KJW

This specific wording is what would lead me to interpretation of your stance as treating the mathematical tool (coordinates) as the creator of physical reality. If that wasn’t your intent, then it is simply a semantic misunderstanding between us, and I am glad we cleared it up. Perhaps what I said had a bit of hyperbole. In fact, I should've said: I don't see the non-covariance that arises due to coordinates as a problem, but as an opportunity. It literally creates the fields of the mathematical descriptions of reality, not just in general relativity, but in gauge theory in general. But then again, pretty much everything I am saying is about mathematical descriptions of reality rather than reality itself, so it is natural that I abbreviate. On the other hand, because of the correspondence between the reality and mathematical descriptions of the reality, the notion that non-covariance creates a field in the mathematical description would imply that non-covariance in a way creates a field in the reality itself also. For example, spacetime curvature exists in physical reality for the same reason it does in mathematical descriptions of reality. And yes, spacetime curvature is real and not just an abstract notion in a mathematical model.

In what sense? Is it not from the fact that space and time form spacetime... Space equivalency to time. Sometimes I feel explanations afterwards provided for GR is complicated than how the thought process of Einstein was as he was deriving it. It seems to me to be a common view that the laws of physics are ad hoc, as if given to us from above. I reject this view. For example, according to Noether's theorem, each conservation law corresponds to a particular symmetry. Also, in the mathematics of general relativity, every scalar functional of the metric tensor and its partial derivatives of any order corresponds to a covariantly-conserved second-order tensor field (the symmetry in this case is the invariance of the scalar functional with respect to a change in coordinates, noting the severe restriction on what can actually be a scalar functional of the metric tensor and its partial derivatives). Such laws of physics arise for purely mathematical reasons. But, that the number of dimensions of physical reality appears to be four rather than some other value is something that also needs to be explained logically. It turns out that in the mathematics of general relativity, four dimensions is rather special compared to other numbers of dimensions. For example: • Four is the smallest number of dimensions for which pure gravitation can exist. • Four is the only number of dimensions for which pure gravitation has the same algebraic freedom as energy-momentum. • Four is the only number of dimensions for which the dual of the Riemann tensor is the same order as the Riemann tensor, and for which the dual of the Weyl tensor is the same as the Weyl tensor. • Four is the only number of dimensions for which the integrand of the generalised Gauss-Bonnet topological invariant is quadratic in the curvature tensor. As significant as these properties of four-dimensional spaces are, it is not clear to me how they actually constrain the number of dimensions.

Bear in mind that we had already discussed in greater detail the logical basis I used to derive general relativity, and that I didn't want to repeat that discussion here, choosing only to give a fairly brief statement relevant to the current discussion. So, in effect, you attacked a strawman. I actually make a point of distinguishing the mathematics from the physics. Yet you chose to interpret my statement: the postulate of a spacetime manifold is reasonable on the basis of observed reality. So, the spacetime manifold becomes a fact. as: treating an abstract mathematical model as a physical entity No, my statement does not lead to your interpretation. I said the mathematical model was reasonable on the basis of the physical entity. I had already outlined earlier in this thread the reason that connects the physical entity to the mathematical model. That is, we measure the physical entity to create a description that is the theoretical basis of a mathematical model. It is a fact that the physical spacetime is amenable to being described in terms of a mathematical spacetime manifold. Although one can challenge the spacetime manifold in terms of how close it is to first principles, one can't challenge its validity. It should be noted that accuracy of the description is a separate issue. I assume that the descriptions are perfectly accurate. I know that isn't true in reality but consider inaccuracies to be outside the scope of my interest. My interest is actually what physical reality must be, based on what mathematical descriptions of it must be. The key requirement is that all constraints are logical rather than ad hoc. Thus, I am interested in why spacetime is four-dimensional, and why the signature of the metric is (+,–,–,–)/(–,+,+,+). Much of the mathematics of Ricci calculus is non-specific in terms of the number of dimensions or the signature of the metric. Some formulae explicitly mention n dimensions. Formulae that specify four dimensions and the signature of the metric are specifically about general relativity. The question of how the metric tensor field arises is a challenge to the key requirement that all constraints are logical rather than ad hoc, the specific constraint in this case being to the connection. Thus, although my interest is in the physics, my focus goes beyond the physics. I don't see the non-covariance that arises due to coordinates as a problem, but as an opportunity. It literally creates the fields of reality, not just in general relativity, but in gauge theory in general.

