A Flaw of General Relativity, a Fix, etc.

[Zanket]

That presupposes that an observer can remain at rest at a fixed altitude. I repeat my initial statement that this is an invalid assumption. Note in particular, that it's not relevant whether there is actually an observer there. If there is no possibility of such, the frame of reference is not a valid one.

I am currently at rest at a fixed altitude, so I assume you’re talking about at and below an event horizon. That presupposes the existence of an event horizon. But section 2 shows that free-fall velocity never reaches c, which is required for event horizons to exist. Since I show that event horizons do not exist, it is invalid reasoning for you to say that my reasoning is flawed because it wouldn’t apply below an event horizon. Your reasoning is akin to saying that any refutation of GR is invalid because it disagrees with GR.

[Grey]

I am currently at rest at a fixed altitude, so I assume you’re talking about at and below an event horizon.

You're presupposing that an observer can remain at rest in any imaginable location. There are obviously many locations for which that's a valid assumption, but it may turn out to not be true for all imaginable locations. And indeed, under general relativity, there are locations for which that is an invalid assumption.

But section 2 shows that free-fall velocity never reaches c, which is required for event horizons to exist.

Only be supposing that you can remain at rest at any arbitrary distance from any given gravitational source, which is equivalent to assuming what you are attempting to prove. Your argument is circular.

Since I show that event horizons do not exist, it is invalid reasoning for you to say that my reasoning is flawed because it wouldn’t apply below an event horizon. Your reasoning is akin to saying that any refutation of GR is invalid because it disagrees with GR.

General relativity specifically limits the domains under which special relativity is valid. That's the point, after all. Finding a circumstance under which general relativity says you cannot use special relativity in the manner you do, applying it anyway, and then arriving at a contradiction does not show that general relativity is inconsistent. To show general relativity inconsistent, you'd have to show that you can reach a contradiction by actually using the rules of general relativity.

[Zanket]

You're presupposing that an observer can remain at rest in any imaginable location. There are obviously many locations for which that's a valid assumption, but it may turn out to not be true for all imaginable locations.

It is true where the escape velocity is < c. Only where I have verified (not assumed) that the escape velocity is < c is where I use that.

And indeed, under general relativity, there are locations for which that is an invalid assumption.

Which section 2 refutes. GR cannot be presupposed to be valid in a refutation of it. And GR cannot be used to refute a refutation of it.

Only be supposing that you can remain at rest at any arbitrary distance from any given gravitational source, which is equivalent to assuming what you are attempting to prove. Your argument is circular.

Where the escape velocity is < c, I can suppose that. I start free-falling at infinity, where the escape velocity is zero, so I can remain at rest at that altitude. Then I determine my free-fall velocity a little lower down. It is < c, so escape velocity is < c, so I can remain at rest at that altitude. Then I determine my free-fall velocity a little lower down, and so on. In this way section 2 shows that the free-fall velocity never reaches c. Never does section 2 assume that the escape velocity is < c at a given altitude so that I can remain at rest at that altitude. Rather, that is a given at infinity, and implicitly verified at each lower altitude before continuing lower still. And so my argument is not circular.

General relativity specifically limits the domains under which special relativity is valid. Finding a circumstance under which general relativity says you cannot use special relativity in the manner you do, applying it anyway, and then arriving at a contradiction does not show that general relativity is inconsistent.

GR cannot limit my usage of SR, a standalone theory, outside of GR. A theory that says, "The maximum velocity in SR is 0.5c, and using SR to show a velocity > 0.5c is an invalid usage of SR", is not irrefutable.

To show general relativity inconsistent, you'd have to show that you can reach a contradiction by actually using the rules of general relativity.

Section 2 uses SR and the equivalence principle, standalone components of GR, to show that GR is inconsistent. Since I'm working outside of GR, I need not follow any rule of GR (like a rule that says, “SR cannot be used to refute this theory”).

[Grey]

It is true where the escape velocity is < c. Only where I have verified (not assumed) that the escape velocity is < c is where I use that.

You have not verified this (more below), since you use this assumption in your verification, and you have not done the math to actually support this conclusion. So, the argument is indeed circular.

Where the escape velocity is < c, I can suppose that. I start free-falling at infinity, where the escape velocity is zero, so I can remain at rest at that altitude. Then I determine my free-fall velocity a little lower down. It is < c, so escape velocity is < c, so I can remain at rest at that altitude. Then I determine my free-fall velocity a little lower down, and so on. In this way section 2 shows that the free-fall velocity never reaches c. Never does section 2 assume that the escape velocity is < c at a given altitude so that I can remain at rest at that altitude. Rather, that is a given at infinity, and implicitly verified at each lower altitude before continuing lower still. And so my argument is not circular.

Except that you don't actually calculate the formula you'd get for escape velocity using this method, you merely provide a handwaving argument. Specifically, I expect you'll find that if you applied these rules as allowed by general relativity (more on why you need to use these rules in the manner general relativity prescribes below), that at the Schwarzschild radius you'll reach an asymptotic value of the speed of light for the escape velocity, and below that, it will give an imaginary value for the velocity of an infalling particle.

GR cannot limit my usage of SR, a standalone theory, outside of GR. A theory that says, "The maximum velocity in SR is 0.5c, and using SR to show a velocity > 0.5c is an invalid usage of SR", is not irrefutable.

It doesn't have to; special relativity puts that limit on itself. In particular, special relativity cannot be applied by itself in any region where gravity cannot be neglected (that's what the special is all about, remember?). General relativity actually expands the applicability of special relativity, by allowing it to be used in situations involving gravity, but in a very specific manner.

