A Flaw of General Relativity, a Fix, etc.

[Zanket]

I meant what do you believe? Mathematically?

I don’t have any mathematical beliefs along those lines. I like to simplify things by abstracting them. Then I can ignore the underlying details.

My thinking in developing the new equation for escape velocity went like this: Section 1 shows that eq. 19 is the relativistic equation for free-fall velocity in a uniform gravitational field. Section 2 shows that free-fall velocity from rest at infinity to a given altitude is the escape velocity at that altitude. All that remains for me to do is to create an equation that integrates eq. 19 from infinity to a given altitude r, using some formula that returns the g for each r. Newton already did that for eq. 1 to derive eq. 4, and I find no fault with his inverse square law of gravity applied to r. There is a one-to-one relationship between Newtonian velocity and relativistic velocity, shown by eq. 3. Then I can implicitly integrate, piggybacking on Newton’s integration, by using eq. 3 to convert Newton’s equation for escape velocity into a relativistic equation, eq. 6.

At that point I have an equation for escape velocity that is the desired integration of eq. 19 from infinity to a given altitude r. At that point I’ve abstracted to a level where I don’t need to know more mathematical details about Newtonian gravity.

[Grey]

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.

A reasonable analogy. Now, to continue, I still maintain that to test a theory for consistency, you're obligated to follow its own rules, whatever they are. For example, you can't show that Newtonian Tuesday (my new name for this little theory, sort of like "casual friday") is inconsistent just by doing a calculation in special relativity and showing that it's different from what this theory would predict on a Tuesday. That just shows that special relativity given unlimited domain contradicts Newtonian Tuesday, but not that Newtonian Tuesday itself is inconsistent. And since Newtonian Tuesday told you up front that it was limiting special relativity, it should be pretty obvious that if you instead assume that it's universally valid that you'll be able to show a contradiction.

So we should agree that general relativity can limit the applicability of special relativity, and that if it does so, using special relativity outside that domain doesn't show anything about the consistency of general relativity. You might still disagree about whether general relativity does limit special relativity in such a way that your derivation is invalid, but we can discuss that (and the next few paragraphs of your post, which touch on this latter question) once we've reached agreement here.

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.

This is worded better, but I'll still disagree with the statement. Sometimes when you actually work out the math, the results can be surprising, and you wouldn't have realized that if you didn't actually work it out. For example, there's the mathematical figure known as "Gabriel's Horn" (the graph of y = 1/x for x >= 1, rotated about the x axis). If you work out the integrals for the surface area and volume of this figure, you'll discover the surprising result that the surface area is infinite, but the volume is finite. If I were to try to paint the surface, I would never have enough paint, but if I turn it upright and fill it with paint, a finite amount will suffice. I'll maintain that we can't be certain that there isn't some surprising result of this integration until we actually check to see. At the very least, we should at least set up the integral itself, so we can see explicitly what the integrand will be and what the bounds of integration are. I'll also maintain that I can make this claim and still remain a "reasonable person", though perhaps you might say that in such case I'm really just disagreeing that the logic used to set up the integral is valid.

[Zanket]

So we should agree that general relativity can limit the applicability of special relativity, and that if it does so, using special relativity outside that domain doesn't show anything about the consistency of general relativity.

I have agreed to that.

You might still disagree about whether general relativity does limit special relativity in such a way that your derivation is invalid, but we can discuss that (and the next few paragraphs of your post, which touch on this latter question) once we've reached agreement here.

If your argument will be that SR was modified by GR to restrict its domain from globally flat to locally flat, that won’t be compelling to me. For the sake of argument, I analyze these two versions of SR. Let’s call the original SR “SR1” and the modified version “SR2”. SR1 is a theory in thought only; there’s no way to experimentally test it, since spacetime is not globally flat. The relativistic rocket equations refer to SR2, the experimentally tested version. Then all instances of “special relativity” in the paper can be replaced with “SR2”, and the inconsistency shown in section 2 holds.

This is worded better, but I'll still disagree with the statement. Sometimes when you actually work out the math, the results can be surprising, and you wouldn't have realized that if you didn't actually work it out. For example, there's the mathematical figure known as "Gabriel's Horn" (the graph of y = 1/x for x >= 1, rotated about the x axis).

The scenario in section 2 is much simpler than that.

At the very least, we should at least set up the integral itself, so we can see explicitly what the integrand will be and what the bounds of integration are.

There are a lot of things in the paper that require a bit of intuition to visualize before they can be accepted. I won’t do the actual integration.

I'll also maintain that I can make this claim and still remain a "reasonable person", though perhaps you might say that in such case I'm really just disagreeing that the logic used to set up the integral is valid.

At some level of explanation, I have to leave it up to the reader to believe it or not. Section 2 is like one of Einstein’s thought experiments that have no accompanying rigorous mathematical proof (at least not in the source being read). Some readers won’t believe parts of my paper for lack of such proofs, which could be done, but which are out of scope for the paper.

[Grey]

I have agreed to that.

Excellent.

If your argument will be that SR was modified by GR to restrict its domain from globally flat to locally flat, that won’t be compelling to me.

Probably true by itself. If, however, I were to show that, in applying special relativity to a curved space, general relativity required one to follow certain mathematical rules which the derivation in section two violates, that would indeed mean that you have not shown an inconsistency in general relativity itself. Rather you would have shown only that general relativity is not consistent with an unmodified version of special relativity. If this were the case, general relativity by itself would be a perfectly consistent theory (or at least, your demonstration of its inconsistency would be flawed; there might yet be some undiscovered inconsistency). If one wished to accept general relativity, one would of course have to view special relativity alone as only a useful approximation in cases where gravity can be ignored, just as Newtonian mechanics are technically incorrect but remain useful in many cases. Correct?

The scenario in section 2 is much simpler than that.

I'm not certain that's the case. "Gabriel's Horn" is a pretty simple mathematical shape, and the integration is quite straightforward. I think the mathematics involved are probably simpler than that of special relativity used to analyze an accelerating reference frame. In any case, it was merely an example to show that the results of working out the math may have results that are not intuitive, even in a situation where it seems like the solution should be obvious.

There are a lot of things in the paper that require a bit of intuition to visualize before they can be accepted. I won’t do the actual integration.

Perhaps this is just poor wording on your part, but it sounds very much like your "rigorous proof" of a contradiction in general relativity relies on "intuition", and that you aren't willing or able to actually do the math that would give it a rigorous foundation.

