of clocks on/near gravitational wells. This contradicts your description of it:
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.
The second method is actually doing what the first one did: Calculates the numbers for a gravitational field. It computes g from r for instance in the GravitationalTimeDilation function, with r being bumped by H each iteration. But in the case of clocks at either end of an accelerating room/box/elevator/rocket which is not in a gravitational field, g would be a constant instead of a function of a nonexistent r.
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.

I'm not disputing that GR does not follow from SR.

Steve3007
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Re: Does Special Relativity contain contradictions?

Post by Steve3007 » October 9th, 2018, 9:42 am

Halc wrote:The second method is actually doing what the first one did
No it's not. The first method uses General Relativity, as described by Schwartzschild.

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Halc
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Re: Does Special Relativity contain contradictions?

Post by Halc » October 9th, 2018, 9:47 am

Tamminen wrote:
October 9th, 2018, 9:27 am
Halc wrote:
October 9th, 2018, 7:34 am
From an inertial frame POV: The rocket guy is accelerating and the planet guy is not.
I have understood that being in free fall in a curved spacetime is being in an inertial frame.
It is equivalent to an inertial frame. Locally, the observer cannot tell the difference. Non-locally, we can. You look in the sky and notice that we're going around the sun once a year. Clearly we're not inertial by this non-local observation.
Light propagates in the same way as in an inertial frame in flat spacetime, so the relative time dilation between the top and bottom of the rocket is 0.
The rocket isn't inertial, but accelerating, so I don't know what you mean by this. It isn't in free fall.
Accelerating in relation to each of these inertial frames at a = 1 g makes the same effect on the relative time dilation, and one example of this acceleration is the rocket standing on the ground in a gravitational field of 1 g. Have I misunderstood the whole situation or is there a terminological confusion?
That part looks good.

Steve3007
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Re: Does Special Relativity contain contradictions?

Post by Steve3007 » October 9th, 2018, 9:47 am

Halc wrote: It computes g from r for instance in the GravitationalTimeDilation function
No, it computes 'g' from 'r' using the standard definition:

g = GM/r2
with r being bumped by H each iteration.
Yes.
But in the case of clocks at either end of an accelerating room/box/elevator/rocket which is not in a gravitational field, g would be a constant instead of a function of a nonexistent r.
It is the equivalent of a uniform gravitational field, in which g is constant. As I've said, in a small volume, with small 'h', g can be regarded as constant within that volume. As I've said, with reference to measuring the area under a graph of a curve using a series of thin rectangles, that's how numerical integration works. The small the value of 'h' the better the approximation but the larger the number of iterations required.

Steve3007
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Re: Does Special Relativity contain contradictions?

Post by Steve3007 » 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.

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?

Steve3007
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Re: Does Special Relativity contain contradictions?

Post by Steve3007 » October 9th, 2018, 9:55 am

Halc, do you get this now?

In the application I use two different methods to calculate the time dilation.

1.
This equation from General Relativity: (1 - 2GM / rc2)0.5

2.
This equation which was derived from Special Relativity: (1 - gh / c2)

Both of those equations are written on the application's GUI to make this easier to see.

Steve3007
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Re: Does Special Relativity contain contradictions?

Post by Steve3007 » October 9th, 2018, 10:04 am

Halc wrote: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. One is the gravity case, in the sense that it uses the equation from General Relativity. The other is the acceleration case in the sense that it uses the equation from Special Relativity which tells us the time dilation between the top and bottom of a box of height 'h' which is accelerating. It then uses the fact that acceleration is exactly equivalent in every measurable way to a uniform gravitational field.

The only reason why you've read people saying that the equivalence principle only works "locally" is because real gravitational fields are radial. So they can only be approximated as a uniform gravitational field in a small volume. i.e. locally. Hence the integration/iteration.

Tamminen
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Re: Does Special Relativity contain contradictions?

Post by Tamminen » October 9th, 2018, 10:06 am

Halc wrote:
October 9th, 2018, 9:47 am
The rocket isn't inertial, but accelerating, so I don't know what you mean by this. It isn't in free fall.
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.

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Halc
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Re: Does Special Relativity contain contradictions?

Post by Halc » October 9th, 2018, 10:11 am

Steve3007 wrote:
October 9th, 2018, 9:27 am
To try to make the above snippets of code clearer, and to try again to explain the process of iterating (numerically integrating) over a long series of small heights, within each of which the gravitational field can be regarded as uniform, here's a simple flowchart which schematically represents that code:
I can read the code. The GR functions are also missing. All you posted was the SR parts. The SR formula for the rocket needs no iteration. Just plug in the acceleration and height of the rocket to get the dilation factor, and then multiply that by the duration of the experiment to get the number of seconds difference after X many years.
As I've said, 'h' is a very small constant value. The smaller it is, the closer each section comes to having a uniform gravitational field within it, and the closer the approximation to the GR result. But each of these small sections is calculated at a different value of 'g' because they are each at a different value of 'r'. This is what happens in a process of numerical integration. That's what numerical integration means.
I know what numerical integration is. I used a similar method to simulate a stable inverted pendulum, a really weird effect of classical physics. You did it correctly, but you did it for varying gravity both times, never constant acceleration like you'd get in the rocket. That's why the numbers came out the same.

To do the rocket, you need to ignore r. There is no r in a rocket. There is just a fixed g, and the computations with GR and SR will still come out the same if you use fixed g instead of g computed as a function of r and m (nit: Mass is m, not M), but with a fixed g, there is no dilation in comparison with the clock at infinity, so that would not be computable, which illustrates that actual dilation is not a function of only g.

Correction of one of my comments: I had mistakenly put the escape velocity from the rocket at infinity, but actually its escape velocity is c. Anything slower than that shot from the front of the rocket, and the rocket will eventually catch up to it.

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Re: Does Special Relativity contain contradictions?

Post by Steve3007 » October 9th, 2018, 10:12 am

Halc wrote:OK, you seem to be using two different methods to compute the dilation of clocks on/near gravitational wells. This contradicts your description of it:
No, it agrees with what I've now said over the last few posts. I have used two different methods and they have given the same numerical result. Method one uses General Relativity's equations that describe time dilation in real, radial gravitational fields. Method two uses Special Relativity's equations that describe time dilation in accelerated reference frames.

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Re: Does Special Relativity contain contradictions?

Post by Halc » October 9th, 2018, 10:16 am

Steve3007 wrote:
October 9th, 2018, 9:47 am
Halc wrote: It computes g from r for instance in the GravitationalTimeDilation function
No, it computes 'g' from 'r' using the standard definition:
g = GM/r2
I thought I just said that. I don't understand the 'No,' part.

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