A Flaw of General Relativity, a New Metric and Cosmological Implications [Technical]

Zanket

Son Goku wrote:

You're missing the point, a set of field equations are useless without an action.
...
What are your new dynamics?

You’re missing the point that the new metric is all that is needed to predict the results of experimental tests; i.e. the dynamics. Experimental tests of general relativity do not directly test the field equations or the Einstein-Hilbert action. Instead they test the metrics. I gave a reference for that by T&W above.

My paper does not derive its metric from field equations or from an action principle. There is no scientific requirement that it do so.

Son Goku

Zanket wrote:

You’re missing the point that the new metric is all that is needed to predict the results of experimental tests; i.e. the dynamics. Experimental tests of general relativity do not directly test the field equations or the Einstein-Hilbert action. Instead they test the metrics. I gave a reference for that by T&W above.

My paper does not derive its metric from field equations or from an action principle. There is no scientific requirement that it do so.

Yes, I know. I'm not saying you have to derive your equation from an action.
Obviously since you are taking away General Relativity you are also taking away it's dynamical content.
What generates the dynamics in your theory?
What is your new rule for the response of matter to curvature?

The metric on its own is useless, even the Schwarschild metric is.

For instance what use is a potential in Newtonian Mechanics without F=ma?

planck2

yes you can test the experiment from the metric, but one can do it from the action too. However, doing so from the metric has a very limited power, doing so from the action is more powerful.

planck2

there is no similarity between the two metrics at large r, why do you belive there is. show me that there is with refering to the graph. Do it with plain mathematics taking the limit as r goes to infinity

planck2

and another point Einstein came up with his field equations before Schwarzscild came up his vacuum solution.
So you must have a set of field equations which this metric satisfies as a vacuum solution, which can also be derived from an action and do as I described above.

Zanket

planck2 wrote:

no i don't disagree with Taylor and Wheeler, the sign convention is +---, Landau and Lif****z. I have it right here in front of me, along with Gravitation by MTW and General Relativity by Wald.

the point is i know how to get your new metric and it doesn't predict anything.

clearly they don't converge as r goes to infinity.

I can’t explain that. From the reference by Taylor and Wheeler above:



This shows that the sign usage is the same as the new metric. It shows that the only difference between the metrics is the difference between eq. 8 and eq. 9 in the paper, the curves of which fig. 3 shows converge as r increases.

you are missing the fundamental point which is that the EH action allows one to predict the equations of motion for the particle which can be tested against experiment. Therefore I need an action to determine the theoretical equations of motion so as to test them against experiment. You don't provide one. So how am I to believe you?

The metric is the equation of motion. It is all that is needed to predict the motion. One does not need anything that sources the metric. See my reply to Son Goku above. And note that the EH action is not employed in Einstein’s original valid version of GR, showing that it is superfluous.

the increment of solid angle is not used any where in the Schwarzschild metric, the metric for the surface of 2-sphere of radius r is though.

That contradicts T&W and this site:

From Schwarzschild geometry:

Schwarzschild's geometry is described by the metric (in units where the speed of light is one, c = 1)

ds^2 = –((1 – (rs / r)) * dt^2) + ((1 – (rs / r))^-1 * dr^2) + (r^2 * do^2)

[Reformatted for clarity. rs = the Schwarzschild radius.]

The quantity ds denotes the invariant spacetime interval, an absolute measure of the distance between two events in space and time, t is a 'universal' time coordinate, r is the circumferential radius, defined so that the circumference of a sphere at radius r is 2 pi r, and do is an interval of spherical solid angle. (boldface mine)

The Schwarzschild metric from the site above matches the spacelike metric by T&W.

planck2

Zanket wrote:

I can’t explain that. From the reference by Taylor and Wheeler above:



This shows that the sign usage is the same as the new metric. It shows that the only difference between the metrics is the difference between eq. 8 and eq. 9 in the paper, the curves of which fig. 3 shows converge as r increases.

Zanket wrote:

to get those signs T&W use the LL convention (+---) and set ds^2 to be > 0 for timelike and < 0 for spacelike, they also set theta equal to pi/2, so particles move in the equatorial plane.

Zanket wrote:

The metric is the equation of motion. It is all that is needed to predict the motion. One does not need anything that sources the metric.

