KJW

According to the Wikipedia article, "Rømer's determination of the speed of light", it wasn't the occultation of Io that was used for the time measurements, but the eclipse of Io by Jupiter's shadow. And it was both immersion, when Io suddenly disappears into Jupiter's shadow, and emergence, when Io suddenly reappears from Jupiter's shadow, that was used (immersion and emergence cannot be observed from Earth for the same eclipse because one or the other will be hidden by Jupiter).

It's worth noting that it is not known if Hawking radiation or Unruh radiation actually exist. Saying that fundamental particles can't be black holes because of Hawking radiation is reminiscent of the claim that electrons can't orbit the nucleus because it would radiate and fall into the nucleus.

I recall reading in an electronics book a while back the author strongly urging its readers not to attempt to build a switching power supply but to purchase a commercial version instead. It is my understanding that there is no isolation between the powered circuit and the mains power. One of the things I have learnt over the years is that real electronic circuits are more complicated than ideal circuits because the non-ideal behaviour of electronic components can cause problems that need to be countered by additional circuitry. A simple example is a switch. An ideal switch transitions power from OFF to ON or from ON to OFF. A real switch bounces, which can be a problem for digital circuitry.

I have wondered if fundamental subatomic particles are black holes. However, photons as black holes seems to be problematic.

The question you ask the person is: "Which road leads to the village where you live?" The answer given is the road to your friend.

Bear in mind that poisonous creatures also have strategies for not being poisoned by their own toxin.

It should be noted that objects are really four-dimensional, each point of the three-dimensional object being a worldline in four-dimensional spacetime, with the appearance of the object in three dimensions at a given time being the section of a three-dimensional slice through the object. In the inertial frame of the object itself, this three-dimensional slice is perpendicular to the worldlines of the object, but from another inertial frame of reference at motion relative to the object, the three-dimensional slice is oblique to the worldlines of the object. For example, an oblique slice through a cylinder is elliptical, with the length of the circular section being elongated in the direction of the obliqueness. However, due to the peculiarity of the geometry of spacetime, lengths of spatial sections are shortened rather than lengthened. So, given that an inertial object is always in its own inertial frame of reference, and all other inertial frames of reference exist regardless of whether there are any observers in them, length contraction is not a real phenomenon. However, it becomes more real for an accelerated object (which is always accelerating upward in its own frame of reference) because the bottom of the object has a greater acceleration than the top of the object. Nevertheless, the natural length of the object in its own accelerated frame of reference will remain constant unless forces associated with the acceleration physically distort the object.

StudioT can probably answer that. Maybe sticking a label or tape over the hole should suffice? I'm only thinking of a pinhole. Or perhaps using the lid from a previous jar from when you were younger and stronger.

I recently saw an episode of QI in which it was suggested that the harsh conditions on the moon would probably have disintegrated the flags.