At some level of explanation, I have to leave it up to the reader to believe it or not. Section 2 is like one of Einstein’s thought experiments that have no accompanying rigorous mathematical proof (at least not in the source being read). Some readers won’t believe parts of my paper for lack of such proofs, which could be done, but which are out of scope for the paper.

I'd maintain that a refutation of a theory, particularly one with a great deal of corroborating evidence, should in fact be quite rigorous. That is, if you expect to have your idea taken seriously, you'd actually need to demonstrate your case completely. Though there are many cases in which Einstein's thought experiments are reproduced with out the mathematical foundation behind them, in the published papers that foundation is always there. Touching on some comments in the thread about how open-minded the mainstream is, had Einstein only submitted the general idea of his theory, I very much doubt that he could have had it published in the journals of the day. Instead, he provided the mathematical details that showed how it worked, and as such made a convincing case.

[Zanket]

If, however, I were to show that, in applying special relativity to a curved space, general relativity required one to follow certain mathematical rules which the derivation in section two violates, that would indeed mean that you have not shown an inconsistency in general relativity itself.

Agreed.

If one wished to accept general relativity, one would of course have to view special relativity alone as only a useful approximation in cases where gravity can be ignored, just as Newtonian mechanics are technically incorrect but remain useful in many cases. Correct?

If by “special relativity alone” you mean the unmodified version of SR (hereafter “SR1”), and if GR modifies SR1, and if one wished to accept GR, then I say this is correct. But in that case I’d be able to show an inconsistency between GR and its equivalence principle, which shows that gravity can be ignored in a local frame in free fall, hence SR1 can be used in those frames. Integration lets results of SR1 in adjacent local frames be summed to calculate, for example, free-fall velocity in a nonuniform gravitational field. The result is correct; it is not technically incorrect. I think for there to be no way for me to show an inconsistency, GR would have to modify both SR1 and the equivalence principle. I am aware of no such modifications.

I'm not certain that's the case. "Gabriel's Horn" is a pretty simple mathematical shape, and the integration is quite straightforward. I think the mathematics involved are probably simpler than that of special relativity used to analyze an accelerating reference frame.

Section 2 shows that the math to determine a limit of free-fall velocity involves just two simple equations. A numerical integration can be done in about ten lines of code. I think most readers can just eyeball the equations to see the limit.

Perhaps this is just poor wording on your part, but it sounds very much like your "rigorous proof" of a contradiction in general relativity relies on "intuition", and that you aren't willing or able to actually do the math that would give it a rigorous foundation.

I didn’t say it was a rigorous proof.

I'd maintain that a refutation of a theory, particularly one with a great deal of corroborating evidence, should in fact be quite rigorous.

I disagree. For example, Einstein’s thought experiment to show relativity of simultaneity, involving the train passenger and the bystander, is convincing to me. A rigorous proof of that would be superfluous to me. My paper is written for the same type of audience.

That is, if you expect to have your idea taken seriously, you'd actually need to demonstrate your case completely. Though there are many cases in which Einstein's thought experiments are reproduced with out the mathematical foundation behind them, in the published papers that foundation is always there. Touching on some comments in the thread about how open-minded the mainstream is, had Einstein only submitted the general idea of his theory, I very much doubt that he could have had it published in the journals of the day. Instead, he provided the mathematical details that showed how it worked, and as such made a convincing case.

I tend to agree with your opinions here. Such is among the reasons why I do not expect my paper to be taken seriously by the scientific community. I won’t cater to their narrowness.

[Celestial Mechanic]

That is, if you expect to have your idea taken seriously, you'd actually need to demonstrate your case completely. Though there are many cases in which Einstein's thought experiments are reproduced with out the mathematical foundation behind them, in the published papers that foundation is always there. Touching on some comments in the thread about how open-minded the mainstream is, had Einstein only submitted the general idea of his theory, I very much doubt that he could have had it published in the journals of the day. Instead, he provided the mathematical details that showed how it worked, and as such made a convincing case.

I tend to agree with your opinions here. Such is among the reasons why I do not expect my paper to be taken seriously by the scientific community. I won’t cater to their narrowness.

You don't have to "cater to [our] narrowness". Just provide the necessary mathematical details and make a convincing case. Oh, that's right, you're going to let this theory die out eventually. Never mind!

[Grey]

If by “special relativity alone” you mean the unmodified version of SR (hereafter “SR1”), and if GR modifies SR1, and if one wished to accept GR, then I say this is correct. But in that case I’d be able to show an inconsistency between GR and its equivalence principle, which shows that gravity can be ignored in a local frame in free fall, hence SR1 can be used in those frames. Integration lets results of SR1 in adjacent local frames be summed to calculate, for example, free-fall velocity in a nonuniform gravitational field.

Unless, of course, general relativity specifies the manner in which you need to integrate over a curved space, and you aren't following those rules for integration. In which case, your derviation would be invalid.

The result is correct; it is not technically incorrect. I think for there to be no way for me to show an inconsistency, GR would have to modify both SR1 and the equivalence principle. I am aware of no such modifications.

Remember that special relativity and the equivalence principle alone do not lead to general relativity. In particular, as I pointed out earlier, it's possible to assume both of these, and yet come up with a theory that makes different predictions than general relativity in some circumstances. That should at least make it obvious that these two principles, without further guidelines to specify how they are to be applied, are not actually consistent! (Since there's more than one possible solution to some problems, depending on what further assumptions are made). So it should be clear that general relativity has to add further requirements in order to be a complete theory.

Section 2 shows that the math to determine a limit of free-fall velocity involves just two simple equations. A numerical integration can be done in about ten lines of code. I think most readers can just eyeball the equations to see the limit.

I don't think it's that simple, since you'll have to deal with the fact that your coordinate system isn't a simple flat one. But if you believe that it is, then why the stark unwillingness to do it?

[Fortis]

There are a lot of things in the paper that require a bit of intuition to visualize before they can be accepted. I won’t do the actual integration.

We're no longer asking for you to actually perform the integration. Grey was asking for you to state the integrand. If you provide us with that, then one of us could do the integral itself (assuming that it is tractable in closed form). This was why I was asking for you to mathematically explain what you believe to be correct about Newtonian gravity.