Zanket wrote:

the metric is not an equation of motion, it describes how the spacetime is curved and how certain geodisics behave. The geodesics or equations of motion can be dervived using the Euler-Lagrange formalism taking the line element as the lagrangian. Or it can be done another way using an action principle which gives the field equations

Zanket wrote:

That contradicts T&W and this site:



The Schwarzschild metric from the site above matches the spacelike metric by T&W.

I am afraid it does not they do not refer to "do" only dphi which is an increment of angle on a circle

planck2

i know how you got the difference in the two metrics, but the point is this if your metric is in fact asymptotically flat then how come the scalar curvature is non zero

Zanket

planck2 wrote:

your theory doesn't need to predict BEC's or any of QM's predictions because neither does Einstein's.

Nice to have confirmation on that, thanks.

yes you can test the experiment from the metric, but one can do it from the action too. However, doing so from the metric has a very limited power, doing so from the action is more powerful.

I don’t deny that “doing so from the action is more powerful”, but I don’t see how the metric has a “very limited power”, given that T&W say:

From T&W; google for it:

Further investigation has shown that the Schwarzschild metric gives a complete description of spacetime external to a spherically symmetric, nonspinning, uncharged massive body (and everywhere around a black hole but at its central crunch point). Every (nonquantum) feature of spacetime around this kind of black hole is described or implied by the Schwarzschild metric. This one expression tells it all! (italics theirs)

Then nothing else is needed to make predictions of GR for Schwarzschild geometry. That is powerful indeed. Elsewhere they say:

From T&W; google for it:

The metric helps to answer every scientific question about (nonquantum) features of spacetime surrounding a black hole, every possible question about trajectories of light and satellites around the black hole as well around more familiar centers of attraction such as Earth and Sun.

It’s extra clear from this that my theory needs no more than a metric.

I am afraid it does not they do not refer to "do" only dphi which is an increment of angle on a circle

I can’t refute this. I don’t know why I chose to use “do”, as in the link above, instead of dphi, used by T&W, my main reference. Either way, you have forced me to change the paper. Congratulations, and thank you. I have changed “do” to dphi throughout the paper, including the meaning (which is now an “increment of an angle in a plane through a center of gravitational attraction”), and re-checked the meanings of all the symbols in my metric to verify that they are now all equivalent to the symbols used in the Schwarzschild metric by T&W in the image above. Now I can support my claim that the Schwarzschild metric and my metric differ only by the difference between eqs. 8 and 9 in the paper.

I would like to acknowledge you in the paper. Is that okay with you?

there is no similarity between the two metrics at large r, why do you belive there is. show me that there is with refering to the graph. Do it with plain mathematics taking the limit as r goes to infinity

Now that I use dphi, the Schwarzschild metric in the image above and the new metric in the paper differ only by the difference between eqs. 8 and 9 in the paper (that is how the new metric is derived, by swapping eq. 8 for eq. 9), a difference shown by fig. 3, which shows that the curves converge as r increases.

More mathematically, the right-hand side of eq. 9 is equivalent to sqrt(1 - (R / (r + R))). Comparing that the right-hand side of eq. 8, sqrt(1 - (R / r)), it is easy to see that the effect of R in the denominator of eq. 9, the only difference between the two equations, diminishes as r increases. Then the curves given by the equations must converge as r increases.

An equivalent explanation is given in section 6.

and another point Einstein came up with his field equations before Schwarzscild came up his vacuum solution.
So you must have a set of field equations which this metric satisfies as a vacuum solution, which can also be derived from an action and do as I described above.

That’s a non sequitur. The order of those events does not show a scientific requirement for field equations. I already have reader comments in the paper for this:

Reader: A metric must be derived from field equations.

Author: The scientific method lets any type of equation be presented without derivation. Then no particular method of derivation is required.

Reader: Without new field equations, your theory is worthless.

Author: Field equations are not required. A metric makes falsifiable predictions.

to get those signs T&W use the LL convention (+---) and set ds^2 to be > 0 for timelike and < 0 for spacelike, they also set theta equal to pi/2, so particles move in the equatorial plane.