***CORRECTION*** This post is a correction of the following text in the post about circular orbits of the Schwarzschild metric: The above contains an error of sorts. I mishandled the [math]r[/math] and [math]\theta[/math] coordinate variables. Specifically, I substituted the location of the object in the rotating coordinate system before obtaining the partial derivatives of [math]g_{tt}[/math]. Anticipating the correct result for [math]a_r[/math] (and [math]a_\theta[/math]) of the object, the error was confined to the above and did not affect the remainder of the post. Below is obtained the acceleration components [math]a_r[/math] and [math]a_\theta[/math] for an arbitrarily located stationary object in the rotating coordinate system. Only then is the particular location of the object substituted. [math](ds)^2 = \Big(c^2 - \dfrac{2GM}{r} - \omega^2 r^2 \sin^2\theta\Big) (dt)^2 - \Big(1 - \dfrac{2GM}{c^2 r}\Big)^{-1} (dr)^2 - r^2 (d\theta)^2 - r^2 \sin^2\theta (d\varphi)^2 - 2\omega r^2 \sin^2\theta\ d\varphi dt[/math] [math]g_{tt} = c^2 - \dfrac{2GM}{r} - \omega^2 r^2 \sin^2\theta[/math] [math]a_r = -\dfrac{c^2}{\sqrt{g_{tt}}} \dfrac{\partial \sqrt{g_{tt}}}{\partial r} = -\dfrac{1}{2} \dfrac{c^2}{g_{tt}} \dfrac{\partial g_{tt}}{\partial r} = \dfrac{\omega^2 r \sin^2\theta - \dfrac{GM}{r^2}}{1 - \dfrac{2GM}{c^2 r} - \dfrac{\omega^2 r^2 \sin^2\theta}{c^2}}[/math] [math]= \dfrac{\omega^2 R - \dfrac{GM}{R^2}}{1 - \dfrac{2GM}{c^2 R} - \dfrac{\omega^2 R^2}{c^2}}\ \ \ \ \ \text{at}\ \left\{\begin{array}{} r=R &;& dr=0\\\theta = \dfrac{\pi}{2} &;& d\theta = 0\\\varphi = 0 &;& d\varphi = 0 \end{array}\right.[/math] [math]a_\theta = -\dfrac{c^2}{\sqrt{g_{tt}}} \dfrac{\partial \sqrt{g_{tt}}}{\partial \theta} = -\dfrac{1}{2} \dfrac{c^2}{g_{tt}} \dfrac{\partial g_{tt}}{\partial \theta} = \dfrac{\omega^2 r^2 \sin\theta \cos\theta}{1 - \dfrac{2GM}{c^2 r} - \dfrac{\omega^2 r^2 \sin^2\theta}{c^2}}[/math] [math]= 0\ \ \ \ \ \text{at}\ \left\{\begin{array}{} r=R &;& dr=0\\\theta = \dfrac{\pi}{2} &;& d\theta = 0\\\varphi = 0 &;& d\varphi = 0 \end{array}\right.[/math]

One isn't going to have a [math]\sqrt{3}[/math] factor difference between observations made from outside and observations made from inside of a non-relativistic system. It is my understanding that it is the virial theorem that indicates the existence of dark matter in galaxies.

[math](ds)^2 = \Big(c^2 - \dfrac{2GM}{r} - \omega^2 r^2 \sin^2\theta\Big) (dt)^2 - \Big(1 - \dfrac{2GM}{c^2 r}\Big)^{-1} (dr)^2 - r^2 (d\theta)^2 - r^2 \sin^2\theta (d\varphi)^2 - 2\omega r^2 \sin^2\theta\ d\varphi dt[/math] [math](ds)^2 = \Big(c^2 - \dfrac{2GM}{r} - \omega^2 r^2 \sin^2\theta\Big) (dt)^2 - \Big(1 - \dfrac{2GM}{c^2 r}\Big)^{-1} (dr)^2 - r^2 (d\theta)^2 - r^2 \sin^2\theta (d\varphi)^2 - 2\omega r^2 \sin^2\theta\ d\varphi dt[/math] [math]g_{tt} = c^2 - \dfrac{2GM}{r} - \omega^2 r^2 \sin^2\theta[/math] [math]a_r = -\dfrac{c^2}{\sqrt{g_{tt}}} \dfrac{\partial \sqrt{g_{tt}}}{\partial r} = -\dfrac{1}{2} \dfrac{c^2}{g_{tt}} \dfrac{\partial g_{tt}}{\partial r} = \dfrac{\omega^2 r \sin^2\theta - \dfrac{GM}{r^2}}{1 - \dfrac{2GM}{c^2 r} - \dfrac{\omega^2 r^2 \sin^2\theta}{c^2}}[/math] [math]= \dfrac{\omega^2 R - \dfrac{GM}{R^2}}{1 - \dfrac{2GM}{c^2 R} - \dfrac{\omega^2 R^2}{c^2}}\ \ \ \ \ \text{at}\ \left\{\begin{array}{} r=R &;& dr=0\\\theta = \dfrac{\pi}{2} &;& d\theta = 0\\\varphi = 0 &;& d\varphi = 0 \end{array}\right.[/math] [math]a_\theta = -\dfrac{c^2}{\sqrt{g_{tt}}} \dfrac{\partial \sqrt{g_{tt}}}{\partial \theta} = -\dfrac{1}{2} \dfrac{c^2}{g_{tt}} \dfrac{\partial g_{tt}}{\partial \theta} = \dfrac{\omega^2 r^2 \sin\theta \cos\theta}{1 - \dfrac{2GM}{c^2 r} - \dfrac{\omega^2 r^2 \sin^2\theta}{c^2}}[/math] [math]= 0\ \ \ \ \ \text{at}\ \left\{\begin{array}{} r=R &;& dr=0\\\theta = \dfrac{\pi}{2} &;& d\theta = 0\\\varphi = 0 &;& d\varphi = 0 \end{array}\right.[/math]

