[confused]

I originally posted this in another thread, but it was kind of a departure from that discussion so I figured I'd just make a new topic outlining my issues with the math in the UA wiki article. I have found some really egregious mathematical and logical errors that I believe demonstrate major problems with UA, and potentially the system of fact-checking (or lack thereof) on the Wiki as a whole.

At the beginning of this Wiki article, it is stated that "the Earth is accelerating 'upward' at a constant rate of 9.8m/s^2."

So with this sentence, you have basically defined the earth's acceleration:
dv/dt=9.8 m/s^2.

OK, cool, no complaints so far. However, in the special relativity section, we see the following differential equation used for the earth's acceleration:
dv/dt=g/ɣ^3.

OK, so dv/dt has been redefined specifically for the special relativity argument-- this is a really disingenuous move, and it gets worse from here. Here are my main contentions:

1. The whole point of this page is to show that the earth's acceleration is constant. If the earth's acceleration is dv/dt=g/ɣ^3, it is very clearly not constant. The fact that no one would notice such an obvious contradiction is somewhat alarming to me.

2. I don't think this is even the correct equation for calculating relativistic acceleration, but my knowledge of special relativity is fairly elementary so I won't argue that point. However, based on my calculations (using separation of variables), the solution given for this differential equation isn't even correct! You can check it yourself by taking the derivative of the "solution"-- it will not give you the original differential equation!

3. Here is the conclusion that is arrived at based on these erroneous calculations: "As you can see, it is impossible for dark energy to accelerate the Earth past the speed of light." This statement actually disproves UA-- if the earth's velocity cannot exceed the speed of light, how can it continue accelerating at a constant rate forever? Based on the solution given, as t approaches infinity and v approaches c, it follows that dv/dt must approach 0, meaning that the earth's acceleration is decreasing.

[Pete Svarrior]

OK, so dv/dt has been redefined specifically for the special relativity argument--

No, it hasn't. Because we're talking about Special Relativity, the frame of reference is extremely important. The acceleration is constant at 9.81ms^-2 from a non-inertial frame of reference (in this case, the only frame of reference that's easily observable in the real world, e.g. an observer who had just jumped up). The argument of accelerating to the speed of light, however, only becomes at all interpretable in an inertial frame of reference - hence the need to rephrase the equations.

this is a really disingenuous move

Nah, it's just physics.

https://en.wikipedia.org/wiki/Special_relativity

Pay special attention to the fact that "the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source".

1. The whole point of this page is to show that the earth's acceleration is constant. If the earth's acceleration is dv/dt=g/ɣ^3, it is very clearly not constant. The fact that no one would notice such an obvious contradiction is somewhat alarming to me.

Addressed above, not a contradiction. Your username is very applicable.

Please read more about the Lorentz Transformations at https://en.wikipedia.org/wiki/Lorentz_transformation

2. I don't think this is even the correct equation for calculating relativistic acceleration, but my knowledge of special relativity is fairly elementary so I won't argue that point. However, based on my calculations (using separation of variables), the solution given for this differential equation isn't even correct! You can check it yourself by taking the derivative of the "solution"-- it will not give you the original differential equation!

Could you show your workings?

3. Here is the conclusion that is arrived at based on these erroneous calculations: "As you can see, it is impossible for dark energy to accelerate the Earth past the speed of light." This statement actually disproves UA-- if the earth's velocity cannot exceed the speed of light, how can it continue accelerating at a constant rate forever? Based on the solution given, as t approaches infinity and v approaches c, it follows that dv/dt must approach 0, meaning that the earth's acceleration is decreasing.

This is simply a restatement of your introduction and of point 1. You have no familiarity with Special Relativity, specifically Lorentz transformations, and you're easily confused by multiple frames of reference. You are not arguing against the Flat Earth Theory here, you are arguing against simple physics, which you should have a good grasp of by the time you finished high school.