OK, thanks for pointing that out. It might not be good for my paper to be limited to a plane. I’ll think about changing that, or at least clarifying it.

the metric is not an equation of motion, it describes how the spacetime is curved and how certain geodisics behave. The geodesics or equations of motion can be dervived using the Euler-Lagrange formalism taking the metric as the action. Or it can be done another way using an action principle which gives the field equations

Given that it answers “every possible question about trajectories of light and satellites around the black hole as well around more familiar centers of attraction such as Earth and Sun”, it sure seems to meet the definition of an equation of motion, given by Wikipedia as “equations that describe the behavior of a system (e.g., the motion of a particle under an influence of a force) as a function of time”. How do you explain that it doesn't meet that definition?

i know how you got the difference in the two metrics, but the point is this if your metric is in fact asymptotically flat then how come the scalar curvature is non zero

Do you still think there is a problem, now that I changed the paper? If yes, then how can the Schwarzschild metric be asymptotically flat yet mine not, when the difference between the two metrics approaches nothing as r goes to infinity?

Zanket

Son Goku wrote:

Yes, I know. I'm not saying you have to derive your equation from an action.
Obviously since you are taking away General Relativity you are also taking away it's dynamical content.
What generates the dynamics in your theory?
What is your new rule for the response of matter to curvature?

The metric on its own is useless, even the Schwarschild metric is.

For instance what use is a potential in Newtonian Mechanics without F=ma?

Given that the Schwarzschild metric and the new metric in the paper make falsifiable predictions on their own, then they cannot be useless, for they serve a prime focus of physics, predicting observations.

A theory of gravity need not show what “generates the dynamics” or show a “rule for the response of matter to curvature”. It is enough to predict observations.

Suppose beings on some other planet have a metric that approximates the Schwarzschild metric for the tests done so far on Earth, plus it accurately predicts phenomena that GR fails to predict (like stars that accelerate away), but it doesn’t have a “rule for the response of matter to curvature”. Is it useless?

Suppose someone creates an equation that consistently and accurately predicts the path of hurricanes, but it has no basis other than it just “felt right” to its author. Should science reject it? Can science even invalidate it?

Professor_Fink

Zanket wrote:

In the new cosmological model proposed in section 7, spacetime always existed; it never arose. There is no requirement that a cosmological model have a beginning of spacetime. There is no direct observational evidence that our universe had a beginning.

Zanket, a couple of points. 1) Stars form. 2) I can't see how this statement can agree with cosmic microwave background anisotropy measurements.

For the spacetime in question to be possible, it cannot be a fixed point of your theory, which it is. So even if it did predict exactly the same results as GR outside the schwarzchild radius (it doesn't), it would still not be able to fit experimental data since we know stars have a finite life.

Zanket wrote:

If the mass is zero, neither r nor R applies. These apply to only material objects.

Zanket, the mass is not zero when you produce a pair which later anihilate nor is the energy. These are not conserved in vacuum fluctuations.

Zanket wrote:

The only place that GR is incompatible with quantum mechanics is at a singularity.

Actually thats not strictly true. If the curvature of space is high compared to the wavelength of the particles involved you cannot do QFT on a curved background.

Zanket wrote:

Then my theory need not predict BECs or the observations that quantum mechanics predicts.

Your theory is not GR. I'm not saying that it should predict them, rather that you rule them out, which is counter to experiments. Well, at least if it is compatible with QM, as you have claimed. GR isn't completely compatible, but I thought you were claiming to have solved that problem.

Actually if no particle can be exchanged between two masses, then it would seem that you have ruled out at least 3 forces which rely on exchange particles 1. Electromagnetism 2. The weak interaction 3. The strong interaction.

So the very fact that you exist to propose the theory disproves your theory.

planck2

Zanket wrote:

I would like to acknowledge you in the paper. Is that okay with you?

No you may not

planck2

Nice to have confirmation on that, thanks.
I don’t deny that “doing so from the action is more powerful”, but I don’t see how the metric has a “very limited power”, given that T&W say:
Then nothing else is needed to make predictions of GR for Schwarzschild geometry. That is powerful indeed. Elsewhere they say:
It’s extra clear from this that my theory needs no more than a metric.

The metric is the fundamental requirement, but you need other things in order to get the equations of motion such as the Euler-Lagrange formalism or the Hamilton-Jacobi method. See any standard graduate text on classical mechanics

Now that I use dphi, the Schwarzschild metric in the image above and the new metric in the paper differ only by the difference between eqs. 8 and 9 in the paper (that is how the new metric is derived, by swapping eq. 8 for eq. 9), a difference shown by fig. 3, which shows that the curves converge as r increases.
More mathematically, the right-hand side of eq. 9 is equivalent to sqrt(1 - (R / (r + R))). Comparing that the right-hand side of eq. 8, sqrt(1 - (R / r)), it is easy to see that the effect of R in the denominator of eq. 9, the only difference between the two equations, diminishes as r increases. Then the curves given by the equations must converge as r increases.

but they don't converge and never will

That’s a non sequitur. The order of those events does not show a scientific requirement for field equations. I already have reader comments in the paper for this:

Reader: A metric must be derived from field equations.