To put this into a more familiar context, a sphere is a two-dimensional surface in three-dimensional Euclidean space that is a constant distance from the origin, with the metric of the three-dimensional Euclidean space inducing a metric on the two-dimensional surface of the sphere: [math]x^2 + y^2 + z^2 = r^2[/math] where [math]r[/math] is a constant. Defining latitude [math]\phi[/math] and longitude [math]\lambda[/math] coordinates on the sphere: [math]x = r \cos\phi \cos\lambda[/math] [math]y = r \cos\phi \sin\lambda[/math] [math]z = r \sin\phi[/math] Differentials: [math]dx = -r\ (\cos\phi \sin\lambda\ d\lambda + \sin\phi \cos\lambda\ d\phi)[/math] [math]dy = r\ (\cos\phi \cos\lambda\ d\lambda - \sin\phi \sin\lambda\ d\phi)[/math] [math]dz = r \cos\phi\ d\phi[/math] Square of differentials: [math](dx)^2 = r^2\ (\cos^2\phi \sin^2\lambda\ (d\lambda)^2 + 2 \cos\phi \sin\lambda \sin\phi \cos\lambda\ d\lambda\ d\phi + \sin^2\phi \cos^2\lambda\ (d\phi)^2)[/math] [math](dy)^2 = r^2\ (\cos^2\phi \cos^2\lambda\ (d\lambda)^2 - 2 \cos\phi \cos\lambda \sin\phi \sin\lambda\ d\lambda\ d\phi + \sin^2\phi \sin^2\lambda\ (d\phi)^2)[/math] [math](dz)^2 = r^2\ \cos^2\phi\ (d\phi)^2[/math] The Euclidean metric of the three-dimensional space induces the (non-Euclidean) metric on the two-dimensional sphere: [math](ds)^2 = (dx)^2 + (dy)^2 + (dz)^2[/math] [math]= r^2\ ((\cos^2\phi \sin^2\lambda + \cos^2\phi \cos^2\lambda)\ (d\lambda)^2 + (\sin^2\phi \cos^2\lambda + \sin^2\phi \sin^2\lambda + \cos^2\phi)\ (d\phi)^2)[/math] [math]= r^2\ (\cos^2\phi\ (d\lambda)^2 + (d\phi)^2)[/math] A sphere is a two-dimensional space of constant curvature. One thing to note is that the notion of a curved differential manifold with coordinate system and metric is actually familiar as we are living on such a thing. And we draw maps of such a thing too. A Mercator projection distorts scale but preserves angles, making it a conformal mapping to a flat two-dimensional surface. Thus, a sphere (like all two-dimensional spaces) is a conformally flat space. Conformally flat spacetimes also exist. A de Sitter space is an example. A Friedmann–Lemaître–Robertson–Walker (FLRW) metric with flat three-dimensional space is another example. Conformally flat spacetimes have the property that the Weyl conformal tensor field, the curvature tensor field that describes gravitation away from energy-momentum, is zero. By contrast, the curvature tensor field of the Schwarzschild black hole is entirely of the Weyl type. The point I'm making is that the mathematics of general relativity is quite extensive and also touches upon familiar notions such as for example the Mercator projection of the world.