[Zanket]

Unless, of course, general relativity specifies the manner in which you need to integrate over a curved space, and you aren't following those rules for integration. In which case, your derviation would be invalid.

If GR does that, then it overrides calculus, at which point it might be a consistent theory, but it would also be a silly theory. Can you point out any such rule?

Remember that special relativity and the equivalence principle alone do not lead to general relativity.

Assuming that some other basics exist, like the gravitational constant and some formula that specifies how the strength of gravity changes with altitude, I see no reason why they should not, for Schwarzschild geometry.

In particular, as I pointed out earlier, it's possible to assume both of these, and yet come up with a theory that makes different predictions than general relativity in some circumstances.

It’s possible, but there’s no reason for a theory of Schwarzschild geometry to have additional assumptions or rules, as shown by section 6, Experimental Confirmation. And when there are none, four major problems of physics go away. So I would like to know from you what these additional assumptions or rules of GR are, for, while they might show that GR is consistent, they will also show me where it went astray. Absent evidence of such, I have to assume that the theory is inconsistent.

That should at least make it obvious that these two principles, without further guidelines to specify how they are to be applied, are not actually consistent! (Since there's more than one possible solution to some problems, depending on what further assumptions are made). So it should be clear that general relativity has to add further requirements in order to be a complete theory.

I disagree. The paper makes no further assumptions, and yet gets an exact answer for, say, the relativistic orbital precession of Mercury. You can always add further requirements to a system to get different answers than before. For example, I can change the result of an equation by requiring that the plus operator do a division operation.

I don't think it's that simple, since you'll have to deal with the fact that your coordinate system isn't a simple flat one. But if you believe that it is, then why the stark unwillingness to do it?

Because it would be overkill. Sections 2 and 3 both discuss escape velocity. Section 2 shows intuitively that its limit is c. Section 3 shows this mathematically, deriving the relativistic equation for escape velocity for the inverse square law of gravity. It is easy to see in section 3 that no matter what law of gravity Newton employed (inverse square, inverse cube, inverse square root, etc.), the conversion of his equation for escape velocity done by eq. 3 would guarantee that the limit of escape velocity is still c. I think that's enough proof of it.

[Zanket]

Grey was asking for you to state the integrand. If you provide us with that, then one of us could do the integral itself (assuming that it is tractable in closed form).

Section 2 shows how the integration would be set up. I think concluding that the limit is c is easy. Actually doing the integration, however, would be harder. For one thing, rather than use the addition operator (+) to add velocities, you’re using eq. 27, the special relativistic velocity addition formula. For another, the integration would need to figure out what r is at each step, so it can use that to get g. Bottom line is, I don’t know what the integrand would be. Nor will I work on that, because:

Section 2 is not about deriving an exact free-fall velocity; it’s about determining the limit of free-fall velocity, hence the limit of escape velocity. Section 2 is designed to be intuitively rather than mathematically convincing. So long as one can agree with section 2 that the free-fall velocity in a nonuniform gravitational field is given by eq. 27 for inputs from eq. 19 for each segment of the gravitational field deemed uniform, then it is easy to see that the limit of free-fall velocity, hence the limit of escape velocity, is c, because eq. 19 returns values always less than c and eq. 27 cannot return >= c for inputs less than c.

Section 3 does the integration implicitly, piggybacking on Newton’s integration, to derive the new equation for escape velocity. The velocity addition formula is incorporated into eq. 19, so using eq. 19 to convert Newton’s equation for escape velocity is the easiest way I know of to do an integration that employs the inverse square law.

See also the last paragraph I wrote to Grey immediately above.

[Grey]

If GR does that, then it overrides calculus, at which point it might be a consistent theory, but it would also be a silly theory. Can you point out any such rule?

Reading over my statement, I see that you might have misinterpreted my meaning. Calculus indeed applies just as well to curved space as to flat space. However, precisely what you are integrating (that is, exactly what the integrand would turn out to be) depends on the metric defining the space you are integrating over. So the rule in question is simply the statement in general relativity that space is curved.

Remember that special relativity and the equivalence principle alone do not lead to general relativity.

Assuming that some other basics exist, like the gravitational constant and some formula that specifies how the strength of gravity changes with altitude, I see no reason why they should not, for Schwarzschild geometry.

Yes, assuming that some other basics exist. For the record, the principles of general relativity are as follows:

General principle of relativity (that the laws of physics are the same for all observers; this extends the version used in special relativity to include accelerated observers)
Principle of general covariance (that the laws of physics must take the same form in all coordinate systems; this may seem to be a repeat of the previous condition, but is slightly different; in practice, it means that the equations of physics should be written as tensors)
Inertial motion follows geodesics (to be precise, timelike geodesics for massive particles, and null geodesics for massless particles)
Local Lorentz invariance (for an inertial observer, the laws of special relativity can be applied locally)
Spacetime is curved (this is the source of gravity)
Curvature is created by stress-energy (so the metric of a given region of spacetime can be determined from the stress-energy tensor)

As it turns out, although general relativity was motivated by the principle of equivalence, this is actually a consequence of the more fundamental general principle of relativity and that inertial motion follows geodesics, so this is actually not needed as a separate postulate.

So, the extra information you said you need is just about right. You need to specify how space is curved, which corresponds more or less to defining the strength of gravity at a given point. Now, it looks like you're assuming Newton's formula for gravity is correct, and that's not precisely true under general relativity. Instead, the curvature is defined by Einstein's Field Equations, which you've specifically stated that you aren't using. Your work is not following the rules of general relativity, and so does not say anything about the consistency thereof.

You might find it interesting to explore some of the alternate theories to general relativity, all of which obey the principle of equivalence and local special relativity, but which make different assumptions about the form gravity takes. These include Brans-Dicke, Rosen's bi-metric theory, Whitehead, Birkhoff, Nordstrom, and others. A little research might help you understand just how far the equivalence principle and special relativity alone get you, what options you have for further assumptions you can make about the nature of gravity, and how the resulting theory works out. Many of these theories match general relativity in a number of tests, though some have been ruled out by observation.

It’s possible, but there’s no reason for a theory of Schwarzschild geometry to have additional assumptions or rules, as shown by section 6, Experimental Confirmation. And when there are none, four major problems of physics go away. So I would like to know from you what these additional assumptions or rules of GR are, for, while they might show that GR is consistent, they will also show me where it went astray. Absent evidence of such, I have to assume that the theory is inconsistent.