Author: The scientific method lets any type of equation be presented without derivation. Then no particular method of derivation is required.

Reader: Without new field equations, your theory is worthless.

Author: Field equations are not required. A metric makes falsifiable predictions.

but then you are just guessing, the point is that Schwarzschild had a set of field equations (which related the curvature to the presence of matter) to work from, he used them to find a possible solution for the spacetime outside a star and this happened to agree with experiment.

Come up with new field equations and solutions to it such as your metric

Given that it answers “every possible question about trajectories of light and satellites around the black hole as well around more familiar centers of attraction such as Earth and Sun”, it sure seems to meet the definition of an equation of motion, given by Wikipedia as “equations that describe the behavior of a system (e.g., the motion of a particle under an influence of a force) as a function of time”. How do you explain that it doesn't meet that definition?

See my answer above.

Do you still think there is a problem, now that I changed the paper? If yes, then how can the Schwarzschild metric be asymptotically flat yet mine not, when the difference between the two metrics approaches nothing as r goes to infinity?

Yes I do, because they don't agree at infinity. Simple.

And Professor Fink is correct on the matter of QFT's on Curved Spacetime

planck2

and if you even dare credit me i'll sue your sorry ass all the way to seattle and back again

Son Goku

Zanket wrote:

Suppose beings on some other planet have a metric that approximates the Schwarzschild metric for the tests done so far on Earth, plus it accurately predicts phenomena that GR fails to predict (like stars that accelerate away), but it doesn’t have a “rule for the response of matter to curvature”. Is it useless?

Yes. Theories cannot have no equations of motion. I don't think you even understand what I'm saying.

How does stuff move in your metric? What is the rule for generating the equations of motion. It can't be Einstein's geodesic rule as you are rejecting GR, so how do the particles move in your theory?

i.e. Given the metric how do you obtain the trajectory of a particle within it, for some initial condition.

The fact that you don't understand this is dreadful.

Zanket

Professor_Fink wrote:

Zanket, a couple of points. 1) Stars form. 2) I can't see how this statement can agree with cosmic microwave background anisotropy measurements.

For the spacetime in question to be possible, it cannot be a fixed point of your theory, which it is. So even if it did predict exactly the same results as GR outside the schwarzchild radius (it doesn't), it would still not be able to fit experimental data since we know stars have a finite life.

What “spacetime in question”? A singularity? The new metric is experimentally confirmed by tests of the Schwarzschild metric; section 6 shows that, and you haven’t refuted it. And so what if stars have a finite life? My paper doesn’t suggest otherwise. Stars can be born and die in a universe that had no beginning. Even GR supports cosmological models in which the universe has no beginning. Is there a larger point of yours that I’m missing? Can you elaborate?

Zanket, the mass is not zero when you produce a pair which later anihilate nor is the energy. These are not conserved in vacuum fluctuations.

That’s outside the scope of my theory, just as it is for GR. It no more affects my theory than it affects GR. It invalidates neither theory.

Actually thats not strictly true. If the curvature of space is high compared to the wavelength of the particles involved you cannot do QFT on a curved background.

Offhand, I can’t refute that, and it makes some sense to me. You seem to be saying that GR is incompatible with QM not just at a singularity, but also in its immediate vicinity.

Your theory is not GR. I'm not saying that it should predict them, rather that you rule them out, which is counter to experiments. Well, at least if it is compatible with QM, as you have claimed. GR isn't completely compatible, but I thought you were claiming to have solved that problem.

I don’t rule them out. The new metric is compatible with QM because, unlike the Schwarzschild metric, it doesn’t demand a minimum radius for a body that is incompatible with QM.

Actually if no particle can be exchanged between two masses, then it would seem that you have ruled out at least 3 forces which rely on exchange particles 1. Electromagnetism 2. The weak interaction 3. The strong interaction.

So the very fact that you exist to propose the theory disproves your theory.

That my theory precludes singularities does not rule out any prediction of QM. You suggest that QM requires singularities. But that would be inconsistent, for it is singularities that create the incompatibility between GR and QM.