Section 2 uses SR and the equivalence principle, standalone components of GR, to show that GR is inconsistent. Since I'm working outside of GR, I need not follow any rule of GR (like a rule that says, “SR cannot be used to refute this theory”).

If you're working outside of general relativity, you cannot prove that general relativity is inconsistent. To show that system X is inconsistent, you need to follow the rules of system X, and show that they lead to a contradiction. If you instead follow the rules of system Y and reach a contradiction, you've shown instead that system Y is inconsistent.

In particular, as I hope you're aware, adopting special relativity and the principle of equivalence is not equivalent to general relativity, which involves further assumptions. It's possible to create theories similar to general relativity by taking those two as axioms, but then making different further assumptions than those made by Einstein. Indeed, a number of people have done so, but to date, all such theories that I am aware of have either been shown to be mathematically equivalent to general relativity, or do not match the experimental data as well. You've essentially created such a theory yourself, by assuming that special relativity has unlimited validity along with the equivalence principle, so what you've really shown* is that "Zanket's Theory of Relativity" leads to a contradiction, thus demonstrating that at least one of your assumptions is false. But one of your assumptions is that special relativity can be applied under any circumstances, and we already know that assumption is false, so this should be no surprise.

* Actually, as I point out above, I'm not even certain that you've shown this, since you haven't provided anything other than a handwaving argument that the escape velocity doesn't reach the speed of light, rather than actually calculating it, as I noted above.

[Zanket]

You have not verified this (more below), since you use this assumption in your verification, and you have not done the math to actually support this conclusion. So, the argument is indeed circular.

I don’t see anything in your post that shows how the argument is circular. Section 2 does not assume that an observer can remain at rest at any altitude; I gave my reason for that above and you haven’t addressed that reason here. Section 2 provides all the equations needed to make its point; no more math is needed.

Except that you don't actually calculate the formula you'd get for escape velocity using this method, you merely provide a handwaving argument.

An argument that escape velocity must always be less than c need not derive the equation for escape velocity, as section 2 shows. All sorts of arguments to show maximum or minimum values can be made without deriving equations for the exact value. For example, if you give me a curve that spans from 0 to 10 on the x-axis, and is always < 10 on the y-axis, I can make a valid argument that the area under that curve above the y-axis is less than 100, without deriving an equation to calculate the exact area.

Specifically, I expect you'll find that if you applied these rules as allowed by general relativity (more on why you need to use these rules in the manner general relativity prescribes below), that at the Schwarzschild radius you'll reach an asymptotic value of the speed of light for the escape velocity, and below that, it will give an imaginary value for the velocity of an infalling particle.

Your reasoning for why I must use those rules is invalid, as shown below.

It doesn't have to; special relativity puts that limit on itself. In particular, special relativity cannot be applied by itself in any region where gravity cannot be neglected (that's what the special is all about, remember?).

The paper uses SR only in flat spacetime where it applies. Section 2 applies SR to each section of a nonuniform gravitational field deemed uniform, and sums the results. The summing is not an invalid usage of SR, just like the summing of the areas of rectangles under a curve (to find the area under the curve) is not an invalid usage of the formula for the area of a rectangle.

General relativity actually expands the applicability of special relativity, by allowing it to be used in situations involving gravity, but in a very specific manner.

I’m not beholden to any rules of GR about the usage of SR. Only if my theory was dependent upon GR—and it isn’t—would I need to follow such rules. SR is a standalone component of GR. I need only follow the rules of SR in using SR.

If you're working outside of general relativity, you cannot prove that general relativity is inconsistent. To show that system X is inconsistent, you need to follow the rules of system X, and show that they lead to a contradiction. If you instead follow the rules of system Y and reach a contradiction, you've shown instead that system Y is inconsistent.

Let me clarify: With the exception of SR and the equivalence principle, standalone components of GR, I am working outside of GR. The paper shows that those components can be used in a valid manner, i.e. a manner which those components allow, to reach a conclusion that contradicts GR. That shows an inconsistency of GR. If you disagree, then tell me how you are not implicitly agreeing that I can create an irrefutable theory that says, "The maximum velocity in SR is 0.5c, and using SR to show a velocity > 0.5c is an invalid usage of SR". That is a theory that uses SR as a component, and has a rule that limits the usage of SR. According to your logic, such theory cannot be shown to be inconsistent. If you think that only GR can place such limits, then tell me your basis for that.

In particular, as I hope you're aware, adopting special relativity and the principle of equivalence is not equivalent to general relativity, which involves further assumptions.

Agreed. The paper certainly does not attempt to recreate GR or an equivalent theory.

It's possible to create theories similar to general relativity by taking those two as axioms, but then making different further assumptions than those made by Einstein.

Agreed.

Indeed, a number of people have done so, but to date, all such theories that I am aware of have either been shown to be mathematically equivalent to general relativity, or do not match the experimental data as well.

My theory is not mathematically equivalent, as shown in section 3, A New Equation for Escape Velocity. It matches the experimental data equally (i.e. to the same degree as does the Schwarzschild metric), as shown in section 6, Experimental Confirmation.

You've essentially created such a theory yourself, by assuming that special relativity has unlimited validity along with the equivalence principle, so what you've really shown* is that "Zanket's Theory of Relativity" leads to a contradiction, thus demonstrating that at least one of your assumptions is false.

You haven't shown how that would demonstrate that any of my assumptions would be false. What is your basis for that conclusion?