So, there are additional assumptions required, and you've done it (add further assumptions), too. As for the four major problems of physics going away, you haven't demonstrated any of that, of course.

I disagree. The paper makes no further assumptions, and yet gets an exact answer for, say, the relativistic orbital precession of Mercury.

As I pointed out above, you do make further assumptions about the way space is curved (that is, the strength of gravity).

Section 2 shows intuitively that its limit is c.

So you believe that you've shown intuitively that there is a flaw in general relativity, but you're unable to actually prove it rigorously?

[Zanket]

Calculus indeed applies just as well to curved space as to flat space. However, precisely what you are integrating (that is, exactly what the integrand would turn out to be) depends on the metric defining the space you are integrating over. So the rule in question is simply the statement in general relativity that space is curved.

“Space is curved” isn’t a rule. Thinking more about this after my last post, I now think that GR must be inconsistent, rather than merely silly, if it has a rule that affects the integration. As you say, calculus applies to both flat and curved spacetime (uniform and nonuniform gravitational fields). The relativistic rocket equations, which are SR equations and the equations of motion for a uniform gravitational field, are derived by a process of integration. Then GR cannot change the rules of integration and be consistent with the unmodified version of SR to which the equivalence principle refers. The only way that GR can be consistent is if it modifies both SR and the equivalence principle. Which it doesn’t.

Yes, assuming that some other basics exist. For the record, the principles of general relativity are as follows:

I’ll keep your list as an excellent reference, thanks.

Now, it looks like you're assuming Newton's formula for gravity is correct, and that's not precisely true under general relativity. Instead, the curvature is defined by Einstein's Field Equations, which you've specifically stated that you aren't using. Your work is not following the rules of general relativity, and so does not say anything about the consistency thereof.

Section 2 of the paper does not assume that Newton’s inverse square law of gravity is valid; it says nothing about that. Section 2 shows that regardless of how the strength of gravity changes with altitude, SR and the equivalence principle will always show that the limit of escape velocity is c. You haven’t made your case here.

A little research might help you understand just how far the equivalence principle and special relativity alone get you, what options you have for further assumptions you can make about the nature of gravity, and how the resulting theory works out. Many of these theories match general relativity in a number of tests, though some have been ruled out by observation.

Section 2 shows that any consistent theory of gravity that incorporates SR and the equivalence principle will have a limit of c for escape velocity. AFAIK, that rules out all such theories of gravity except for the one described in the paper, which is not ruled out by observation, as shown in section 6.

So, there are additional assumptions required, and you've done it (add further assumptions), too.
...
As I pointed out above, you do make further assumptions about the way space is curved (that is, the strength of gravity).

By “additional assumptions or rules” I meant those beyond the “basics” I had added by that point in the post.

As for the four major problems of physics going away, you haven't demonstrated any of that, of course.

The paper does show that. The four major problems of physics that the paper obviates are:
- Singularities
- Flatness problem
- Horizon problem
- Lack of a non-ad hoc explanation for observed accelerating cosmic expansion

So you believe that you've shown intuitively that there is a flaw in general relativity, but you're unable to actually prove it rigorously?

Section 3 shows it mathematically. And section 2 makes more than just an intuitive argument; it makes a logical argument, with equations provided. It’s not that I’m unable to prove it rigorously, it’s that I won’t. If you are the type of person who needs a rigorous proof to convince you that Einstein’s thought experiment about relativity of simultaneity (say) is valid, then you are not my target audience.

[Fortis]

Section 3 does the integration implicitly, piggybacking on Newton’s integration, to derive the new equation for escape velocity. The velocity addition formula is incorporated into eq. 19, so using eq. 19 to convert Newton’s equation for escape velocity is the easiest way I know of to do an integration that employs the inverse square law.

See also the last paragraph I wrote to Grey immediately above.

If you are doing an integral (even implicitly) what is the integrand?

You are assuming something about Newtonian gravity to be correct within the framework of SR, and it isn't completely clear what that is. You say that you assume that the 1/r^2 law is correct for the force (which in Newtonian terms is a 3-vector), but how does that relate to, for example, the derivatives of the 4-momentum?

i.e. In Newtonian mechanics we can write

dp/dt = -GMmr/r^3 (1)

where p is the usual 3-momentum of Newtonian mechanics and r is the spatial position vector.

If we want an equation of motion that is consistent with SR then you need to be able to write it in terms of objects such as 4-vectors, e.g. the velocity 4-vector, or the momentum 4-vector. Obviously we can't just equate the derivative (with respect to the proper time) of the momentum 4-vector to the right hand side of equation 1 because the right hand side is a 3-vector.

For example, in relativistic electrodynamics, the proper time (tau) derivative of the momentum 4-vector, p of a charged particle with 4-velocity u is given by (following Misner et al)

dp/dtau = qF(u)

where F is the electromagnetic field tensor and q is the charge on the particle.

[Fortis]

Section 2 of the paper does not assume that Newton’s inverse square law of gravity is valid; it says nothing about that. Section 2 shows that regardless of how the strength of gravity changes with altitude, SR and the equivalence principle will always show that the limit of escape velocity is c. You haven’t made your case here.
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Section 2 shows that any consistent theory of gravity that incorporates SR and the equivalence principle will have a limit of c for escape velocity. AFAIK, that rules out all such theories of gravity except for the one described in the paper, which is not ruled out by observation, as shown in section 6.

You keep ignoring curvature. You can't.

GR involves curved spacetime. The reason that inertial mass and gravitational mass are equivalent is because all we have are bodies tracing geodesics in a curved spacetime.

It has been demonstrated that the curvature of the Scwarzschild solution is perfectly well behaved all the way through the event horizon. If this well behaved spacetime geometry leads to "escape velocities greater than c", then clearly curvature is important. (Even if the Schwarzschild metric turned out to be invalid.)

[Zanket]

If you are doing an integral (even implicitly) what is the integrand?

I don’t know. The power of abstraction is that it reduces what you need to know. I don’t need to know what Newton did to derive his equation for escape velocity. I needed only know how to validly convert his equation to derive the new equation.

You are assuming something about Newtonian gravity to be correct within the framework of SR, and it isn't completely clear what that is. You say that you assume that the 1/r^2 law is correct for the force (which in Newtonian terms is a 3-vector), but how does that relate to, for example, the derivatives of the 4-momentum?