Professor_Fink

Zanket wrote:

What “spacetime in question”? A singularity? The new metric is experimentally confirmed by tests of the Schwarzschild metric; section 6 shows that, and you haven’t refuted it. And so what if stars have a finite life? My paper doesn’t suggest otherwise. Stars can be born and die in a universe that had no beginning. Even GR supports cosmological models in which the universe has no beginning. Is there a larger point of yours that I’m missing? Can you elaborate?

Yes, you are completely missing the point. The spacetime I am talking about is the manifold described by your metric.

The schwarzchild solution is not a cosmological model, so I'm not sure where all this talk of cosmology fits in (but for what it's worth the CMB is extremely compelling evidence in favour of a big bang).

I am talking about stars, blackholes, and anything else described by the schwarzchild solution. If you say that nothing can ever reach r=0 in any frame how can such objects form? Must they all be hollow?

Zanket wrote:

That’s outside the scope of my theory, just as it is for GR. It no more affects my theory than it affects GR. It invalidates neither theory.

My point is that your theory makes predictions which prevent anything in such a geometry from interacting with a central mass. If you apply this on a microscopic scale, you prevent particles from interacting. This messes up quantum field theory. But QFT is _extremely_ accurate (read >13 significant digits in some circumstances), so there most be a problem with one.

Of course GR predicts some very weird stuff when you reach the Planck scale, but you are claiming that your theory is completely consistant with quantum mechanics. I am saying that it is not.

Zanket wrote:

Offhand, I can’t refute that, and it makes some sense to me. You seem to be saying that GR is incompatible with QM not just at a singularity, but also in its immediate vicinity.

Yes I am, but I am also saying it is incompatible (or at least the curved background approach is incompatible) in isolated areas of high curvature, even if they are no accompanied by a black hole. Hence all the interest in quantum gravity.

Zanket wrote:

I don’t rule them out. The new metric is compatible with QM because, unlike the Schwarzschild metric, it doesn’t demand a minimum radius for a body that is incompatible with QM.

No, it's not. It doesn't work on a small scale as the curvature becomes large compared to the wavelength of the particles. Yours is infact worse, since it completely prevents them from interacting.

That my theory precludes singularities does not rule out any prediction of QM. You suggest that QM requires singularities. But that would be inconsistent, for it is singularities that create the incompatibility between GR and QM.

I'm not saying that the singularities are necessary for QM. I'm not talking about singularities at all. I'm saying that in your theory a particle at r=0 cannot interact with any other particle. If we apply this to subatomic particles, this causes a major problem. NOTHING INTERACTS WITH ANYTHING ELSE!!!

Of course if you back off on claiming that your theory is compatable with QM and QFT then the problem goes away, but you can no longer claim that your theory has any advantage over GR in terms of how nicely it can play with quantum mechanics (as I have already mentioned, your proposal is actually less compatible).

Zanket

planck2 wrote:

No you may not

As you wish.

The metric is the fundamental requirement, but you need other things in order to get the equations of motion such as the Euler-Lagrange formalism or the Hamilton-Jacobi method. See any standard graduate text on classical mechanics

Then how do you explain T&W’s ultra-clear comment that only the metric is needed to predict motion? How do you explain that their book Exploring Black Holes is full of predictions of motion, all using the metric exclusively?

but they don't converge and never will

So what? The difference between them becomes arbitrarily small as r goes to infinity. They converge as r increases.

but then you are just guessing, the point is that Schwarzschild had a set of field equations (which related the curvature to the presence of matter) to work from, he used them to find a possible solution for the spacetime outside a star and this happened to agree with experiment.

You’re just describing the history of the metric, that’s all. You’re not showing that field equations are a scientific requirement. You don’t think Einstein was “just guessing” when he formulated his field equations using trial and error?

Come up with new field equations and solutions to it such as your metric

Nah, it’s not required.

See my answer above.

See my question to your answer above.

Yes I do, because they don't agree at infinity. Simple.

So if the metrics differ only beyond the trillionth significant digit beyond some r, you think my metric is not asymptotically flat whereas the Schwarzschild metric is? Or do you think there’s some other problem with that?

and if you even dare credit me i'll sue your sorry ass all the way to seattle and back again

Take a deep breath. Now, why do you suppose I asked if it was okay?

Son Goku

Zanket wrote:

Then how do you explain T&W’s ultra-clear comment that only the metric is needed to predict motion? How do you explain that their book Exploring Black Holes is full of predictions of motion, all using the metric exclusively?