I don't agree with this assessment of what general relativity must postulate. Mass is not a postulate. It is a physical quantity that is part of the connection between the pure mathematics of Ricci calculus and physical reality. Obtaining the Schwarzschild solution involves solving a first-order ordinary differential equation, which produces a single arbitrary constant. Physically, it is directly proportional to the mass of the black hole, but to obtain the proportionality constant, one needs to compare the behaviour of the Schwarzschild solution under weak-field conditions, with the Newtonian formula. You have chosen to use the Schwarzschild radius to represent mass. That's ok until you need to specify mass in terms of mass units. Note that using the Schwarzschild radius as the arbitrary constant of integration is purely mathematical, and therefore requires some form of physical measurement to connect your Schwarzschild solution to physical reality. What do you mean by a background 4D spacetime manifold? Actually, the fundamental fields of general relativity describe a spacetime manifold rather than sit upon a background. Fields that sit upon a general relativistic manifold are probably not themselves part of general relativity. For example, the metric tensor field is a set of coefficients of the metric that describe the distance between infinitesimally separated points of the spacetime manifold. And the various spacetime curvature fields are mathematically derived from the metric tensor field. The existence of the curvature fields is guaranteed given the existence of the metric tensor field. It is not a postulate that the curvature fields exist. And because it is not assumed that the metric tensor field is special, it is not assumed that the curvature fields are zero. Although it is desirable to minimise the number of postulates, the postulate of a spacetime manifold is reasonable on the basis of observed reality. So, the spacetime manifold becomes a fact. And from this fact, I can derive guaranteed theoretical results by deliberately avoiding assumptions about physical reality. I've already discussed this earlier in this thread, so I won't repeat myself here. The method I have used to derive general relativity, I have so far been unable to derive the metric tensor field from first principles. The problem isn't the invertible matrix field itself. That emerges naturally from the connection. The problem is that given an arbitrary connection, it is not guaranteed that a covariantly constant invertible matrix field exists. And it is covariant constancy that distinguishes the metric tensor field from an arbitrary invertible matrix field. However, the postulate of the existence of the metric tensor field can be justified by the notion of magnitude. Nevertheless, I have a keen eye on what can and can't be derived without the metric tensor field. As for "spatial curvature as a distinct geometric entity", that is not a postulate of general relativity. That is a notion that emerges from the Schwarzschild solution as well as an understanding of the nature of familiar gravity. Although the theory of general relativity doesn't use coordinates, specific solutions of the Einstein field equations do use coordinates, and often those coordinates are space and time coordinates. Often, interesting solutions possess symmetries that render splitting the spacetime into space and time quite natural (because symmetry is manifestly covariant). And with that natural splitting, the notion of spatial curvature is also natural.