No. Four-dimensional de Sitter space is defined as a four-dimensional "surface" in five-dimensional Minkowskian spacetime that is a constant distance from the origin, with the Minkowskian metric of the five-dimensional spacetime inducing a metric on the four-dimensional de Sitter space. From this metric, one directly obtains the cosmological horizon. But also from the metric, one obtains the various curvature fields. A de Sitter space is a space of constant curvature and is therefore a space with a cosmological constant. Thus, we have the relationship between the cosmological constant and the cosmological horizon. The factor of "3" does depend on the dimensionality of the de Sitter space (for n-dimensional de Sitter space, [math]\Lambda = \dfrac{(n-1)(n-2)}{2 \alpha^2}[/math]). No energy is involved. The cosmological constant is on the geometric side of the Einstein field equations. You have it backwards. [math]\beta^2 = \dfrac{GM}{c^2 R} \Big(1 - \dfrac{2GM}{c^2 R}\Big)^{-1}[/math] is the local value of [math]\beta^2[/math]. [math]\beta_\infty^2 = \dfrac{GM}{c^2 R}[/math] is the value observed from infinity.

I'm currently writing replies to other things you've written, but I want to address this now: I know the answer to the Photon sphere question is a radius while the answer to the Lensing question is a deflection angle, but what is the result of inserting the Photon sphere radius [math]1.5 r_s[/math] into the Lensing formula? Have you read the circular orbit derivations in my previous post? How would you derive the radius of the circular orbit for an object moving at [math]\beta = \dfrac{v}{c}[/math]?