But one of your assumptions is that special relativity can be applied under any circumstances, and we already know that assumption is false, so this should be no surprise.

You basis for that assumption being false is an invalid assumption that GR is valid. GR is a theory; its validity need not be assumed by another theory of gravity. Nor can GR place any limitation on the usage of its standalone components, such that other theories of gravity need follow those rules in using those components. In the years between 1905 and 1915, there was SR and not GR. Anybody could have used SR to create a relativistic theory of gravity. Do you think that once GR existed, the doors were shut to theories of gravity that do not adhere to limits on the usage of SR that GR places?

* Actually, as I point out above, I'm not even certain that you've shown this, since you haven't provided anything other than a handwaving argument that the escape velocity doesn't reach the speed of light, rather than actually calculating it, as I noted above.

Section 3, A New Equation for Escape Velocity, derives an equation for escape velocity that returns only values < c.

[Grey]

Let me just address one important portion of your reply at the moment. I'll get to the rest later.

I’m not beholden to any rules of GR about the usage of SR. Only if my theory was dependent upon GR—and it isn’t—would I need to follow rules of GR about the usage of SR.

You are if you wish to show that general relativity is inconsistent. I'll discuss that further below.

SR is a standalone component of GR. I need only follow the rules of SR in using SR.

The rules of special relativity say that you cannot use it at all in the presence of gravity.

Let me clarify: With the exception of SR and the equivalence principle, standalone components of GR, I am working outside of GR.

Then you cannot show a contradiction in general relativity using this method. Again, more below.

The paper shows that those components can be used in a valid manner, i.e. a manner which those components allow, to reach a conclusion that contradicts GR. That shows an inconsistency of GR.

No, it would merely show that general relativity is inconsistent with the use of those components in that manner, and that's already more or less clear from the theory itself. Since special relativity alone cannot be used in the presence of gravity at all, you cannot be using it in a manner which it allows.

If you disagree, then tell me how you are not implicitly agreeing that I can create an irrefutable theory that says, "The maximum velocity in SR is 0.5c, and using SR to show a velocity > 0.5c is an invalid usage of SR". That is a theory that uses SR as a component, and has a rule that limits the usage of SR. According to your logic, such theory cannot be shown to be inconsistent.

You're confusing two terms here, irrefutable and inconsistent. The small theory you've proposed here (perhaps with a few slight modifications of wording) is in fact consistent! To show it to be inconsistent, you'd actually have to follow its own rules and show that they lead to a contradiction. But is it irrefutable? Of course not. It can be refuted by showing that experimental evidence does not agree with it (in particular, that objects can travel at velocities faster than half that of light).

You've claimed to show that general relativity must be incorrect because it is inconsistent. But since you haven't followed general relativity's rules, you've shown nothing at all about its consistency.

[Zanket]

The rules of special relativity say that you cannot use it at all in the presence of gravity.

The equivalence principle shows that a local frame (i.e. a frame throughout which the tidal force is negligible) in free fall is an inertial frame, within which SR applies. The spacetime within such frame is flat. Then SR applies in flat spacetime. The paper holds the equivalence principle to be valid, and uses SR only in flat spacetime.

No, it would merely show that general relativity is inconsistent with the use of those components in that manner, and that's already more or less clear from the theory itself. Since special relativity alone cannot be used in the presence of gravity at all, you cannot be using it in a manner which it allows.

Since SR can be used in flat spacetime, and since curved spacetime (a nonuniform gravitational field) is flat locally, SR can be used in curved spacetime in each local segment, and the results of those segments be summed without violating SR to calculate a result that applies to curved spacetime, just like the areas of rectangles under a curve can be summed to find the area under the curve without invalidly using the formula for the area of a rectangle. Section 2 uses SR in this way, and reaches a different conclusion than GR reaches. GR cannot be consistent and have components that can used by themselves in a valid way to reach a conclusion that differs from what GR itself reaches. Nor can GR restrict the way in which those components can be used by themselves.

The small theory you've proposed here (perhaps with a few slight modifications of wording) is in fact consistent! To show it to be inconsistent, you'd actually have to follow its own rules and show that they lead to a contradiction.

A theory cannot enforce consistency by limiting how a standalone component of itself can be used. In my small theory, a rule is given that “using SR to show a velocity > 0.5c is an invalid usage of SR”. But SR has no such rule, and the small theory cannot force the rule to be followed outside of the small theory. Then SR can be used by itself without following the rule to show that the maximum velocity in SR is > 0.5c, showing an inconsistency of the small theory. The inconsistency is obvious in the following sentence implied by the small theory: “The maximum velocity in SR, in which the maximum velocity can be > 0.5c, is 0.5c.”

But is it irrefutable? Of course not. It can be refuted by showing that experimental evidence does not agree with it (in particular, that objects can travel at velocities faster than half that of light).

I agree with you here. My choice of words was bad. By “irrefutable” I meant “cannot be refuted by being shown to be inconsistent”.

You've claimed to show that general relativity must be incorrect because it is inconsistent. But since you haven't followed general relativity's rules, you've shown nothing at all about its consistency.

By your logic, any theory can enforce consistency by including rules that disallow showing the inconsistency of claims like “the maximum velocity in SR is 0.5c”. I disagree, for the reason given above. Do you also think that any theory can be made irrefutable by adding the rule, “this theory cannot be shown to be refuted”? If not, what makes that rule invalid, whereas a rule that disallows showing an inconsistency is valid?