Don't know if am wrong but the opposite signs in spacetime metric... signature -1,+1,+1,+1 or +1,-1,-1,-1 makes sure the spacetime is unified as it's also stated by □=0. ( To whoever get concerned...Sorry I couldn't not comment in another thread...when experts discuss, sometimes it's reasonable to keep quite). The three sides of de'Alembert operator magnitude(space axis x,y,z) being equivalent to magnitude of one side of the de'Alembert operator ( square,four sided, x,y,z,t) opposite sign ensure the total magnitude is equal to Zero...each degree of freedom is equivalent...t equivalent to -x,t equivalent to -y,t equivalent to -z and t equivalent to -(x,y,z). Sorry corrections, each degree of freedom is equivalent...t = x,t = y,t = z and t = (x,y,z). □=0. It seems there is no need of correction,it depends on which signature you are using signature -1,+1,+1,+1 or +1,-1,-1,-1. It is the metric being expressed in terms of all four dimensions that ensures spacetime is a unified notion. The signature of the metric provides finer detail about the geometry of spacetime, giving rise to the distinction between space and time, even though spacetime is a unified notion. In four dimensions, there are three distinct signatures: (+,+,+,+)/(–,–,–,–), (+,–,–,–)/(–,+,+,+), and (+,+,–,–). In the case of (+,+,+,+)/(–,–,–,–), there is no distinction at all between the dimensions, and there is no notion of a speed of light. Space is four-dimensional and time does not exist. In the case of (+,+,–,–), there are two space dimensions and two time dimensions, and the notion of a speed of light. However, because changing all the signs of a signature doesn't change the geometry, there is nothing to indicate which dimensions are space and which are time. But it is the case of (+,–,–,–)/(–,+,+,+) that is our spacetime in which there is the notion of a speed of light, and there are unequivocally three dimensions of space and one dimension of time. Solving the equation [math]\pm\ x^2 \pm y^2 \pm z^2 \pm t^2 = 0[/math] for the various sign combinations shows how the partition between space and time depends on the signature.

I don't think even criminals are an exception which wouldn't have thin wedge problems. After all, what is a "criminal"? Because "criminals" are defined by the government, then disenfranchising them becomes an example of "conflict of interest" that I mentioned in the other thread. To many people, "criminals" means murderers, rapists, etc, but in many places, "criminals" can also mean political opponents.

Why should they be not familiar? Or in what sense....I think spacetime is a unified thing,what happens in either dimensions should affect the other...If it's not like that then there might be some aspect of either dimension that we don't fully understand...and according to me specifically what time is(nature of time). Yes, spacetime is a unified thing. But that's not how we experience reality. Our experience of space and time are very distinct. Firstly, there are three space dimensions and one time dimension. The spacetime metric itself separates space and time into separate notions by their opposite signs in the signature. This creates a notion called the speed of light which acts as an impenetrable barrier between space and time, forcing us to exist as timelike worldlines in spacetime. Thus, we exist in space but experience time. At human scale, one second in spacetime is very close to three hundred thousand kilometres in the time direction. This disparity in the perceived magnitudes of space and time intervals is connected to why the speed of light seems so fast to us. Actually, why the speed of light is so fast is an interesting question in its own right, but the consequence of this is that the time components of quantities have an exaggerated existence compared to the space components of the same quantities. This can also lead to the notion of magnetic quantities, space components that arise due to the motion of time components. For example, the electric field vector is the time components of the electromagnetic field tensor. But when an electric field vector is put into motion, that motion produces space components of the electromagnetic field tensor that is the magnetic field pseudovector. With regards to curvature, it is my understanding that when considered in terms of the same units, the space and time components of the curvature of the Schwarzschild spacetime are essentially the same in magnitude. But we do not see the space curvature. Pythagoras theorem is assumed to be true. Yet for time, we see objects fall to the ground, planets orbiting the sun, and moons orbiting their planets. In spacetime, the earth has a helical trajectory around the sun in spacetime. But in spacetime, the trajectory of an object in orbit is a straight line. However, because the time of an orbit of the earth around the sun is one year, the length of the helical trajectory in spacetime is about one light-year, and at this length, no longer seems significantly curved.

You’re absolutely right, and it was meant to be that, I once again forgot the conversion. This is what happens when you don’t do this stuff every day. Thanks for picking up on it 👍 Thanks @Markus Hanke. I'll write out the corrected formula: \[\varphi =\int_{r_{\min}}^{\infty}\frac{\gamma v_{\infty} b dr}{r^2 \sqrt{\gamma^2 c^2 - \left(1 - \dfrac{2GM}{rc^2}\right) \left( c^2 + \dfrac{\gamma^2 v^{2}_{\infty}b^2}{r^2} \right)}}-\pi\] I’m not entirely sure what “to second post-Newtonian order” actually means, but I presume this is an approximation of some kind? The full integral looks elliptic, so there shouldn’t be a closed-analytic form for the exact result. The formula itself was said to be "classical general relativity". I'm not sure what the authors meant by “to second post-Newtonian order”. My initial thought was that they were applying modified theories to the problem and that "classical general relativity" was just one theory that was applied, since they also mentioned "semiclassical general relativity". But I suspect you may be right about the formula I gave being an approximation of some kind.