I don’t know. I didn’t need to know that to derive the new equation for escape velocity.

If we want an equation of motion that is consistent with SR then you need to be able to write it in terms of objects such as 4-vectors, e.g. the velocity 4-vector, or the momentum 4-vector.

Section 3 shows that an equation of motion that is consistent with SR can be derived without needing to be able to do that. (That doesn't mean that it cannot be done.)

You keep ignoring curvature. You can't.

The clause you quoted, “regardless of how the strength of gravity changes with altitude”, accounts for curvature; it doesn’t ignore it.

GR involves curved spacetime.

The new theory of the paper also involves curved spacetime. The difference in the curvature respectively predicted by the Schwarzschild metric and the new metric is slight in weak gravity, as shown in fig. 2. The difference is so slight that the Schwarzschild metric and the new metric are equally confirmed by all experimental tests of Schwarzschild geometry, as shown in section 6.

It has been demonstrated that the curvature of the Scwarzschild solution is perfectly well behaved all the way through the event horizon.

Section 2 shows that the Schwarzschild metric is inconsistent with SR and the equivalence principle. It shows that event horizons do not exist in Schwarzschild geometry.

If this well behaved spacetime geometry leads to "escape velocities greater than c", then clearly curvature is important.

I agree that curvature is important in a theory of gravity. I doubt I could derive a metric that is experimentally confirmed while ignoring curvature.

[worzel]

Ok, way out of my depth here...

The clause you quoted, “regardless of how the strength of gravity changes with altitude”, accounts for curvature; it doesn’t ignore it.

Does it though? You say that you can piggy back off of Newton's integration without needing to do any yourself. But g changes with altitude in Newton's and there is no curvature there. I don't see how you can take an integration of an equation over flat space and just assume that applying gamma to one variable implicitly converts it into an integration over curved spacetime. And you seem rather reticent to justify this.

[Grey]

“Space is curved” isn’t a rule.

To be fair, I'd almost agree that "space is curved" isn't a rule. The full rule is that the curvature of space can be determined from the stress-energy tensor by using Einstein's Field Equations. It's just that in most treatments, the fact that spacetime is curved is considered important enough to mention as a separate point on its own, since it's a serious departure from previous theories.

Thinking more about this after my last post, I now think that GR must be inconsistent, rather than merely silly, if it has a rule that affects the integration. As you say, calculus applies to both flat and curved spacetime (uniform and nonuniform gravitational fields).

I hadn't thought this would be one of the little points that we'd have to go over really carefully. Say that I'm doing a path integration to find the length of a path. Now suppose that instead of my path being on a Euclidean plane, it happens to lie on the surface of the Earth. Will I, or will I not, have to take into account the curvature of the surface in performing my integration in order to find the correct result for the length of the path?

[Zanket]

You say that you can piggy back off of Newton's integration without needing to do any yourself. But g changes with altitude in Newton's and there is no curvature there.

There is curvature in Newtonian mechanics. What Newton called a nonuniform gravitational field, Einstein called curved spacetime. These are synonymous. And where a gravitational field is uniform, spacetime is flat.

I don't see how you can take an integration of an equation over flat space and just assume that applying gamma to one variable implicitly converts it into an integration over curved spacetime.

Eq. 3 converts Newtonian velocity to relativistic velocity regardless of curvature (i.e. regardless of the nonuniformity of the gravitational field). The only input to the equation is the Newtonian velocity k at a given moment. Then eq. 3 can be used to convert Newton’s equation for escape velocity, eq. 4, into a relativistic equation, eq. 6. When you input values into eq. 6, you get the same result as eq. 3 returns when you input into it the value k returned by eq. 5 for the values you input into eq. 6.

And you seem rather reticent to justify this.

I have justified the derivation of eq. 6 many times in many ways. To justify it, I need not answer every question put to me.

[Zanket]

The full rule is that the curvature of space can be determined from the stress-energy tensor by using Einstein's Field Equations.

I translate this to mean that in GR the curvature of spacetime must be determined from the stress-energy tensor by using Einstein's Field Equations.

It's just that in most treatments, the fact that spacetime is curved is considered important enough to mention as a separate point on its own, since it's a serious departure from previous theories.

The way I see it is, flat spacetime is curved spacetime with curvature deemed zero, which is synonymous with saying that a uniform gravitational field is a nonuniform gravitational field deemed uniform.

Say that I'm doing a path integration to find the length of a path. Now suppose that instead of my path being on a Euclidean plane, it happens to lie on the surface of the Earth. Will I, or will I not, have to take into account the curvature of the surface in performing my integration in order to find the correct result for the length of the path?

You will. But section 2 is not concerned with finding the correct result (i.e. a precise value) for the escape velocity. It is concerned with finding the limit of escape velocity. Section 2 shows that the escape velocity given by SR and the equivalence principle, while employing whatever curvature GR calls for at each altitude, has a limit of c. Then GR is inconsistent.

[Grey]

I translate this to mean that in GR the curvature of spacetime must be determined from the stress-energy tensor by using Einstein's Field Equations.

Of course. And we've agreed that, in order to show general relativity to be inconsistent, you'd actually have to follow the rules thereof and show that they lead to a contradiction.

You will. But section 2 is not concerned with finding the correct result (i.e. a precise value) for the escape velocity. It is concerned with finding the limit of escape velocity. Section 2 shows that the escape velocity given by SR and the equivalence principle, while employing whatever curvature GR calls for at each altitude, has a limit of c.

And we hopefully agree that in this instance, the fact that the integration will be different for the curved surface does not violate or modify the rules of calculus. And that working with a curved metric does not in itself cause a contradiction, even though it changes the way the result of the integration needs to be worked out.

What if it turns out that, taking the curvature into account, the formula for the escape velocity reaches an asymptotic limit of c at the Schwarzschild radius, and that the formula has mathematical problems if you try to use it at a smaller radius than that? Your intuition tells you that can't happen, but you haven't actually proven that this isn't the case, since you won't work it out. You don't even know what the curvature is, let alone how it affects the formula that you're integrating.

[Zanket]

And we've agreed that, in order to show general relativity to be inconsistent, you'd actually have to follow the rules thereof and show that they lead to a contradiction.

Section 2 follows all the relevant rules and still shows an inconsistency.

And we hopefully agree that in this instance, the fact that the integration will be different for the curved surface does not violate or modify the rules of calculus.

Agreed.