Your failure here basically boils down to the fact that UA would indeed be impossible under classical mechanics, which is clearly your default state of mind when thinking about mechanics. However, classical mechanics is just a simplification of reality which produces reasonably accurate results as long as we're not dealing with (for example) things that approach the speed of light.

[confused]

OK, so dv/dt has been redefined specifically for the special relativity argument--

No, it hasn't. Because we're talking about Special Relativity, the frame of reference is extremely important. The acceleration is constant at 9.81ms^-2 from a non-inertial frame of reference (in this case, the only frame of reference that's easily observable in the real world, e.g. an observer who had just jumped up). The argument of accelerating to the speed of light, however, only becomes at all interpretable in an inertial frame of reference - hence the need to rephrase the equations.

This isn't a simple rephrasing. If the acceleration changes with speed (as the equation given for dv/dt suggests) it is not constant, regardless of reference frame. There is no Lorentz transform in the calculations done on the Wiki page, we are not talking about multiple frames of reference.

2. I don't think this is even the correct equation for calculating relativistic acceleration, but my knowledge of special relativity is fairly elementary so I won't argue that point. However, based on my calculations (using separation of variables), the solution given for this differential equation isn't even correct! You can check it yourself by taking the derivative of the "solution"-- it will not give you the original differential equation!

Could you show your workings?

If v is indeed the solution of dv/dt, then taking the derivative of v should give us the original differential equation. However, when taking the derivative of the given v, dv/dt=g/(ɣ^3*t^3). Uh-oh, a t^-3 has appeared!

3. Here is the conclusion that is arrived at based on these erroneous calculations: "As you can see, it is impossible for dark energy to accelerate the Earth past the speed of light." This statement actually disproves UA-- if the earth's velocity cannot exceed the speed of light, how can it continue accelerating at a constant rate forever? Based on the solution given, as t approaches infinity and v approaches c, it follows that dv/dt must approach 0, meaning that the earth's acceleration is decreasing.

This is simply a restatement of your introduction and of point 1. You have no familiarity with Special Relativity, specifically Lorentz transformations, and you're easily confused by multiple frames of reference. You are not arguing against the Flat Earth Theory here, you are arguing against simple physics, which you should have a good grasp of by the time you finished high school.

Do you even math bro? The statement I'm making here is purely mathematical, and it is only the logical progression of the mathematical conclusion made in the Wiki article. Even if I didn't know anything about special relativity it would not affect this point. The author says that as t-->∞, v-->c. I'm saying that based on this statement that the author has made, it follows that dv/dt must approach 0 as t-->∞.

You have no familiarity with Special Relativity, specifically Lorentz transformations, and you're easily confused by multiple frames of reference. You are not arguing against the Flat Earth Theory here, you are arguing against simple physics, which you should have a good grasp of by the time you finished high school.

Your failure here basically boils down to the fact that UA would indeed be impossible under classical mechanics, which is clearly your default state of mind when thinking about mechanics. However, classical mechanics is just a simplification of reality which produces reasonably accurate results as long as we're not dealing with (for example) things that approach the speed of light.

Tsk, tsk, so many assumptions about my background. Please point out which part of my argument hinges on a classical assumption.

Edit: Actually, I see what you're saying in the context of the article, but I'm still not convinced that its interpretation of proper acceleration is correct. Gotta do some reading

[Pete Svarrior]

If the acceleration changes with speed (as the equation given for dv/dt suggests) it is not constant, regardless of reference frame.

It is constant in the reference frame that matters to us locally. It obviously wouldn't be constant in an inertial frame, because we're talking about SR.

There is no Lorentz transform in the calculations done on the Wiki page, we are not talking about multiple frames of reference.

I'm sorry, I can't say anything other than "Yes, we are talking about multiple frames of reference, and the Lorentz transformations are referenced directly in the article."

See https://en.wikipedia.org/wiki/Lorentz_transformation#Physical_formulation_of_Lorentz_boosts for more information.