The factor of 2 is the result of the curvature of the three-dimensional space of the Schwarzschild spacetime (as a static spacetime). Notice that in my explanation of your error, I shifted from double the Newtonian result to double the equivalence principle result. When Einstein first calculated the deflection of light, he obtained a result that was half the correct result. This is the result one obtains from applying the equivalence principle. But this doesn't mean that the equivalence principle is invalid because the equivalence principle is only valid locally, and it is valid locally. For example, the gravitational deflection of a laser beam across a room does correspond to that expected in an accelerated frame of reference. And this occurs at all locations along the entire trajectory of light. But when one joins together all the many local deflections along the entire trajectory of light, the result is no longer a local deflection corresponding to the equivalence principle. The extra deflection, which would not occur in an ordinary flat space in which Pythagoras theorem is obeyed, is the result of the curvature of the three-dimensional space. Earlier in this thread, I gave a formula for familiar gravity in terms of time dilation. This formula actually describes an accelerated frame of reference and applies the equivalence principle in describing familiar gravity. Thus, familiar gravity and the equivalence principle is about time dilation and what happens in the time dimension. What happens in the space dimensions do not contribute to familiar gravity. For one thing, space isn't amplified by the speed of light like time is. Thus, we don't notice the curvature of three-dimensional space because it is so small, whereas we notice the effects of curvature in the time dimension because they give rise to impossible to ignore gravity. Basically, what I'm saying is that spatial curvature is an entirely new thing in physics. There is nothing in Newtonian physics that is a manifestation of spatial curvature. Spatial curvature is a purely general relativistic effect without any correspondence in Newtonian physics. No, the [math]c^4[/math] should be [math]c^2[/math]. Then the units will be correct.

Thanks again. Yes, I wasn't sure what E and L were, so this reworked formula is much better. I also come across this formula: [math]\Phi_C = \dfrac{2GM}{v_0^2 b} [1 + \dfrac{v_0^2}{c^2} + \dfrac{3\pi}{2} \dfrac{GM}{c^2 b} + \dfrac{3\pi}{8} \dfrac{GM}{c^2 b} \dfrac{v_0^2}{c^2} + 9(\dfrac{GM}{c^2 b})^2][/math] from https://ui.adsabs.harvard.edu/abs/2002CQGra..19.5429A/abstract Finding such formulae for massive particles is difficult because everyone wants to tell you about the formula for light. I think there's an error in this formula. It's not dimensionally correct. Should that [math]\dfrac{1}{r^2}[/math] be [math]\dfrac{1}{r^2 c^2}[/math]? Actually, on closer look, the two [math]b^2 v_{\infty}^2[/math] should be [math]b^2 v_{\infty}^2/c^2[/math].

It is: \[\varphi =\int_{r_{\min}}^{\infty}\frac{dr}{r^2 \sqrt{\dfrac{E^2}{L^2} - \left(1 - \dfrac{2GM}{rc^2}\right) \left( \dfrac{1}{L^2} + \dfrac{1}{r^2} \right)}}\] For non-relativistic speeds and weak fields, this reduces to the Newtonian scattering formula. For v=c and massless test particles, you get the Schwarzschild light deflection formula. For strong fields and massive particles, the integral can be evaluated numerically. Thanks. +1