I want to revisit the perihelion shift, particularly for circular orbits. Thus, this post will investigate circular trajectories centred around a Schwarzschild black hole. However, to avoid working with Christoffel symbols, I use a formula that I've already mentioned on the forum some time ago, the relationship between an accelerated frame of reference and time dilation. [math]a_\mu = -\dfrac{c^2}{T}\dfrac{\partial T}{\partial x^\mu}[/math] [math]a^\mu = -g^{\mu\nu} \dfrac{c^2}{T}\dfrac{\partial T}{\partial x^\nu}[/math] [math]a = \sqrt{|a_\mu a^\mu|} = \sqrt{|g_{\mu \nu} a^\mu a^\nu|} = \sqrt{|g^{\mu \nu} a_\mu a_\nu|}[/math] [math]= \dfrac{c^2}{T} \sqrt{\Big|g^{\mu \nu}\dfrac{\partial T}{\partial x^\mu}\dfrac{\partial T}{\partial x^\nu}\Big|}[/math] [math]T[/math] is the magnitude of a timelike Killing vector, so this formula assumes a stationary frame of reference. But, note that an object on a circular trajectory centred around a Schwarzschild black hole is a stationary object. However, to make the stationary nature of this object explicit, the Schwarzschild metric is coordinate-transformed to a rotating coordinate system in which the object is at rest. The angular velocity of the object in the original non-rotating coordinate system is [math]\dfrac{d\phi}{dt} = \omega[/math]. [math]\text{Schwarzschild metric:}\ \ \ (ds)^2 = \Big(1 - \dfrac{2GM}{c^2 r}\Big) c^2 (dt)^2 - \Big(1 - \dfrac{2GM}{c^2 r}\Big)^{-1} (dr)^2 - r^2 (d\theta)^2 - r^2 \sin^2\theta (d\phi)^2[/math] [math]\text{Let:}\ \ \ \phi = \varphi + \omega t\ \ \ ;\ \ \ d\phi = d\varphi + \omega dt[/math] [math](ds)^2 = \Big(1 - \dfrac{2GM}{c^2 r}\Big) c^2 (dt)^2 - \Big(1 - \dfrac{2GM}{c^2 r}\Big)^{-1} (dr)^2 - r^2 (d\theta)^2 - r^2 \sin^2\theta (d\varphi + \omega dt)^2[/math] [math](ds)^2 = \Big(1 - \dfrac{2GM}{c^2 r}\Big) c^2 (dt)^2 - \Big(1 - \dfrac{2GM}{c^2 r}\Big)^{-1} (dr)^2 - r^2 (d\theta)^2 - r^2 \sin^2\theta (d\varphi)^2 - 2\omega r^2 \sin^2\theta\ d\varphi dt - \omega^2 r^2 \sin^2\theta (dt)^2[/math] [math](ds)^2 = \Big(\Big(1 - \dfrac{2GM}{c^2 r}\Big) c^2 - \omega^2 r^2 \sin^2\theta\Big)(dt)^2 - \Big(1 - \dfrac{2GM}{c^2 r}\Big)^{-1} (dr)^2 - r^2 (d\theta)^2 - r^2 \sin^2\theta (d\varphi)^2 - 2\omega r^2 \sin^2\theta\ d\varphi dt[/math] [math](ds)^2 = \Big(c^2 - \dfrac{2GM}{r} - \omega^2 r^2 \sin^2\theta\Big) (dt)^2 - \Big(1 - \dfrac{2GM}{c^2 r}\Big)^{-1} (dr)^2 - r^2 (d\theta)^2 - r^2 \sin^2\theta (d\varphi)^2 - 2\omega r^2 \sin^2\theta\ d\varphi dt[/math] Specifying the location of the object at rest in the rotating coordinate system, the acceleration of the object is derived, noting that in this coordinate system, the magnitude of the Killing vector can be taken to be [math]\sqrt{g_{tt}}[/math], and