And that working with a curved metric does not in itself cause a contradiction, even though it changes the way the result of the integration needs to be worked out.

It changes the way the result of the integration needs to be worked out, but not in a way that prevents an inconsistency.

What if it turns out that, taking the curvature into account, the formula for the escape velocity reaches an asymptotic limit of c at the Schwarzschild radius, and that the formula has mathematical problems if you try to use it at a smaller radius than that?

Section 2 already takes every possible curvature into account, and shows that the limit of escape velocity is c.

Your intuition tells you that can't happen, ...

Not just my intuition tells me that; simple logic tells me that when I look at the equations. As I said above, so long as one can agree with section 2 that the free-fall velocity in a nonuniform gravitational field is given by eq. 27 for inputs from eq. 19 for each segment of the gravitational field deemed uniform, then it is easy to see that the limit of free-fall velocity, hence the limit of escape velocity, is c, because eq. 19 returns values always less than c and eq. 27 cannot return >= c for inputs less than c.

... but you haven't actually proven that this isn't the case, since you won't work it out.

Section 2 does prove its case, with equations. I do not claim that it is a rigorous proof. It sounds like your whole argument is boiling down to a claim that the paper doesn’t offer rigorous proof. About rigorous Mathworld says, “A proof or demonstration is said to be rigorous if the validity of each step and the connections between the steps is explicitly made clear in such a way that the result follows with certainty.” The paper is rigorous in that regard, and that’s sufficient for my target audience. But I say that the paper is not rigorous in the regard of other more stringent definitions of the word that I’ve seen.

You don't even know what the curvature is, let alone how it affects the formula that you're integrating.

I know that whatever the curvature is anywhere, the limit of escape velocity will still be c, because eq. 19 returns values always less than c for any value g input and eq. 27 cannot return >= c for inputs less than c.

[Edit to add:] Worzel pointed out to me that I’m misusing the terms “curvature” and “curved spacetime” in this thread. Typically I’ve been using these terms to mean “nonuniformity of the gravitational field” (i.e. how g changes with altitude) and “nonuniform gravitational field” respectively. In section 4, curved spacetime is predicted by a derivation that uses the new equation for escape velocity, so it isn’t valid for me to use that term when talking about a derivation of escape velocity, even one that just finds its limit, as section 2 does. It seems to me that you have also been misusing the term “curvature”, since you talk about it in terms of a rule of GR, despite that the Schwarzschild metric, a solution of GR, predicts the degree of curvature. Something predicted by a solution of GR cannot be a rule of it. Rather than change anything above, I’ll just use the terms correctly henceforth. But if you think you can use this info to strengthen your argument against me, please do.

[worzel]

There is curvature in Newtonian mechanics. What Newton called a nonuniform gravitational field, Einstein called curved spacetime. These are synonymous.

No they're not. Of course g is not uniform in Newtonian mechanics, but space is Euclidean. So according to Newton the angles of a triangle will always equal 180 degrees whereas in GR this is never true except in perfectly flat spacetime, i.e. in a universe without mass.

And where a gravitational field is uniform, spacetime is flat.

And where there is a body of any mass (including those as massive as black holes) spacetime is not flat. Then you cannot treat spacetime as flat and claim that a Newtonian integration accounts for the curvature if v is substitued for gamma*v, which is itself only SR in style, and as we know, SR only deals with flat spacetime.

Eq. 3 converts Newtonian velocity to relativistic velocity regardless of curvature (i.e. regardless of the nonuniformity of the gravitational field). The only input to the equation is the Newtonian velocity k at a given moment. Then eq. 3 can be used to convert Newton’s equation for escape velocity, eq. 4, into a relativistic equation, eq. 6. When you input values into eq. 6, you get the same result as eq. 3 returns when you input into it the value k returned by eq. 5 for the values you input into eq. 6.

Why does it follow that all v's in Newtonian mechanics can be replaced with gamma*v? It may work out to be that way in some cases, but it needs more justification than "I've seen it in this equation so it must be true in that one".

I have justified the derivation of eq. 6 many times in many ways. To justify it, I need not answer every question put to me.

You have admitted that your paper is not rigorous but intuitive. Then you have not justified anything, and my intuition tells me you are wrong. Not least because what I see as the probable fundamental flaw in your reasoning you handwave away first with claims that it is a side issue, then that your power of abstraction removes the necessity for you to be rigorous, and then with a claim that varying g in Newtonian mechanics is synomymous with curved spacetime.

For this last claim to be true, GR would make exactly the same predictions that Newtonian mechanics makes regarding free fall, orbits, etc.

[Zanket]

No they're not. Of course g is not uniform in Newtonian mechanics, but space is Euclidean. So according to Newton the angles of a triangle will always equal 180 degrees whereas in GR this is never true except in perfectly flat spacetime, i.e. in a universe without mass.

I agree. Although spacetime is curved in GR only where the gravitational field is nonuniform, the two are not synonymous. You’ve exposed a problem with the definitions in the paper, which I’ve also been using in this thread. Thanks, I appreciate that! I dropped the paper’s definition for “curved spacetime” (which wasn’t referenced), and changed the definition for “flat spacetime” to “the spacetime of a local frame”. I acknowledged your help as “Worzel (for exposing bad definitions)”. If that isn’t okay with you, please let me know.

And where there is a body of any mass (including those as massive as black holes) spacetime is not flat.

That may be strictly true, but spacetime is deemed flat and a gravitational field is deemed uniform in a local frame, a frame throughout which the tidal force is negligible. So the spacetime along the height of a building, say, may be deemed flat, concomitant with the gravitational field in the same region being deemed uniform, even though the spacetime is not perfectly flat, nor is the gravitational field perfectly uniform. The equivalence principle treats any infinitesimally small frame as a local frame. So in GR, the universe comprises infinitesimally small local frames within which the spacetime is flat in the limit.

Then you cannot treat spacetime as flat and claim that a Newtonian integration accounts for the curvature if v is substitued for gamma*v, which is itself only SR in style, and as we know, SR only deals with flat spacetime.

The equivalence principle lets me treat spacetime as flat locally, which is everywhere within infinitesimally small regions. SR applies in flat spacetime. Then SR can be integrated across any region of spacetime regardless of size or the nonuniformity of its gravitational field. While I can’t say that the derivation accounts for the curvature (that’s a mistake, as you pointed out), I can say that the derivation accounts for the nonuniformity of the field (i.e. it accounts for g changing with altitude), which is what I meant. Section 4 predicts curved spacetime, by using the new equation for escape velocity to derive the new equation for gravitational distortion.