[Grey]

The equivalence principle shows that a local frame (i.e. a frame throughout which the tidal force is negligible) in free fall is an inertial frame, within which SR applies. The spacetime within such frame is flat. Then SR applies in flat spacetime. The paper holds the equivalence principle to be valid, and uses SR only in flat spacetime.

This is, of course, an extension of special relativity. General relativity makes a similar extension, though it does not extend it as far as you seem to. General relativity's extension of special relativity is certainly as valid as your own. Remember, by itself, special relativity cannot be applied when gravity is involved at all. It's only through an extension such as one of these that we can use special relativity in these cases.

A theory cannot enforce consistency by limiting how a standalone component of itself can be used.

Again, general relativity expands the applicability of special relativity. Besides, consistency of a theory can only be judged within that theory itself. If, following the rules of a system, you can come up with a contradiction, then that system is inconsistent. If you come up with a contradiction by not following the rules of a system, you haven't shown anything about whether those rules or consistent or not.

In my small theory, a rule is given that “using SR to show a velocity > 0.5c is an invalid usage of SR”. But SR has no such rule, and the small theory cannot force the rule to be followed outside of the small theory. Then SR can be used by itself without following the rule to show that the maximum velocity in SR is > 0.5c, showing an inconsistency of the small theory. The inconsistency is obvious in the following sentence implied by the small theory: “The maximum velocity in SR, in which the maximum velocity can be > 0.5c, is 0.5c.”

I did say that the theory you proposed would probably need to have its wording tweaked. I didn't think that I'd need to, since I assumed we both understood, but perhaps I was wrong. Saying "the maximum velocity allowed in special relativity is 0.5c" is not a theory. Rather it's a statement about special relativity, and you're correct that it's a false one. However, I could create a theory that says "use special relativity for all calculations where velocity is less than 0.5c, and any velocity which special relativity suggests is higher than that will be observed as being 0.5c", which is what I'd assumed you intended. Note that this theory doesn't say anything about what special relativity itself says, it merely limits its domain. This theory is internally consistent. Similarly, I could create a theory that says that special relativity is universally applicable and there is no such thing as gravity. Such a theory is also internally consistent. For that matter, Newtonian mechanics is internally consistent. All of these can be refuted by showing that they contradict experimental evidence, but they are consistent nevertheless.

I agree with you here. My choice of words was bad. By “irrefutable” I meant “cannot be refuted by being shown to be inconsistent”.

...

By your logic, any theory can enforce consistency by including rules that disallow showing the inconsistency of claims like “the maximum velocity in SR is 0.5c”. I disagree, for the reason given above. Do you also think that any theory can be made irrefutable by adding the rule, “this theory cannot be shown to be refuted”? If not, what makes that rule invalid, whereas a rule that disallows showing an inconsistency is valid?

Assuming you're not confusing irrefutable and inconsistent again, I never claimed that one could meaningfully add such a rule to a theory. For that matter, you can't necessarily just add a rule saying "this theory cannot be shown to be inconsistent", and that this would make the theory consistent, and I never claimed that you could. Whether a theory is consistent or not can only be judged by following the rules of that theory and seeing if you can come to a contradiction.

General relativity doesn't just say "you can't use special relativity to show that this theory is inconsistent". That's a clear straw man on your part, Zanket. Rather, it extends the applicability of special relativity in a specific way. It does not say that special relativity can be freely applied anywhere, but special relativity never made that claim either. Special relativity also never made the claim that it could be applied in small local regions of curved space as though they were flat. That's an extension made by general relativity.

You're trying to do a standard Reductio ad Absurdum argument hear, which is perfectly reasonable. To do that, you assume that the premises are true, you reason logically from them, and arrive at a contradiction, thus showing your premises false. But your premises at the moment are "general relativity is valid and I can use special relativity in ways that general relativity does not allow". If you reach a contradiction from those premises, you're correct that would show at least one of those premises to be false. But it's perfectly possible to decide that your second premise is the false one, leaving general relativity consistent.

[Zanket]

Remember, by itself, special relativity cannot be applied when gravity is involved at all. It's only through an extension such as one of these that we can use special relativity in these cases.

Agreed. The paper holds the equivalence principle to be valid.

However, I could create a theory that says "use special relativity for all calculations where velocity is less than 0.5c, and any velocity which special relativity suggests is higher than that will be observed as being 0.5c", which is what I'd assumed you intended. Note that this theory doesn't say anything about what special relativity itself says, it merely limits its domain. This theory is internally consistent.

This theory is consistent, but rather than the rule limiting SR’s domain, it enforces a modified version of SR to be applied in all domains, that caps SR’s velocity at 0.5c.

Similarly, I could create a theory that says that special relativity is universally applicable and there is no such thing as gravity. Such a theory is also internally consistent.

Agreed.

Assuming you're not confusing irrefutable and inconsistent again, I never claimed that one could meaningfully add such a rule to a theory.

I know you didn’t claim that. I was just asking to test your logic.

For that matter, you can't necessarily just add a rule saying "this theory cannot be shown to be inconsistent", and that this would make the theory consistent, and I never claimed that you could.

Agreed on both.

Whether a theory is consistent or not can only be judged by following the rules of that theory and seeing if you can come to a contradiction.

A theory cannot enforce consistency by having a rule that effectively alters a standalone component of itself, like your small theory above does.

General relativity doesn't just say "you can't use special relativity to show that this theory is consistent".

I assume you mean “inconsistent”. I agree that GR doesn’t say that.

Special relativity also never made the claim that it could be applied in small local regions of curved space as though they were flat. That's an extension made by general relativity.

