(Alpha) A new equation for escape velocity

(The “Alpha” in the title indicates that the Alpha rules apply to this thread.)

In another thread I showed that GR is self-inconsistent. The proof there stands unrefuted. In this thread I lay some groundwork for a new theory of gravity.

The other thread showed that a theory of gravity that incorporates SR cannot predict black holes, in particular their horizons, or else the theory will be self-inconsistent. The supporting info in the other thread notes that “You can think of the horizon as the place where the escape velocity equals the velocity of light”. Then part of the solution for a new theory of gravity becomes obvious: the escape velocity must always be less than c. This is compatible with the following conclusion in the other thread: “Then there is no reason—in a self-consistent theory of gravity—why A and B could not remain essentially at rest with respect to each other for an arbitrarily long time as measured in their common frame, as both particles move outward toward r=infinity”. When the escape velocity is always less than c, then the particles A and B can remain essentially at rest with respect to each other for an arbitrarily long time as measured in their common frame, as they both move outward (escape) toward r=infinity.

So a new equation for escape velocity is needed.

Einstein’s equation for escape velocity, in geometric units, is the same as Newton’s:

v = sqrt(2M / r)

The “sqrt” is the square root function. The r-coordinate r must be at or above the surface of the body. The velocity v is a fraction of c (unity); e.g. v = 0.5 = 0.5c = half of the speed of light. (See also andrewgray’s derivation of the equation here.)

The scientific method lets an equation be presented without a derivation. The validity of an equation is determined solely by whether its predictions are self-consistent and agree with observations within their margins of error. For example, a self-consistent equation that accurately predicts the paths of tornadoes is scientifically valid regardless of its derivation. (Note that Einstein’s field equations, the source of the equation above, are not derived; see pg. 2-20 here.) Although I could derive the new equation for escape velocity, I present it here without a derivation:

v = sqrt(2M / (r + 2M))

Notice that v is less than c (unity) when r is greater than 0. There is no reason to believe that r can be zero (a singularity), since a body need not implode to r=0 when the escape velocity is less than c above r=0. When objects can escape to r=infinity from anywhere above r=0, then the surface of a body can stay above r=0, for the body is but a collection of objects.

Experimental confirmation of the new equation for escape velocity: GR has been confirmed to all significant digits by every experimental test of it to date. GR has been experimentally confirmed to no more than five significant digits, one less digit than the current best precision of the gravitational constant G, which is factored into M (one significant digit is reserved for calculations, according to the rules for handling significant digits). In the strongest-gravity test of GR to date, the minimum r is 210000M. When r is 100000M or greater, the results of the equations above match to at least five significant digits (for r=100000M, both equations return 0.0044721c). Then the new equation is confirmed to all significant digits by every experimental test of GR’s equation to date, because the equations above return the same result for every experimental test of GR’s equation to date, after rounding to five significant digits.

 

Log in or Sign up to hide all adverts.

Then your equation is useless.

This is like saying---"I have data which confirms that life eixists on Mars. I will not show you the data, though, I will just present the conclusion."

 

Log in or Sign up to hide all adverts.

I'd like to point out in my opinion, the alleged proof is neither confirmed nor refuted in this forum, and that it would be more accurately called an "argument" rather than a "proof".
Other forum members, including BenTheMan, have asserted that the alleged proof has been thoroughly refuted. Ben specifically claims that zanket's alleged proof relies on laymen's definitions of fundamental concepts and does not attempt to show a mathematical inconsistency.
Refer to zanket's link if you're interested.

 

Log in or Sign up to hide all adverts.

Then you must agree that Einstein's field equations are useless too, for they weren’t derived either, as Taylor and Wheeler note in the link I gave. Einstein's equation for escape velocity is ultimately derived from his field equations. A derivation that ultimately sources from something made up doesn’t add any validity per se. It’s like making up the equation y=x+x and saying that the derived equation y=2x is somehow made valid by the derivation. Einstein called his field equations “the free invention of the human mind”.

Newton’s laws of motion weren’t derived either. Newton made them up, fitting them to observations.

No, it’s not like that at all. It’s like making up an equation that fits the observed paths of tornadoes, or the orbits of the planets. Weather forecasting models are filled with underived equations; nevertheless, those models are certainly useful.

 

Anyone can assert anything they want. What counts is only what can be supported, as usual.

Ben’s assertion that an inertial frame must be zero-sized blatantly disagrees with all of Einstein, Taylor, Thorne and Wheeler, and doesn’t even allow SR to be experimentally tested. What more could possibly need to be said against this claim? He asserts that all of these top-notch physicists intentionally used incorrect definitions in the texts I got them from. An outlandish claim, but should be easy to prove: just show nonequivalent definitions from a “valid” text. But did he prove it? Nope. He just made an empty claim, with no attempt to support it. Yet here you are suggesting that it’s my problem.

The proof does show a mathematical inconsistency, but it need not contain math when the math has already been done by hundreds of people and the same results published in hundreds of texts. In that case I can just use the published predictions to show a self-inconsistency (contradicting published predictions), which is I what I do. It sure sounds damning to cry “there’s no math!”, but notice there’s no math in Einstein’s thought experiments either and they’re no less valid for the omission. It takes more than “there’s no math!” to show a problem. I’ve been waiting a long time for someone to show me proof that math is required to refute a theory.