That's because the fact that you need a law of motion is blatantly obvious and they've already given it.
What is your law of motion?

Look at the potential V(x) = (kx^2)/2, on its own it can't tell you anything about motion, you need F=ma, or a Euler-Langrange equations in that formalism or the Hamilton Equations in Hamiltonian Dynamics.

What are your equations of motion?

breadmonkey

Please forgive my spamminess. I was just thinking it would be hilarious if Zanket's theory turned out to revolutionary but gets dismissed only to be taken seriously in 20 years time (this has happened quite a lot, right?). Then, in the future, someone will write about how his theory was originally rubbished by a bunch of guys on an internet message board!

Professor_Fink

breadmonkey wrote:

Please forgive my spamminess. I was just thinking it would be hilarious if Zanket's theory turned out to revolutionary but gets dismissed only to be taken seriously in 20 years time (this has happened quite a lot, right?). Then, in the future, someone will write about how his theory was originally rubbished by a bunch of guys on an internet message board!

No, it hasn't happened often at all.

But we're not just saying its unphysical or something vague like that, there are specific errors in Zankets paper where he has made incorrect assumptions (such as where he concludes that curvature along geodesics has no effect). General relativity doesn't make these additional assumptions, and so is more rigorous. I have yet to hear of a theory being rplaced by another theory which makes more assumptions. Thats just not the way science works.

And for what it's worth, all three of us (me, Planck2 and Son Goku) are theoretical physicists. I know this because I met Son Goku while doing my PhD and Planck2 was in my undergraduate class (theoretical physics).

planck2

Zanket wrote:

Then how do you explain T&W’s ultra-clear comment that only the metric is needed to predict motion? How do you explain that their book Exploring Black Holes is full of predictions of motion, all using the metric exclusively?

Of course they say the metric is all you need because that is were you start from. And then you you use the fact that you are dealing with spacelike/timelike/lightlike(null) geodesics.
but you still need a methodology in order to get the geodesics. And how do you explain the fact that the Schwarzschild metric approximates to Newtonian gravity and hence classical mechanics in the weak field limit where as your's doesn't.

Zanket wrote:

Nah, it’s not required.

Einstein said field strength=curvature, so what are you saying?
you just say Einstein is wrong, Schwarzschild metric is not unique and give a metric as a replacement which is not spherically symmetric. I'm sure I can get the geodesics but your not telling me anything with these. The metric describes a curved manifold and I can obtain how the particles move on it if one postulates that particles move along geodesics, but how do you relate curvature to gravity? This is what the field equations do.

Zanket wrote:

So what? The difference between them becomes arbitrarily small as r goes to infinity. They converge as r increases

So if the metrics differ only beyond the trillionth significant digit beyond some r, you think my metric is not asymptotically flat whereas the Schwarzschild metric is? Or do you think there’s some other problem with that?

but they actually don't converge and it is not by some mere trillionith significant figure difference. your metric diverges from flat spacetime whereas the Schwarzschild one converges to flat spacetime in the limit as r goes to infinity. So your metric is not asymptotically flat and never will be.

Zanket

Son Goku wrote:

Yes. Theories cannot have no equations of motion. I don't think you even understand what I'm saying.

I do understand you. I agree that a theory of gravity should have an equation of motion. I say that the metric is an equation of motion. By itself it predicts motion, so it must be an equation of motion. The new metric in my paper is the only expression that is needed to predict any motion for Schwarzschild geometry, the scope of my paper.

How does stuff move in your metric? What is the rule for generating the equations of motion. It can't be Einstein's geodesic rule as you are rejecting GR, so how do the particles move in your theory?

i.e. Given the metric how do you obtain the trajectory of a particle within it, for some initial condition.

I suggest you get the book Exploring Black Holes by T&W. For example, in chapter 4 (which is not online) they use the principle of extremal aging and the Schwarzschild metric, and only those as a basis, to compute an orbit. Throughout the book they prove their claim that “this one expression [the metric] tells it all!” (boldface mine).

Look at the potential V(x) = (kx^2)/2, on its own it can't tell you anything about motion, you need F=ma, or a Euler-Langrange equations in that formalism or the Hamilton Equations in Hamiltonian Dynamics.