Yes, it’s an extension made by the equivalence principle, a standalone component of GR.

But your premises at the moment are "general relativity is valid and I can use special relativity in ways that general relativity does not allow".

No, in section 2 it goes like this: “SR and the equivalence principle are valid components of GR, and I can use those components in a way that those components allow, while ignoring any other rules of GR, to return a result that differs from what GR predicts, showing an inconsistency of GR”.

Now, so that we’re on the same page, below I summarize and paraphrase your points against the paper, and summarize the rebuttal I’ve given above for each of your points. Please correct me where I’ve misstated your points:

Grey: You handwave in section 2 because you don’t calculate the equation for escape velocity.
Z: Calculating the equation for escape velocity isn’t necessarily required to show a maximum value for escape velocity, as section 2 shows.

G: The argument in section 2 is circular because you assume that an observer can remain at rest at a fixed altitude to show that an observer can remain at rest at a fixed altitude.
Z: An observer can remain at rest at a given altitude if the escape velocity there is < c. It is a given that the escape velocity at infinity, where the free-falling particle starts, is zero; hence an observer can remain at rest there. The escape velocity at a given altitude is the free-fall velocity from rest at infinity there. The equations used in section 2 to calculate free-fall velocity, starting from infinity and falling to any given lower altitude, always return less than c, hence escape velocity is less than c everywhere, hence an observer can remain at rest at any altitude, hence this is a conclusion rather than an assumption, and hence the argument is not circular.

G: Section 2 doesn’t show that GR is inconsistent, because you haven’t followed the rules of GR for the region at and below the Schwarzschild radius that, were they followed, would cause the result there to match what GR predicts.
Z: In using SR and the equivalence principle, standalone components of GR, I need not follow any rule of GR outside of the rules of those components. GR does not have a rule that says that objects must fall at and below the Schwarzschild radius. Rather, this is an interpretation of GR’s prediction that the escape velocity is >= c there. Section 2, validly using components of GR, shows that the escape velocity is < c there, thus showing an inconsistency of GR.

G: You’re using SR in the presence of gravity, where SR doesn’t apply.
Z: The equivalence principle lets SR be used in flat spacetime (a uniform gravitational field), and the paper holds the equivalence principle to be valid.

You've essentially created such a theory yourself, by assuming that special relativity has unlimited validity along with the equivalence principle, so what you've really shown* is that "Zanket's Theory of Relativity" leads to a contradiction, thus demonstrating that at least one of your assumptions is false.

Z: The inconsistency of GR shown in section 2 is shown by validly using components of GR.

My position at this point is that you haven’t shown a bona fide problem of the paper.

[worzel]

G: You’re using SR in the presence of gravity, where SR doesn’t apply.
Z: The equivalence principle lets SR be used in flat spacetime (a uniform gravitational field), and the paper holds the equivalence principle to be valid.

So you proved that if there were no gravity (or only uniform gravity), then there would be no black holes

[Fortis]

Zanket, if you have already answered this question then you're welcome to blow a raspberry in my direction. (We were talking about this back when it seemed as if you might have been assuming that the Newtonian potential was correct, but after deciding that potential was a tricky concept in GR I don't think that we came to any conclusions.)

What bit of Newtonian gravity do you assume to be correct?

Just want to make sure that your assumption here is absolutely explicit.

[Zanket]

So you proved that if there were no gravity (or only uniform gravity), then there would be no black holes

A nonuniform gravitational field (a region of curved spacetime) comprises uniform gravitational fields (regions of flat spacetime). Then SR can be integrated across a nonuniform gravitational field, applied to each segment of the field deemed uniform. The paper shows that there are no black holes in Schwarzschild geometry, a geometry in which spacetime is curved.

[Zanket]

What bit of Newtonian gravity do you assume to be correct?

I assume, like Newton did, that the inverse square law of gravity applies to the r-coordinate radius (the reduced circumference). That assumption is implied in the derivation of eq. 6, the new equation for escape velocity.

[worzel]

A nonuniform gravitational field (a region of curved spacetime) comprises uniform gravitational fields (regions of flat spacetime). Then SR can be integrated across a nonuniform gravitational field, applied to each segment of the field deemed uniform. The paper shows that there are no black holes in Schwarzschild geometry, a geometry in which spacetime is curved.

I don't think that follows from the equivalence principle. That you can integrate the way you did, that is. Could you be specific about how you did the integration and justify why it is ok. IIRC there is no integration in your paper.

[Fortis]

I assume, like Newton did, that the inverse square law of gravity applies to the r-coordinate radius (the reduced circumference). That assumption is implied in the derivation of eq. 6, the new equation for escape velocity.

What is the the "inverse square law"? This sounds silly (we all think that we know what this means), but let's but a bit more math on the bones.

Is it,

F = - G.m_1.m_2.r/r^3,

combined with F = m_2.(d^2/dt^2)r ?

Or does it now relate to the rate of change of the four momentum with respect to the proper time of the test particle?

[Zanket]

Could you be specific about how you did the integration and justify why it is ok.

What is the the "inverse square law"?

These two are related, so I answer with one post. The inverse square law of gravity is: “the force between two masses decreases as the square of the distance r between them”. At some level the derivation of Newton’s escape velocity equation, eq. 4, involves an integration that applies the inverse square law. The derivation of eq. 6 converts eq. 4 to be relativistic, thus implicitly utilizing Newton’s integration. I didn’t need to know any more details about the inverse square law to make the points in the paper, and I don’t offer anything more about it. (In fact, I didn't need to know anything about the inverse square law, nor did I need to know whether or not Newton used integration at some level in his derivation of eq. 4. These topics are side discussions that don't affect the validity of the paper.) The derivation of eq. 6 is mathematically and logically valid, and it leads to a metric that is confirmed by experimental tests and resolves four major problems of physics (as noted in the paper); that justifies it. The equivalence principle cannot prevent SR from being integrated across a nonuniform gravitational field, any more than it can prevent the use of addition or subtraction.