that this magnitude depends only on the radial coordinate. Thus, the acceleration vector has only the radial non-zero component. Only the covariant form of the vector and the invariant magnitude of the acceleration are provided. [math]\text{For:}\ \ \ r = R\ \ \ ;\ \ \ dr = 0\ \ \ ;\ \ \ \theta = \dfrac{\pi}{2}\ \ \ ;\ \ \ d\theta = 0\ \ \ ;\ \ \ \varphi = 0\ \ \ ;\ \ \ d\varphi = 0[/math] [math](ds)^2 = \Big(c^2 - \dfrac{2GM}{R} - \omega^2 R^2\Big) (dt)^2[/math] [math]-a_r = \dfrac{c^2}{\sqrt{g_{tt}}} \dfrac{\partial \sqrt{g_{tt}}}{\partial r} = \dfrac{1}{2} \dfrac{c^2}{g_{tt}} \dfrac{\partial g_{tt}}{\partial r}[/math] [math]= \dfrac{\dfrac{GM}{R^2} - \omega^2 R}{1 - \dfrac{2GM}{c^2 R} - \dfrac{\omega^2 R^2}{c^2}}[/math] [math]a = \dfrac{|a_r|}{\sqrt{|g_{rr}|}} = \left|\dfrac{\dfrac{GM}{R^2} - \omega^2 R}{1 - \dfrac{2GM}{c^2 R} - \dfrac{\omega^2 R^2}{c^2}}\right| \sqrt{\left|1 - \dfrac{2GM}{c^2 R}\right|}[/math] While the above provides the acceleration of circular trajectories centred around a Schwarzschild black hole, the derivations below are specifically for circular orbits for which the acceleration of the object is zero. Kepler's third law is obtained, specifying the orbital period in terms of time measured in the non-rotating coordinate system, both at infinity and at the orbit radius. Also, the radial coordinate of the orbit is obtained for a lightlike orbit, as well as for timelike orbits specified by the speed of the object as a fraction [math]\beta[/math] of [math]c[/math]. And just as [math]R[/math] is obtained in terms of [math]\beta^2[/math], [math]\beta^2[/math] is obtained in terms of [math]R[/math]. [math]\text{Let:}\ \ \ \omega = \dfrac{2\pi}{T_\infty}\ \ \ \text{where}\ \ T_\infty\ \ \text{is the orbital period observed from}\ \ r = \infty[/math] [math]-a_r = \dfrac{\dfrac{GM}{R^2} - \dfrac{4\pi^2 R}{T_\infty^2}}{1 - \dfrac{2GM}{c^2 R} - \dfrac{4\pi^2 R^2}{c^2 T_\infty^2}}[/math] [math]a_r = 0\ \ \text{for}\ \ \dfrac{GM}{R^2} = \dfrac{4\pi^2 R}{T_\infty^2}[/math] [math]\dfrac{R^3}{T_\infty^2} = \dfrac{GM}{4\pi^2}[/math] [math]T_R^2 = T_\infty^2 \Big(1 - \dfrac{2GM}{c^2 R}\Big)\ \ \ \text{where}\ \ T_R\ \ \text{is the orbital period observed from}\ \ r = R[/math] [math]\dfrac{R^3 \Big(1 - \dfrac{2GM}{c^2 R}\Big)}{T_R^2} = \dfrac{GM}{4\pi^2}[/math] [math]\text{For a lightlike trajectory:}\ \ \ T_R = \dfrac{2\pi R}{c}\ \ \ ;\ \ \ \dfrac{T_R^2}{R^2} = \dfrac{4\pi^2}{c^2}[/math] [math]R^3 - \dfrac{2GM}{c^2} R^2 = \dfrac{GM}{4\pi^2} T_R^2[/math] [math]R = \dfrac{2GM}{c^2} + \dfrac{GM}{4\pi^2} \dfrac{T_R^2}{R^2}[/math] [math]= \dfrac{2GM}{c^2} + \dfrac{GM}{c^2}[/math] [math]= \dfrac{3GM}{c^2}[/math] [math]\text{For a timelike