Thank you very much. I can now reveal where you have made your error. I don't know what the actual formula is for the non-zero mass object, but I can see that the above two formulae do not agree for [math]\beta = 1[/math]. For a non-zero mass object travelling at 0.999999999c, the deflection should be the same as for light. That is, your error was to assume that the factor of 2 arose because light is massless. The trajectory of an object in a gravitational field does not depend on the mass of a test mass (a mass that is sufficiently small as to not affect the surrounding spacetime) but depends on the speed of the object. Thus, the trajectory of a non-zero mass object travelling at 0.999999999c will be negligibly different to the trajectory of light, and therefore have deflections that are the same (negligibly different). As I see it, the ratio of the actual deflection angle to that predicted by the equivalence principle will itself depend on [math]\beta[/math]. For [math]\beta \approx 0[/math], the spacetime trajectory is mostly governed by time, and therefore the deflection will correspond to that predicted by the equivalence principle. But for [math]\beta \approx 1[/math], the spacetime trajectory is governed more-or-less equally by time and space, and the contribution by space equals the contribution by time, thus doubling the deflection corresponding to that predicted by the equivalence principle.

That's because I see value in it. Nevertheless, I'm still scrutinising you and your work. And the nature of your work has made it very difficult to scrutinise in the way I feel is necessary. I'm not sure that what value I see is what you intend me to see as I'm not in full agreement with your philosophy, although some of it does align with relativity. What I would like to see is some form of mathematical proof that your theory fully agrees with general relativity. What you have provided so far is not such a proof, and I'm not sure what such a proof would look like. Thanks. This looks more detailed than what I saw earlier. I want to look through it to see if you have addressed the problem I see in your explanation. Bear in mind that I know why the defection of light under general relativity is twice that of Newtonian theory. It's not straightforward and I believe you have chosen the wrong explanation. I don't wish to reveal what I believe to be your mistake because I think it is important for your theory that it be able to derive things without the guidance that exists when deriving preexisting results. I feel that what I'm asking for may be too difficult for the piece of information I'm looking for from you. While I asked for a complete formula, in fact I want to know how you handle a specific scenario that I don't wish to reveal, but maybe I should. I apologise for that. Anyway, if you can preempt why I am asking you about non-zero mass, perhaps you can address my concern without deriving the formula.

A little bit off topic but still broadly relevant is that voting is compulsory in Australia. That is, voting is not just a right, it's an obligation.

Doesn't it just. (Gerrymandering etc.) I was actually thinking of gerrymandering as another example.

That seems to come under the notion of "conflict of interest".

But why the impatience? Why is waiting for the elderly to die such a problem that they need to be disenfranchised early?

@Anton Rize, I've being looking at your first PDF, starting from the beginning, in order to gain insight as to how to solve a physics problem using your theory. I believe I found an error with regards to your explanation of the deflection of light by a source of gravitation, specifically the factor of 2 which distinguishes Einstein's prediction from a Newtonian prediction. I would like you to derive a formula for the deflection of an object with non-zero mass by a source of gravitation. That formula should include all its dependencies. You may specify the gravitation as: [math]\dfrac{r}{r_s}[/math] and you may assume the mass of the deflected object, though non-zero, is sufficiently small that any gravitational radiation is negligible. It is up to you to determine what the deflection angle of the object depends upon.

That doesn't mean they're not mathematical geniuses, although I have no opinion concerning the topic of this thread.

I decided to investigate this because of the mention of the fine-structure constant. I'm actually quite impressed with how close the derived Hubble constant matches the measured value. Unravelling the formula, what you appear to me to be saying is that: [math]\dfrac{\rho_{\text{electromagnetic}}}{\rho_{\text{critical}}} = \alpha^2 \approx \dfrac{1}{18779}[/math] However, I don't agree with your suggestion that you've figured out the fine-structure constant. The mystery of its particular numerical value remains, even though its relation to other physical notions is well known to physicists. In particular, it is known to be the value of [math]\beta[/math] of the electron in the lowest orbit of the Bohr model of the atom.