[Grey]

A theory cannot enforce consistency by having a rule that effectively alters a standalone component of itself, like your small theory above does.

Where do you get this idea? If I'm creating a theory, I can decide that the rules of that theory are anything I want them to be. If I want to create a theory that says special relativity is valid except on Tuesdays, when Newtonian mechanics should be used, I'm free to do so (well, if I can come up with some unambiguous way of deciding when it's Tuesday). Or I could create a theory that says special relativity applies only to baryons, and some other method should be used for the kinematics of leptons. If what you suggest were the case, I couldn't create a theory like general relativity at all, since it alters the way special relativity can be applied (by expanding it to domains that have gravitational forces). I'm completely free to create any theory I'd like, and I'm free to use any pre-existing theories as components, extending or limiting them in any way that I might wish.

I assume you mean “inconsistent”. I agree that GR doesn’t say that.

Correct, thanks for interpreting that correctly. I'll edit the original post so this is clear.

Now, so that we’re on the same page, below I summarize and paraphrase your points against the paper, and summarize the rebuttal I’ve given above for each of your points. Please correct me where I’ve misstated your points:

We can revisit most of these after we've come to some agreement on what limits I have in creating a theory, and in what it means for a theory to be self-consistent. Until then, it's pointless to try to discuss anything more meaningful.

G: Section 2 doesn’t show that GR is inconsistent, because you haven’t followed the rules of GR for the region at and below the Schwarzschild radius that, were they followed, would cause the result there to match what GR predicts.
Z: In using SR and the equivalence principle, standalone components of GR, I need not follow any rule of GR outside of the rules of those components.

Again, the consistency of a system is determined solely by whether you can arrive at a contradiction (proving both P and not-P) by following the rules of the system. If you reason outside the rules of the system, you can't say anything about the consistency of the system itself, since you haven't shown that your conclusion follows from those rules.

In extending the appicability of special relativity GR does not have a rule that says that objects must fall at and below the Schwarzschild radius.

Actually, it does, though it's not that explicit. But if you work out the math according to the rules given in general relativity, that's exactly the result you'll reach.

G: You’re using SR in the presence of gravity, where SR doesn’t apply.
Z: The equivalence principle lets SR be used in flat spacetime (a uniform gravitational field), and the paper holds the equivalence principle to be valid.

But as we've established, the equivalence principle together with special relativity alone does not lead to general relativity. So, assuming your derivation is valid, you've shown that the equivalence principle and special relativity as you're applying it lead to a contradiction. That does not, however, show that general relativity is also inconsistent, since you've assumed a different set of postulates than those used by general relativity.

When you do a proof by contradiction, you show that the set of postulates you've assumed cannot be valid. You haven't shown that some other (prehaps very similar) set of postulates cannot be valid.

[blueshift]

What is the the "inverse square law"? This sounds silly (we all think that we know what this means), but let's but a bit more math on the bones.

Is it,

F = - G.m_1.m_2.r/r^3,

combined with F = m_2.(d^2/dt^2)r ?

Or does it now relate to the rate of change of the four momentum with respect to the proper time of the test particle?

From what my reading has suggested to me was that Schwarzschild was trying to describe surface curvature.

Surface curvature = bending of 2 geodesics/ area swept out by those geodesics.

He used a perfect sphere because the curvature is the same everywhere on a perfect sphere. Therefore, he didn't need to have two nearby geodesics. They could be quite far apart. Both geodesics were put at the equator of the sphere and moved to its north pole and met at a 90* angle,
1/4 of the 2pi radians of a circle.

1/4 x 2pi= (pi/2) radians = the bending of the geodesics..

For the area swept out? This area = 1/4 of the northern hemisphere or 1/8 of the surface of the sphere.

Area of a sphere= 4pi (r)^2
1/8 x 4pi(r)^2 = (pi/2) (r)^2

Surface curvature then= pi/2 divided by (pi/2)(r)^2 = 1/r^2..The curvature of a sphere or any portion of a sphere-or even of any spherelike portion of a surface of the most irregular shape = the inverse square of the radius of the curvature at that locale according to Wheeler's view of Schwarzschild's view..

1/r^2 also suggested to Schwarzschild ( after writing it as Descartes would have...1/(r)(r)....) that more than one curvature could exist. Central mass could be shaped like a cigar or a football or a paraboloid, each with 2 curvatures.

Further, it is under my understanding (or misunderstanding) that Schwarzschild was interested in the greatest amount of curvature potential that each mass had and, thus, considered the density increasing to a limit, an event horizon where the photon sphere was held by escape speed that reached c.

Therefore, the term "radius" didn't make any sense and had no use to him. Measuring the diameter of an object (BH) whose space is breaking up into pieces that are in free-fall is not possible..but describing a "reduced circumference" is possible.

[Zanket]

Where do you get this idea? If I'm creating a theory, I can decide that the rules of that theory are anything I want them to be. If I want to create a theory that says special relativity is valid except on Tuesdays, when Newtonian mechanics should be used, I'm free to do so (well, if I can come up with some unambiguous way of deciding when it's Tuesday).