trajectory:}\ \ \ T_R = \dfrac{2\pi R}{c \beta}\ \ \ ;\ \ \ \dfrac{T_R^2}{R^2} = \dfrac{4\pi^2}{c^2 \beta^2}[/math] [math]R = \dfrac{2GM}{c^2} + \dfrac{GM}{4\pi^2} \dfrac{T_R^2}{R^2}[/math] [math]= \dfrac{2GM}{c^2} + \dfrac{GM}{c^2 \beta^2}[/math] [math]= \Big(2 + \dfrac{1}{\beta^2}\Big) \dfrac{GM}{c^2}[/math] [math]\dfrac{1}{\beta^2} = \dfrac{c^2 R}{GM} - 2 = \Big(\dfrac{GM}{c^2 R}\Big)^{-1} \Big(1 - \dfrac{2GM}{c^2 R}\Big)[/math] [math]\beta^2 = \dfrac{GM}{c^2 R} \Big(1 - \dfrac{2GM}{c^2 R}\Big)^{-1}[/math] Finally, the time dilation of the orbit is obtained compared to time at infinity. This comes from the [math]tt[/math]-component of the rotating metric. And because the [math]t[/math]-coordinate was unchanged by the coordinate transformation to the rotating metric, the rotating metric can be directly compared to the non-rotating metric. [math]\text{Also:}\ \ \ 1 - \dfrac{2GM}{c^2 R} - \dfrac{\omega^2 R^2}{c^2} = 1 - \dfrac{2GM}{c^2 R} - \dfrac{4\pi^2 R^2}{c^2 T_\infty^2}[/math] [math]= 1 - \dfrac{2GM}{c^2 R} - \dfrac{GM}{c^2 R}[/math] [math]= 1 - \dfrac{3GM}{c^2 R}[/math] [math]\text{Therefore, the time dilation for an object in a circular orbit:}\ \ \ \dfrac{\Delta t_R}{\Delta t_\infty } = \sqrt{1 - \dfrac{3GM}{c^2 R}}[/math] From this, and comparing to the [math]tt[/math]-component of the non-rotating metric, gravitation contributes [math]\dfrac{2GM}{c^2 R}[/math] to the time dilation, and speed contributes [math]\dfrac{GM}{c^2 R}[/math] to the time dilation. But, we also obtained [math]\beta^2 = \dfrac{GM}{c^2 R} \Big(1 - \dfrac{2GM}{c^2 R}\Big)^{-1}[/math]. That is because speed [math]c \beta[/math] was specified in terms of time at [math]r = R[/math] rather than time at [math]r = \infty[/math], as required by the invariance of the local speed of light, and as observed from [math]r = \infty[/math], it is slower by the time dilation factor [math]\sqrt{1 - \dfrac{2GM}{c^2 R}}[/math]. Therefore, the contribution of [math]\beta^2[/math], which is not time dilated as it is a ratio, to the time dilation of the object is reduced by factor [math]1 - \dfrac{2GM}{c^2 R}[/math], resulting in a contribution of [math]\dfrac{GM}{c^2 R}[/math], in agreement with the time dilation obtained for the object in a circular orbit. Thus, in terms of [math]T_\infty[/math], defining [math]\beta_\infty[/math] accordingly: [math]\text{For a timelike trajectory:}\ \ \ T_\infty = \dfrac{2\pi R}{c \beta_\infty}\ \ \ ;\ \ \ \dfrac{T_\infty^2}{R^2} = \dfrac{4\pi^2}{c^2 \beta_\infty^2}[/math] [math]\dfrac{R^3}{T_\infty^2} = \dfrac{GM}{4\pi^2}[/math] [math]R = \dfrac{GM}{4\pi^2} \dfrac{T_\infty^2}{R^2} = \dfrac{GM}{c^2 \beta_\infty^2}[/math] [math]\beta_\infty^2 = \dfrac{GM}{c^2 R}[/math]