Quadraphonics’ claim that the “definitions you're using are vague and heuristic” is a red herring. The definitions I’m using source from Taylor, Thorne, and Wheeler. No definitions exist from more definitive sources. No nonequivalent definitions exist from these sources. Quadraphonics showed no problems with the definitions, mind you; he or she only railed against them. Doing that should really be against the Alpha rules, if it isn’t already.

The bottom line is that the “refutations” to which you refer were unscientifically made. They were empty claims that only look valid on the surface. The claimants did not back them up.

Nobody need be convinced by my proof of course. Called it “alleged” if you will. But let’s keep in mind that refutations of my proof can be alleged too. Your comment here seems to give the impression that I have ignored valid refutations in other threads. I amply refuted them when they were brought up the first time.

 

So you've compared yourself to Einstein and Newton in the second post. Nice.

Einstein's equations CAN be derived from differential geometry. They can be derived from Newton's laws.

You have made the statement that "I can derive this equation, but I won't do it," more or less. So either you can derive the equation, or you can't. IF you can't, then fine---retract the statement and we can continue. If you can derive the equation, then please do so, so that I can see how on earth you've convinced yourself that you are right.

 

If so, so what?

I decline, especially for your attitude. Scientists need not give a derivation for their equations. I chose to exclude the derivation here because, for this particular equation, it would add no value. When you prove that an equation can be valid only when it can be derived, then I’ll show my derivation here.

 

Occam’s razor doesn’t say we ought to prefer illogic, does it? I showed that a theory that incorporates SR and Einstein’s equation for escape velocity will be self-inconsistent. My equation is proven to be superior. I have shown a convincing theoretical consideration.

 

What about that whole side-thread Andrew Grey started, where he works out all of the gory details? If that's not a refutation, I don't know what is.

 

Read it all the way through to see that, in the end, he just reinforces my proof of GR’s self-inconsistency. He proved that there’s a Rindler horizon at r=2M for an observer hovering just above (arbitrarily close to) r=2M. That proves that nothing can pass outward through r=2M (the well-published prediction of GR), which my proof depends on. In the end, nothing in his thread refuted my proof.

 

Surely, this is the case. Surely you are refusing to post proof of your equation because of ME, and not because you don't know why this equation should be right. This is CLEARLY not the case.

I have a new equation for escape velocity as well, which I will post without derivation:

\(v_{esc} = \sqrt{\frac{2M}{r+\pi}}\).

Now prove that this equation is wrong.

 

Pete---I would also point out that zanket's refusal to answer the question is in direct conflict with the alpha rules. He must either show how one derives his equation, or be forced to retract the statement that his equation CAN be derived.

 

The same is true for:

\(v = \frac{1}{2 \sqrt{M}} \sqrt{ -r + \sqrt{ r^2 + 16M^2 } }\),​

which I present without derivation.

 

I need not refute your equations here. Start your own thread about that if you want, and then maybe I’ll refute your equation there.

The rules say “The relevance of a question must be demonstrated on request.” The request for relevance was implied in “When you prove that an equation can be valid only when it can be derived, then I’ll show my derivation here”.

What could you do with any derivation I give? It wouldn’t necessarily mean anything. Here’s a derivation for you, to show that I can derive the new equation for escape velocity:

2v = 2 * sqrt(2M / (r + 2M))
Simplifying,
v = sqrt(2M / (r + 2M))

I made up the first equation, but hey, the final equation is derived. Likewise, you say that Einstein’s equations can be derived from Newton’s laws, and Newton made up those laws. This satisfies your request that I defend this statement of mine: “Although I could derive the new equation for escape velocity, ...”.

 

It is perfectly okay to do this, according to the scientific method, which says (boldface mine) “Scientists are free to use whatever resources they have — their own creativity, ideas from other fields, induction, Bayesian inference, and so on — to imagine possible explanations for a phenomenon under study.” An explanation forms a hypothesis, which can be expressed as an equation.

You’d be following in the footsteps of Newton and Einstein, among others. There’s nothing invalid about your equation that I can see (it is even fully experimentally confirmed), but Occam’s razor gives the nod to mine. The principle paraphrases as "All things being equal, the simplest solution tends to be the best one."

 

No. Occam's Razor gives the nod to

\(v_{esc}=\sqrt{\frac{2M}{r}}\).

I told you two different ways to derive Einsten's Equations. You've simply pulled this equation for escape velocity out of your ass.

 

The problem is, there's no reason to BELIEVE your equaiton. I can derive the other equation using conservation of energy. So you're telling me conservation of energy is wrong. Is this correct?

If this is correct, then please show me which line of the following derivation is flawed.

An object (of mass m) sitting a distance R away from the center of mass of a body of mass M has total potential energy given by:
\( U = \frac{Mm}{R}\).
In order to escape from the gravitational field of M, m must convert all of its potential energy to kinetic energy. Hence, the total energy of m at a distance of infinitiy is
\(U = \frac{1}{2}mv^2\)
where v is the ``escape velocity''. Then, by conservation of energy
\(\frac{Mm}{R} = \frac{1}{2}mv^2 \).
This implies
\(v = \sqrt{\frac{2M}{R}}. \)

So are you telling me that you have rejected conservation of energy, zanket?

 

"Simplest" in this context loosely means "based on the least axioms", and the "solutions" being compared are complete, structured theories. All you and I have presented are stand-alone equations with ad-hoc patches to get them to satisfy a property, and with no supporting theory. It makes little sense to apply Occam's razor in specific cases like this. This is why you should post a derivation if you have one.

 

przyk---

I have derived a new equation which I present here without derivation.

\(zanket=wrong\)