Let me restate: When a theory has a rule that effectively alters a standalone component of itself, like your small theory above does, it is using a modified form of that theory. A theory that says that SR is valid except on Tuesdays uses a modified version of SR, a derivative of SR, that does not apply on Tuesdays. This theory is consistent with its own derivative of SR, but it is not consistent with the base version of SR, which is of course valid on Tuesdays. The rule cannot enforce consistency with the base version of SR.

If what you suggest were the case, I couldn't create a theory like general relativity at all, since it alters the way special relativity can be applied (by expanding it to domains that have gravitational forces).

SR’s applicability isn’t altered by GR. Rather, the equivalence principle shows that a local frame in free fall is an inertial frame, the type of frame that SR requires. There is effectively no gravity in this frame.

Again, the consistency of a system is determined solely by whether you can arrive at a contradiction (proving both P and not-P) by following the rules of the system. If you reason outside the rules of the system, you can't say anything about the consistency of the system itself, since you haven't shown that your conclusion follows from those rules.

Where the equivalence principle says that SR applies, it is an unmodified version of SR being referenced. There is no rule in GR that effectively modifies SR, so I’m free to use the unmodified version of SR with the equivalence principle, and where my results conflict with GR’s predictions, an inconsistency of GR is shown.

Actually, it does, though it's not that explicit. But if you work out the math according to the rules given in general relativity, that's exactly the result you'll reach.

It’s not at all explicit because it’s just a consequence of the prediction that the escape velocity is >= c there. Einstein didn’t know that his field equations predicted black holes until Schwarzschild solved those equations. That’s why it’s called a “Schwarzschild radius”, and that proves that there’s no explicit rule in GR that forces the escape velocity to be >= c there or requires that objects must fall there. It becomes an implicit rule only after looking at the predictions of Schwarzschild’s solution.

But as we've established, the equivalence principle together with special relativity alone do not lead to general relativity. So, assuming your derivation is valid, you've shown that the equivalence principle and special relativity as you're applying it lead to a contradiction. That does not, however, show that general relativity is also inconsistent, since you've assumed a different set of postulates than those used by general relativity.

There is no postulate of GR that effectively modifies SR or the equivalence principle, even at and below the Schwarzschild radius. So in section 2 I’m following all the requirements I need to follow by simply using these components in their standalone form. These components are used to calculate a maximum escape velocity that differs from GR’s prediction. That shows an inconsistency of GR.

When you do a proof by contradiction, you show that the set of postulates you've assumed cannot be valid. You haven't shown that some other (prehaps very similar) set of postulates cannot be valid.

What you are referring to as a postulate of GR, I say is simply a consequence of GR’s prediction that the escape velocity is >= c at and below the Schwarzschild radius. Since there is no postulate that forces me to make that same prediction, when I predict differently using GR’s components, I’ve shown an inconsistency of GR.

If you still disagree, please tell me what postulate or rule of GR you think I'm not following in section 2, that I would need to follow to show an inconsistency of GR. I've shown that Einstein didn't know that GR predicted black holes when it was published, so it can't be a postulate or rule related to the Schwarzschild radius.

[blueshift]

A nonuniform gravitational field (a region of curved spacetime) comprises uniform gravitational fields (regions of flat spacetime). Then SR can be integrated across a nonuniform gravitational field, applied to each segment of the field deemed uniform. The paper shows that there are no black holes in Schwarzschild geometry, a geometry in which spacetime is curved.

This is a bit confusing here. I do think that you could integrate across a nonuniform gravitational field in the case of objects moving in spacetime with respect to locally inertial frames. Inside the event horizon, however, objects are not falling. It is space that is in a free-fall and it can exceed speed c.

[Zanket]

This is a bit confusing here. I do think that you could integrate across a nonuniform gravitational field in the case of objects moving in spacetime with respect to locally inertial frames. Inside the event horizon, however, objects are not falling. It is space that is in a free-fall and it can exceed speed c.

Section 2 does such an integration to show that there is no such thing as an event horizon. The paper does not assume that GR is valid, and so the paper cannot assume that event horizons exist, to prevent an integration to show that they don’t exist.

[Grey]

Let me restate: When a theory has a rule that effectively alters a standalone component of itself, like your small theory above does, it is using a modified form of that theory. A theory that says that SR is valid except on Tuesdays uses a modified version of SR, a derivative of SR, that does not apply on Tuesdays. This theory is consistent with its own derivative of SR, but it is not consistent with the base version of SR, which is of course valid on Tuesdays.

Very good. I'm glad that we can agree that I'm free to create any theory I might like. And we agree that such a theory can easily be self-consistent, even if in doing so it might no longer be consistent with the theories that were used to create it, right?

SR’s applicability isn’t altered by GR. Rather, the equivalence principle shows that a local frame in free fall is an inertial frame, the type of frame that SR requires. There is effectively no gravity in this frame.

Where the equivalence principle says that SR applies, it is an unmodified version of SR being referenced. There is no rule in GR that effectively modifies SR, so I’m free to use the unmodified version of SR with the equivalence principle...

Except of course that special relativity in its unmodified form applies to globally flat, Minkowski-type spacetime. Your extension of it using the equivalence principle instead restricts it to applying only piecewise to infinitesimal local regions, and of course we'll need to know something about the overall curvature and establish some rules for connecting those piecewise regions in order to find out what's going on in any macroscopic region. General relativity provides rules for that, and presumably you have some rules in mind as well. So we are dealing with a modified version. Unless you'd like to show some evidence that the original theory of special relativity involved integrating over infinitesimal local regions of spacetime?