Why does it follow that all v's in Newtonian mechanics can be replaced with gamma*v? It may work out to be that way in some cases, but it needs more justification than "I've seen it in this equation so it must be true in that one".

Eq. 3, the equation that shows that “all v's in Newtonian mechanics can be replaced with gamma*v” (or at least you figured that out), takes only k (Newtonian velocity) as its input. Then there would need to be justification to show why the equation would not work in all cases.

You have admitted that your paper is not rigorous but intuitive.

I said that section 2 is designed to make an intuitive argument. I did not say that the whole paper makes only an intuitive argument. Even section 2 employs equations to make its point. I say that the paper is not rigorous, because, although generally accepted proofs are seldom formally rigorous, I am confident that “rigorous” in its formal sense would be argued against me if I were to say that my paper is a rigorous proof. In other words, “rigorous” is a term that anybody can use to claim that a paper is inadequate, save against the rare formally rigorous proofs. I claim about the paper that “the validity of each step and the connections between the steps is explicitly made clear in such a way that the result follows with certainty”, which is the definition of rigorous at Mathworld.

Then you have not justified anything, and my intuition tells me you are wrong.

You are of course free to be unconvinced. But the paper justifies all its points (even a purely intuitive argument can do that), and you haven’t shown how I am wrong other than for the definitions I was misusing (more on that below).

Not least because what I see as the probable fundamental flaw in your reasoning you handwave away first with claims that it is a side issue, ...

If you don’t think I’ve answered it in this post, and still think that it is not a side issue, then please repeat it. I’m trying to be open-minded here, while also endeavoring to stay on topic. If I think an issue is a side issue that can’t lead to a flaw of the paper being revealed, I’ll say so. But you are free to argue against my opinion.

... then that your power of abstraction removes the necessity for you to be rigorous, ...

I said that the power of abstraction is that it reduces what you need to know. I stand by that comment.

... and then with a claim that varying g in Newtonian mechanics is synomymous with curved spacetime.

For this last claim to be true, GR would make exactly the same predictions that Newtonian mechanics makes regarding free fall, orbits, etc.

I agree with you that they are not synonymous, and thanks again for pointing that out. The paper isn’t materially affected by that mistake; I needed change only its definitions. Section 4 predicts curved spacetime. Where I’ve misused “curved spacetime” or “curvature” in this thread, my arguments can hold valid after restating them to use the terminology correctly (such as by replacing “curvature” with “nonuniformity of the gravitational field”, which is what I typically meant).

[Fortis]

Section 2 already takes every possible curvature into account, and shows that the limit of escape velocity is c.

And yet you claim that this isn't true for the curvature of the Schwarzschild metric?

[Zanket]

And yet you claim that this isn't true for the curvature of the Schwarzschild metric?

First let me restate the comment you quoted, as Worzel pointed out that I’m using the term “curvature” incorrectly. I meant that section 2 shows that the limit of escape velocity is c for every possible law of gravity (i.e. a law that specifies how the strength of gravity changes with altitude, like the inverse square law).

As to your question, GR must specify only one law of gravity—whatever it is—for Schwarzschild geometry. Otherwise the Schwarzschild metric would return multiple answers for a given set of inputs.

[Grey]

It changes the way the result of the integration needs to be worked out, but not in a way that prevents an inconsistency.

You're missing the point of this claim. Earlier you seemed to be claiming that, by changing the way the integration needs to be worked out, it was somehow rewriting calculus, and therefore "silly". I was trying to make it clear that this did not change the rules of calculus, and was not, in itself, the source of any contradiction. From your previous statement, I see that you agree.

Section 2 already takes every possible curvature into account, and shows that the limit of escape velocity is c.

This is obviously false. For example, if the curvature were to become infinite at any point along the path, integrating your function along that path would simply not be possible. Since the curvature can be fairly complex mathematically, there may be non-infinite curvature which also causes a problem in performing the integration. Since you haven't even examined the curvature involved, you certainly can't claim that you've taken every possible curvature into account.

Section 2 does prove its case, with equations. I do not claim that it is a rigorous proof. It sounds like your whole argument is boiling down to a claim that the paper doesn’t offer rigorous proof. About rigorous Mathworld says, “A proof or demonstration is said to be rigorous if the validity of each step and the connections between the steps is explicitly made clear in such a way that the result follows with certainty.” The paper is rigorous in that regard, and that’s sufficient for my target audience. But I say that the paper is not rigorous in the regard of other more stringent definitions of the word that I’ve seen.

Except that you haven't show that integrating over the space in question is even valid, since you haven't worked out what it is you're integrating. That can hardly be said to mean that "the connections between the steps is explicitly made clear in such a way that the result follows with certainty". A particularly crucial step is not explicit at all.

[Edit to add:] Worzel pointed out to me that I’m misusing the terms “curvature” and “curved spacetime” in this thread. Typically I’ve been using these terms to mean “nonuniformity of the gravitational field” (i.e. how g changes with altitude) and “nonuniform gravitational field” respectively. In section 4, curved spacetime is predicted by a derivation that uses the new equation for escape velocity, so it isn’t valid for me to use that term when talking about a derivation of escape velocity, even one that just finds its limit, as section 2 does. It seems to me that you have also been misusing the term “curvature”, since you talk about it in terms of a rule of GR, despite that the Schwarzschild metric, a solution of GR, predicts the degree of curvature. Something predicted by a solution of GR cannot be a rule of it. Rather than change anything above, I’ll just use the terms correctly henceforth. But if you think you can use this info to strengthen your argument against me, please do.

I've been using the term curvature as used within general relativity, to talk about the metric used to describe spacetime, as opposed to it being Euclidean or flat. General relativity doesn't define what the curvature of space is, rather, it defines how to compute the curvature for any given region of space. So, the rule of general relativity is that spacetime is curved, and that the degree and nature of the curvature can be worked out if you know the value for the stress-energy tensor, using Einstein's field equations. That's what I said back here. I hadn't corrected your usage because I didn't think it was crucial, and thought you were just being imprecise (I really am trying to avoid nitpicking things that don't actually matter), but perhaps the fact that you didn't understand what curvature means and how it works in general relativity is part of the problem.