I think that's as much about class as anything else, although there is the "first in, best dressed" aspect as well. Bear in mind that the life of the elderly wasn't easy when they were young, either. That's not a reason to deny the elderly the vote for the short time they have left. That's why I said it was a fundamental problem. Although minorities having a vote doesn't prevent their interests from being trampled on, at least a vote gives them some sort of voice which would be silenced if they were disenfranchised. No, I'm looking to stop the disenfranchisement of the elderly.

Given that electoral terms are only a few years, and even old people expect to be still alive after the one they're voting in has completed, to suggest that old people no longer have a stake in the outcome of an election is wrong. A fundamental problem with democracy is that the majority don't always act in the interest of minorities and often act contrary to that interest, so to disenfranchise a minority only exacerbates that problem.

A little bit of background from https://en.wikipedia.org/wiki/Hadronization The top quark does not hadronize The top quark, however, decays via the weak force with a mean lifetime of 5×10−25 seconds. Unlike all other weak interactions, which typically are much slower than strong interactions, the top quark weak decay is uniquely shorter than the time scale at which the strong force of QCD acts, so a top quark decays before it can hadronize. The top quark is therefore almost a free particle.

You are overcomplicating my friend. If we strip the problem of any specific details and other additional anthropocentric components we will see the foundation of this phenomena. let's test the hypothesis that the age difference is manifested by the asymmetric observation of Doppler shifts (the optical delay of the turnaround) by removing the delay entirely. Consider a modified scenario: Observer A remains at rest. Observer B does not travel to a distant star, but instead orbits Observer A at a negligible, constant distance with a kinematic projection of [math]\beta = 0.8[/math]. No. The explanation I provided was specifically for the twin paradox scenario. If you want to modify the scenario, then the analysis of the scenario has to be modified as well. You can't say my explanation of the twin paradox scenario is incorrect or even overcomplicated because (although it is a thought experiment) my explanation is purely in terms of what is observed. What is observed by each twin is the redshift and blueshift of the other twin's clock, and the amount of time the redshift and blueshift are observed. As for your modified scenario, I'd like to first point out that you said nothing about how the observer in circular motion observes the observer in the centre. Your discussion of the scenario suggests that you are only concerned with the time dilation of the circularly moving observer that is observed by the central observer, without addressing the question of consistency between the perspectives of the two observers. And it is the question of consistency between the perspectives of the two observers that lies at the heart of the twin paradox, as well as precisely why the twins end up with different ages. Unfortunately, so far, I have been unable to provide an explanation of your modified scenario in terms I used for the twin paradox scenario. I'm not sure such an explanation even exists. However, I can say that the central observer observes the observer in circular motion as transverse Doppler redshifted, and that the observer in circular motion observes the central observer as accelerationally blueshifted. In the case of the twin paradox scenario, both legs of the travelling twin's journey were inertial, and although the turnaround is an acceleration, it wasn't treated as such and was merely a time and location where the redshift becomes a blueshift. Thus, the explanation I gave for the twin paradox scenario was the natural explanation. I believe (I have a possibly false recollection of doing the maths) that my explanation of the twin paradox scenario can be extended to arbitrary longitudinal motion of the travelling twin. Although such a case has the travelling twin in an accelerated frame of reference, ultimately all redshifts and blueshifts based on time dilation are Doppler effects. Thus, your modified scenario ought to be able to be treated like the generalised twin paradox scenario, although I am presently unable to connect the accelerated frame of reference associated with circular motion to the Doppler effect. That is, I am presently unable to explain the blueshift of the central observer that is observed by the circularly moving observer in terms of the Doppler effect. Nevertheless, the blueshift can be explained by invoking the metric. A problem I see with your explanation is that it seems to be unconnected to the actual physics of the situation. And by being unconnected to the actual physics, it is difficult to see how it can deal with subtleties present in the actual physics. For example, suppose I am in circular motion at constant speed around some object at the centre. I face the direction of my inward acceleration. Am I looking at the object at the centre?

In SI units: E is in joules m is in kilograms c is in metres per second There are a number of different systems of units, but as you appear to have discovered, for a formula to make sense, it is necessary for all the units to belong to the same system of units.