If v is indeed the solution of dv/dt, then taking the derivative of v should give us the original differential equation. However, when taking the derivative of the given v, dv/dt=g/(ɣ^3*t^3). Uh-oh, a t^-3 has appeared!

Ah, okay, you're mistaking ∂ for d. I had originally assumed that your notation is incorrect just for the sake of simplicity (or your inability to use the [tex] tag), but it seems that you're actually mixing the two concepts up. That's pretty silly of you.

Do you even math bro?

I'm seemingly good enough at it to tell the difference between partial derivatives and total derivatives. That puts me at an unfair advantage over you.

Tsk, tsk, so many assumptions about my background.

I'm not making assumptions, I'm drawing conclusions.

Please point out which part of my argument hinges on a classical assumption.

That would be points 1 and 3. Your inability to understand the differences between two frames of reference is perfectly understandable if you operate under classical mechanics. It is not at all understandable if you have a rudimentary understanding of SR. Therefore, it follows that you do not have a fundamental understanding of SR and are thus resorting to Newtonian mechanics.

[confused]

If v is indeed the solution of dv/dt, then taking the derivative of v should give us the original differential equation. However, when taking the derivative of the given v, dv/dt=g/(ɣ^3*t^3). Uh-oh, a t^-3 has appeared!

Ah, okay, you're mistaking ∂ for d. I had originally assumed that your notation is incorrect just for the sake of simplicity (or your inability to use the [tex] tag), but it seems that you're actually mixing the two concepts up. That's pretty silly of you.

As it is written, v is only a function of t, and there don't seem to be any other partials present, so treating it as an ODE should work, no? I'm admittedly a bit rusty on my DE game.

As for the reference frames, the article specifically says "Differential equation for velocity on earth," so this should be acceleration in the same reference frame as "dv/dt=g," no?

[rabinoz]

That would be points 1 and 3. Your inability to understand the differences between two frames of reference is perfectly understandable if you operate under classical mechanics. It is not at all understandable if you have a rudimentary understanding of SR. Therefore, it follows that you do not have a fundamental understanding of SR and are thus resorting to Newtonian mechanics.

Without going through all the points you say, I am quite prepared to say that you are correct and the a number of "confused's" points are not valid.

But, from any practical standpoint for essentially all calculations and observations on earth and even (with a few proviso's) within the Solar system, So what!.

For the ordinary person there are no calculations for which Newtonian Laws of Motion and Gravitation are not perfectly adequate.
On earth for speeds of up to 420,000 m/s (1.53x10⁹ km/hr) the change in mass, time, etc is under 1 part in 1,000,000.

The nearest connection any ordinary person has to anything that is slightly affected by time dilation (due to velocity and gravitation) is the slight correction is the on-board atomic clocks in GPS satellites.
Of course particle physicists do meet very significant relativistic effects.

On gravitation for any earthbound calculations that any ordinary person the Newtonian calculations are perfectly adequate.

And, as I have asked many times before:
What force did Henry Cavendish (and numerous others in the couple of centuries since) measure.
Though there are, so far unexplained small variations, they certainly measured a force that

leads to an accepted value of G = 6.673 x 10^-11 m³ kg^-1 s^-2

Most of these have used apparatus similar in principle to the torsion balance apparatus designed and constructed by geologist John Michell, and performed by Henry Cavendish (John Michell died before he could do the experiment himself). But some later quite different experiments are using quantum properties of cold Rubidium atoms and a 500 kg test mass. So far the results from this method differ a little from the accepted value, but later refinements are getting better agreement.

As far as I can find there was never any thought given to GR in all the calculations here.

So yes, if you want "perfect accuracy" go do your tensor analysis and use GR for all your simple calculations.
But, I simply cannot fathom FE supporters who can be so pedantic over things like this,
yet happily accept distortions in country dimensions of thousands of kilometres!
and do bother about sunrise and sunset directions being in error around 45° at the equator and much more further south!

Come on show a bit of consistency!