Then how do you explain that T&W use no equation other than the metric to compute an orbit?

planck2

Zanket wrote:

Then how do you explain that T&W use no equation other than the metric to compute an orbit?

they assume like Einstein that particles move along geodesics and use the metric to obtain the geodesics. Since you are rejecting Einstein's geodesic assumption how do you use the metric to obtain the equations of motion?

Zanket

Professor_Fink wrote:

If you say that nothing can ever reach r=0 in any frame how can such objects form? Must they all be hollow?

All right, now I think I see what you’re getting at. That r=0 is invalid does not mean that an object is hollow. It means that the surface of the object cannot be at r=0; i.e. the object cannot be contained in zero volume. For the Earth, r is > 0 and matter can be at r=0 within the Earth. My paper handles only Schwarzschild geometry, which is defined in the paper as "The geometry of empty spacetime around a Schwarzschild object". So r refers to an r-coordinate at or above the surface of an object.

No, it's not. It doesn't work on a small scale as the curvature becomes large compared to the wavelength of the particles. Yours is infact worse, since it completely prevents them from interacting.

I assume that you think it “doesn't work on a small scale...” because of the hollow argument above. Now that I have refuted that, do you still think this? If so, why? The new metric doesn’t require a minimum r for a body. So there’s no reason to believe that the curvature must become too large compared to the wavelength of the particles.

I'm not saying that the singularities are necessary for QM. I'm not talking about singularities at all. I'm saying that in your theory a particle at r=0 cannot interact with any other particle. If we apply this to subatomic particles, this causes a major problem. NOTHING INTERACTS WITH ANYTHING ELSE!!!

When r is > 0 for an object, a particle at r=0 within the object can interact with other particles. This should resolve all of your objections regarding QM.

But we're not just saying its unphysical or something vague like that, there are specific errors in Zankets paper where he has made incorrect assumptions (such as where he concludes that curvature along geodesics has no effect).

You’re twisting my words again. Please stop. Nowhere did I conclude that. My theory is one of curved spacetime. I said that spacetime curvature has no effect on the conclusion in section 2. As in, the degree of spacetime curvature does not change the conclusion.

General relativity doesn't make these additional assumptions, and so is more rigorous. I have yet to hear of a theory being rplaced by another theory which makes more assumptions. Thats just not the way science works.

My paper makes no new assumptions; you haven't shown otherwise. The paper shows that GR is not rigorous. It is shown to be inconsistent (in sections 2 and 7), and that has not been refuted in this thread or elsewhere.

Zanket

planck2 wrote:

Of course they say the metric is all you need because that is were you start from. And then you you use the fact that you are dealing with spacelike/timelike/lightlike(null) geodesics.
but you still need a methodology in order to get the geodesics.

Ah, so the metric is an equation of motion after all. No other equation is needed to predict motion. We’ve made progress.

Geodesics are implied by the metric itself. An object simply goes straight, and the curved spacetime that is completely described by the metric curves its path. So one doesn’t “need a methodology in order to get the geodesics”. As T&W emphasize, the metric is a complete description of spacetime around a Schwarzschild object. More on that below.

And how do you explain the fact that the Schwarzschild metric approximates to Newtonian gravity and hence classical mechanics in the weak field limit where as your's doesn't.

Newtonian gravity approximates my metric, because, as the paper shows, the Schwarzschild metric approximates my metric. You haven’t refuted that. You’ve made only empty claims.

Einstein said field strength=curvature, so what are you saying?

I don’t need to say anything to that effect. I give a metric that makes falsifiable predictions of observations. That’s enough.

you just say Einstein is wrong, Schwarzschild metric is not unique ...

I don’t say that the Schwarzschild metric is not unique, which implies that I think there is more than one solution to Einstein’s field equations for Schwarzschild geometry, which I don’t. Rather I say that the Schwarzschild metric is invalid, for it is inconsistent with the finding in section 1, which was inferred by means GR allows. Then Einstein’s field equations, because they are proven to yield only the Schwarzschild metric for Schwarzschild geometry, must be invalid.

... and give a metric as a replacement which is not spherically symmetric.

What is your evidence?

I'm sure I can get the geodesics but your not telling me anything with these. The metric describes a curved manifold and I can obtain how the particles move on it if one postulates that particles move along geodesics, ...

There’s nothing magical about geodesics; it doesn't need to be a postulate. An object goes straight in curved spacetime that curves its path, that’s all. Does my paper need to explain that? No. Geodesics are implied by the metric.

... but how do you relate curvature to gravity? This is what the field equations do.