[worzel]

The equivalence principle lets me treat spacetime as flat locally, which is everywhere within infinitesimally small regions. SR applies in flat spacetime. Then SR can be integrated across any region of spacetime regardless of size or the nonuniformity of its gravitational field. While I can’t say that the derivation accounts for the curvature (that’s a mistake, as you pointed out), I can say that the derivation accounts for the nonuniformity of the field (i.e. it accounts for g changing with altitude), which is what I meant. Section 4 predicts curved spacetime, by using the new equation for escape velocity to derive the new equation for gravitational distortion.

I'm going to risk getting way out of my depth now, hopefully Grey will pick me up on my mistakes.

It is probably true that SR can be integrated across any region of curved spacetime, I believe GR dictates how this is done. But integration across curved spacetime is not done using simple vector calculus over rectangular coordinates. What you're doing is modifying a Newton equation and then integrating it over flat space which implicitly assumes that space is Euclidean on the large scale. Or to put it another way, that a large rectangle's area is its length times its width. Or yet another, that the angles of a triangle add up to 180 degrees. These last two are not quite true in curved spacetime and so integration becomes a lot more complicated (tensors, Reimannian manifolds, etc. - not that I understand them).

Your claim that an equation derived from a Newtonian integration in flat space implicity integrates over curved spacetime just seems totally incredulous to me I'm afraid. So much for an intuitive proof

I hope I haven't made too many blunders in that lot!

[Zanket]

Earlier you seemed to be claiming that, by changing the way the integration needs to be worked out, it was somehow rewriting calculus, and therefore "silly". I was trying to make it clear that this did not change the rules of calculus, and was not, in itself, the source of any contradiction.

OK. I still say that if GR specifies the manner in which you need to integrate over a curved space, then it is inconsistent. More on this below.

This is obviously false. For example, if the curvature were to become infinite at any point along the path, integrating your function along that path would simply not be possible. Since the curvature can be fairly complex mathematically, there may be non-infinite curvature which also causes a problem in performing the integration. Since you haven't even examined the curvature involved, you certainly can't claim that you've taken every possible curvature into account.

I misspoke when said I “curvature”. I meant that section 2 already takes every possible law of gravity (i.e. a law that states how the strength of gravity changes with altitude) into account, and shows that the limit of escape velocity for every possible law is c. When I say “every possible law”, I am excluding silly laws, like those with infinite gravity. Every possible law is accounted for, because eq. 19 returns a value less than c for any value for g at a given altitude for the particle, so eq. 27 returns a value less than c for escape velocity regardless of the law of gravity. More on curvature below.

Except that you haven't show that integrating over the space in question is even valid, since you haven't worked out what it is you're integrating.

Section 2 explicitly states that you’re using eq. 27 to sum results of eq. 19 for changing values of g. That’s integration.

General relativity doesn't define what the curvature of space is, rather, it defines how to compute the curvature for any given region of space. So, the rule of general relativity is that spacetime is curved, and that the degree and nature of the curvature can be worked out if you know the value for the stress-energy tensor, using Einstein's field equations.

I translate and simplify your comment to: the rule of GR is that, when deriving an equation of motion, something about the curvature must be worked out, and that something must be incorporated into the derivation. And I presume that you think that if I followed this rule, then I would see that GR is consistent. Do I have that right?

I say that no such rule can keep GR consistent. The reason is simple: SR does not have such a rule. The equivalence principle lets me use SR in infinitesimal regions of spacetime. Then I can use calculus to integrate SR’s equations of motion to derive an equation of motion for any larger region of spacetime. That is, unless GR modifies both SR and the equivalence principle—but it doesn’t.

A theory that requires me to follow a rule in any region except infinitesimal regions is inherently inconsistent, because any region comprises infinitesimal regions. Then in any region the rule both needs to be followed and doesn’t need to be followed. I challenge you to create an example that disproves that, without resorting to the usage of infinities or other silliness. If you are right, then you should be able to show it by, say, modifying your example of doing integration to find the length of a path.

[Edit to add:] In this post to Worzel below, I put the argument about GR's inconsistency the simplest terms I can think of, starting at "Take the simplest case". Please see if you can dispute it.

That's what I said back here. I hadn't corrected your usage because I didn't think it was crucial, and thought you were just being imprecise (I really am trying to avoid nitpicking things that don't actually matter), but perhaps the fact that you didn't understand what curvature means and how it works in general relativity is part of the problem.

My understanding of what the term “curved spacetime” means was flawed. But what it means is what section 4 predicts. I understand the concept fine. I reject the notion that I need to understand how curvature works in GR to see how GR is consistent. But if you can come up with an example as described above, you might convince me.

[Fortis]

First let me restate the comment you quoted, as Worzel pointed out that I’m using the term “curvature” incorrectly. I meant that section 2 shows that the limit of escape velocity is c for every possible law of gravity (i.e. a law that specifies how the strength of gravity changes with altitude, like the inverse square law).

As to your question, GR must specify only one law of gravity—whatever it is—for Schwarzschild geometry. Otherwise the Schwarzschild metric would return multiple answers for a given set of inputs.

In GR the effects that we attribute to gravity are really due to the underlying geometry of spacetime, i.e. bodies move in the curved spacetime equivalents of straight lines, called geodesics.

If you do not include curved spacetimes in your argument about "escape velocities", then you aren't considering every possible "law of gravity".

[worzel]

Exactly, Fortis. By "all laws of gravity" Zanket means "all vector fields for g in rectilinear coordinates".

A theory that requires me to follow a rule in any region except infinitesimal regions is inherently inconsistent, because any region comprises infinitesimal regions. Then in any region the rule both needs to be followed and doesn’t need to be followed. I challenge you to create an example that disproves that, without resorting to the usage of infinities or other silliness. If you are right, then you should be able to show it by, say, modifying your example of doing integration to find the length of a path.

How about using the fact that infinitesimal regions of the surface of a sphere or a cylinder are flat to integrate an infinte number of rectangles to give the surface area. The way I understand it is that doing this requires a different form of the general integral for the surface in question. You can use rectangular coords to provide approximate answers over small regions, but not over the whole surface. In GR, spacetime can be viewed as a hyper-surface in some higher dimensionality, and its curvature is not quite as simple as a sphere or cylinder, it curves this way and that depending on the distribution of mass. And its curvature will affect integrals performed on that surface.