...and where my results conflict with GR’s predictions, an inconsistency of GR is shown.

Why, since general relativity does not assume an unmodified version of special relativity?

Section 2 does such an integration to show that there is no such thing as an event horizon.

Just a nitpick here. You don't actually do the integration. You merely state the general result. I still think we need to settle issues of what it means for a theory to be consistent before discussing these issues, but I just couldn't let that comment pass.

[Zanket]

Very good. I'm glad that we can agree that I'm free to create any theory I might like. And we agree that such a theory can easily be self-consistent, even if in doing so it might no longer be consistent with the theories that were used to create it, right?

Yes, agreed. Since you are a software designer, we can use this analogy: If a class inherits a base class and overrides some of its methods, an application that uses the derived class is consistent with the derived class but may not be consistent with the base class.

Except of course that special relativity in its unmodified form applies to globally flat, Minkowski-type spacetime. Your extension of it using the equivalence principle instead restricts it to applying only piecewise to infinitesimal local regions, and of course we'll need to know something about the overall curvature and establish some rules for connecting those piecewise regions in order to find out what's going on in any macroscopic region. General relativity provides rules for that, and presumably you have some rules in mind as well. So we are dealing with a modified version. Unless you'd like to show some evidence that the original theory of special relativity involved integrating over infinitesimal local regions of spacetime?

If a theory tells you that the formula for the area of rectangle requires globally Euclidian geometry, does it really require that for a given rectangle? SR in its unmodified form applies to experiments within the confines of a region of flat spacetime. The curvature of spacetime beyond the confines of the experiment doesn’t matter, and SR can’t dictate differently. For example, in section 8 of the paper I say that the relativistic rocket equations apply to a “gantry that floats in flat spacetime of indefinite extent”. But that’s just a convenient way to ensure that any given experiment is confined within a region of flat spacetime.

The equivalence principle restricts the region within which experiments of SR in an unmodified form can be done. Practically speaking, the region need not be infinitesimal in size, or else we’d not have any experimental confirmation of SR. The rules for connecting piecewise regions are just those of basic calculus, just like I need no more than calculus to tell me that I can sum up the areas of tiny rectangles under a curve to find the approximate area under the curve (in fact, I need only employ basic logic for that, and employ calculus only to get a precise answer). Section 2 doesn’t need to know anything specific about the overall curvature, for the equations show that, whatever the curvature, the free-fall velocity, hence the escape velocity, is always less than c.

Why, since general relativity does not assume an unmodified version of special relativity?

I say that it does assume that, as above.

Just a nitpick here. You don't actually do the integration. You merely state the general result.

That is a nitpick! If I always used precise wording, I’d be here, well, longer. By “does such an integration” I mean “shows how such an integration could be formed”. If that logic about how to set up the integration is valid, then the “general result” (that a velocity returned by eq. 27 is always less than c) is apparent, despite that the integration isn’t actually done.

[Grey]

The “general result” is valid—no reasonable person could deny it—despite that the integration isn’t actually done.

I'll get back to this later (as you've probably noticed, I usually don't have time for more than one or two messages a day), but I had to respond to this. Let's keep this civilized, rather than toss about insults at anyone who doesn't agree with you, hm?

[Zanket]

Let's keep this civilized, rather than toss about insults at anyone who doesn't agree with you, hm?

I meant no insult. You took that differently than I intended. I modified that comment to be more clear.

[blueshift]

Zanket,

Perhaps going over section 1 might help clear up a few things. Take your time responding since several of us are probing here..

You "deem a relativistic rocket stationary at an arbitrary altitude of zero while its gantry 'free-falls' below it (relative to the rocket, not necessarily anything else).."

Now, are you assuming that Newton's Laws can be measured from within an accelerating frame of reference? Or, am I misreading something here?

[Fortis]

From what my reading has suggested to me was that Schwarzschild was trying to describe surface curvature.
.
.
.
Therefore, the term "radius" didn't make any sense and had no use to him. Measuring the diameter of an object (BH) whose space is breaking up into pieces that are in free-fall is not possible..but describing a "reduced circumference" is possible.

I think that you've misunderstood what I was trying to get at. (I probably wasn't sufficently clear.)

I'm trying to pin down what you believe to be correct about Newtonian gravity and how does this relate to the dynamics of a test particle.

How do you write the 4-vector equation that is the analogue of what happens with Newtonian mechanics, i.e. dp/dt = -GMm/r^2 ?

If you can write this down then it would make explicit exactly what you are assuming about the properties of the gravitational field.

[Zanket]

You "deem a relativistic rocket stationary at an arbitrary altitude of zero while its gantry 'free-falls' below it (relative to the rocket, not necessarily anything else).."

Now, are you assuming that Newton's Laws can be measured from within an accelerating frame of reference? Or, am I misreading something here?

Section 1 shows that the equations of motion for a uniform gravitational field are the relativistic rocket equations, in section 8. The relativistic rocket equations are not Newton’s; they are the current relativistic equations of SR. SR handles noninertial uniform acceleration, like that of a rocket where the crew feels a constant acceleration.

[Zanket]

I'm trying to pin down what you believe to be correct about Newtonian gravity and how does this relate to the dynamics of a test particle.

Are you trying to pin down what I believe, or what blueshift, who you quoted, believes? Just checking.

[Fortis]

Are you trying to pin down what I believe, or what blueshift, who you quoted, believes? Just checking.

Oops.
I meant what do you believe? Mathematically?