In GR, spacetime curvature is the sole indicator of gravity. They are inseparable. That is how the Schwarzschild metric (and my metric) can be a complete description of Schwarzschild geometry.

but they actually don't converge and it is not by some mere trillionith significant figure difference. your metric diverges from flat spacetime whereas the Schwarzschild one converges to flat spacetime in the limit as r goes to infinity. So your metric is not asymptotically flat and never will be.

The Schwarzschild metric and the new metric do converge as r increases. That does not mean that they eventually meet; it means that they approach each other ever more closely. They are asymptotic to each other. You haven’t shown otherwise. You just make an empty claim. I gave the only difference between the two metrics, which are simple equations, the curves of which can be easily seen to converge as r increases. How do you expect to be a scientist when you make unsupported claims that are so easily seen to be wrong?

they assume like Einstein that particles move along geodesics and use the metric to obtain the geodesics. Since you are rejecting Einstein's geodesic assumption how do you use the metric to obtain the equations of motion?

They assume that a particle would go straight were it not for the curved spacetime that curves its path. That’s not something that needs to be explained, for if the spacetime were not curved, it would be flat, and then the particle would surely go straight. There are lots of things they can assume without the need for explanation. For example, they can assume that the particle continues to exist.

I do not reject the notion of geodesics. You’re reading that into the paper; it’s not there. My theory is built on parts of GR, like SR and the equivalence principle. Unless I say or imply that something is invalid, the reader can assume that I hold it to be valid. The paper shows a specific flaw of GR. It does not reject GR in its entirety. For example, do I think field equations are worthless? No. Someone can take my paper as a starting point, figure out where the flaw is in Einstein’s field equations, and rebuild GR in all its glory. (The new metric for Schwarzschild geometry that the new field equations yield should be consistent with section 1 in my paper. It need not match my metric, but I doubt there’s a simpler metric that is consistent with section 1, and nature seems to prefer simplicity.)

Son Goku

I suggest you get the book Exploring Black Holes by T&W. For example, in chapter 4 (which is not online) they use the principle of extremal aging and the Schwarzschild metric, and only those as a basis, to compute an orbit. Throughout the book they prove their claim that “this one expression [the metric] tells it all!” (boldface mine).

I suggest you read Wald or at least Schutz.

Ah, so the metric is an equation of motion after all. No other equation is needed to predict motion. We’ve made progress.

Geodesics are implied by the metric itself. An object simply goes straight, and the curved spacetime that is completely described by the metric curves its path. So one doesn’t “need a methodology in order to get the geodesics”. As T&W emphasize, the metric is a complete description of spacetime around a Schwarzschild object. More on that below.

Geodesics are implied in General Relativity, all we wanted all along was what your rule was for particle dynamics. Your rule is the geodesics as you just stated above although it took along time to get that out of you.

Zanket did you understand what our question meant?
The part in bold is totally incorrect and I think at this point you're being triumphant on purpose.

I do not reject the notion of geodesics. You’re reading that into the paper; it’s not there.

You reject GR though, so it's difficult to understand how you accept the generator of its dynamics.

The Schwarzschild metric and the new metric do converge as r increases. That does not mean that they eventually meet; it means that they approach each other ever more closely. They are asymptotic to each other.

Zanket, do you not think it's possible that you simply haven't read enough physics and are biting off more than you can chew?
You must admit that your inability to understand basic dynamical questions indicates a alck of familiarity with a lot of the sunject.

Tell me, what did you read before you read Taylor's book.

planck2

look it's pointless arguing with this guy. He clearly doesn't understand what is going on. We are wasting our time

planck2

Zanket wrote:

What is your evidence?

That's easy to prove. The proof of this is given by Werner Israel[1967], who I know.

and besides I am not emotionally attached to GR, you are just saying things that you cant back up.

You have derived stuff which you say GR allows which disagree with the Schwarzschild solution and Einstein's field equations.

It is most likely that there is a serious flaw in your arguement

planck2

Zanket wrote:

They assume that a particle would go straight were it not for the curved spacetime that curves its path. That’s not something that needs to be explained, for if the spacetime were not curved, it would be flat, and then the particle would surely go straight. There are lots of things they can assume without the need for explanation. For example, they can assume that the particle continues to exist.

They assume that particles do not move along a straight line in flat spacetime, but along a geodesic in any spacetime manifold, curved or flat?

What is your understanding of a geodesic?