I want to revisit the perihelion shift, particularly for circular orbits. Thus, this post will investigate circular trajectories centred around a Schwarzschild black hole. However, to avoid working with Christoffel symbols, I use a formula that I've already mentioned on the forum some time ago, the relationship between an accelerated frame of reference and time dilation. [math]a_\mu = -\dfrac{c^2}{T}\dfrac{\partial T}{\partial x^\mu}[/math] [math]a^\mu = -g^{\mu\nu} \dfrac{c^2}{T}\dfrac{\partial T}{\partial x^\nu}[/math] [math]a = \sqrt{|a_\mu a^\mu|} = \sqrt{|g_{\mu \nu} a^\mu a^\nu|} = \sqrt{|g^{\mu \nu} a_\mu a_\nu|}[/math] [math]= \dfrac{c^2}{T} \sqrt{\Big|g^{\mu \nu}\dfrac{\partial T}{\partial x^\mu}\dfrac{\partial T}{\partial x^\nu}\Big|}[/math] [math]T[/math] is the magnitude of a timelike Killing vector, so this formula assumes a stationary frame of reference. But, note that an object on a circular trajectory centred around a Schwarzschild black hole is a stationary object. However, to make the stationary nature of this object explicit, the Schwarzschild metric is coordinate-transformed to a rotating coordinate system in which the object is at rest. The angular velocity of the object in the original non-rotating coordinate system is [math]\dfrac{d\phi}{dt} = \omega[/math]. [math]\text{Schwarzschild metric:}\ \ \ (ds)^2 = \Big(1 - \dfrac{2GM}{c^2 r}\Big) c^2 (dt)^2 - \Big(1 - \dfrac{2GM}{c^2 r}\Big)^{-1} (dr)^2 - r^2 (d\theta)^2 - r^2 \sin^2\theta (d\phi)^2[/math] [math]\text{Let:}\ \ \ \phi = \varphi + \omega t\ \ \ ;\ \ \ d\phi = d\varphi + \omega dt[/math] [math](ds)^2 = \Big(1 - \dfrac{2GM}{c^2 r}\Big) c^2 (dt)^2 - \Big(1 - \dfrac{2GM}{c^2 r}\Big)^{-1} (dr)^2 - r^2 (d\theta)^2 - r^2 \sin^2\theta (d\varphi + \omega dt)^2[/math] [math](ds)^2 = \Big(1 - \dfrac{2GM}{c^2 r}\Big) c^2 (dt)^2 - \Big(1 - \dfrac{2GM}{c^2 r}\Big)^{-1} (dr)^2 - r^2 (d\theta)^2 - r^2 \sin^2\theta (d\varphi)^2 - 2\omega r^2 \sin^2\theta\ d\varphi dt - \omega^2 r^2 \sin^2\theta (dt)^2[/math] [math](ds)^2 = \Big(\Big(1 - \dfrac{2GM}{c^2 r}\Big) c^2 - \omega^2 r^2 \sin^2\theta\Big)(dt)^2 - \Big(1 - \dfrac{2GM}{c^2 r}\Big)^{-1} (dr)^2 - r^2 (d\theta)^2 - r^2 \sin^2\theta (d\varphi)^2 - 2\omega r^2 \sin^2\theta\ d\varphi dt[/math] [math](ds)^2 = \Big(c^2 - \dfrac{2GM}{r} - \omega^2 r^2 \sin^2\theta\Big) (dt)^2 - \Big(1 - \dfrac{2GM}{c^2 r}\Big)^{-1} (dr)^2 - r^2 (d\theta)^2 - r^2 \sin^2\theta (d\varphi)^2 - 2\omega r^2 \sin^2\theta\ d\varphi dt[/math] Specifying the location of the object at rest in the rotating coordinate system, the acceleration of the object is determined, noting that in this coordinate system, the magnitude of the Killing vector can be taken to be [math]\sqrt{g_{tt}}[/math], and that this magnitude depends only on the radial component. Thus, the acceleration vector has only the radial non-zero component. Only the covariant form of the vector and the invariant magnitude of the acceleration are determined. [math]\text{For:}\ \ \ r = R\ \ \ ;\ \ \ dr = 0\ \ \ ;\ \ \ \theta = \dfrac{\pi}{2}\ \ \ ;\ \ \ d\theta = 0\ \ \ ;\ \ \ \varphi = 0\ \ \ ;\ \ \ d\varphi = 0[/math] [math](ds)^2 = \Big(c^2 - \dfrac{2GM}{R} - \omega^2 R^2\Big) (dt)^2[/math] [math]-a_r = \dfrac{c^2}{\sqrt{g_{tt}}} \dfrac{\partial \sqrt{g_{tt}}}{\partial r} = \dfrac{1}{2} \dfrac{c^2}{g_{tt}} \dfrac{\partial g_{tt}}{\partial r}[/math] [math]= \dfrac{\dfrac{GM}{R^2} - \omega^2 R}{1 - \dfrac{2GM}{c^2 R} - \dfrac{\omega^2 R^2}{c^2}}[/math] [math]a = \dfrac{-a_r}{\sqrt{-g_{rr}}} = \dfrac{\Big(\dfrac{GM}{R^2} - \omega^2 R\Big) \sqrt{1 - \dfrac{2GM}{c^2 R}}}{1 - \dfrac{2GM}{c^2 R} - \dfrac{\omega^2 R^2}{c^2}}\ \ \ \text{for}\ \ R \ge \text{. . . (whatever)}[/math] While the above provides the acceleration of circular trajectories centred around a Schwarzschild black hole, the derivations below are specifically for circular orbits for which the acceleration of the object is zero. Kepler's third law is obtained, specifying the orbital period in terms of time measured in the non-rotating coordinate system, both at infinity and at the orbit radius. Also, the radial coordinate of the orbit is obtained for a lightlike orbit, as well as for timelike orbits specified by the speed of the object as a fraction [math]\beta[/math] of [math]c[/math]. And just as [math]R[/math] is obtained in terms of [math]\beta^2[/math], [math]\beta^2[/math] is obtained in terms of [math]R[/math]. [math]\text{Let:}\ \ \ \omega = \dfrac{2\pi}{T_\infty}\ \ \ \text{where}\ \ T_\infty\ \ \text{is the orbital period observed from}\ \ r = \infty[/math] [math]-a_r = \dfrac{\dfrac{GM}{R^2} - \dfrac{4\pi^2 R}{T_\infty^2}}{1 - \dfrac{2GM}{c^2 R} - \dfrac{4\pi^2 R^2}{c^2 T_\infty^2}}[/math] [math]a_r = 0\ \ \text{for}\ \ \dfrac{GM}{R^2} = \dfrac{4\pi^2 R}{T_\infty^2}[/math] [math]\dfrac{R^3}{T_\infty^2} = \dfrac{GM}{4\pi^2}[/math] [math]T_R^2 = T_\infty^2 \Big(1 - \dfrac{2GM}{c^2 R}\Big)\ \ \ \text{where}\ \ T_R\ \ \text{is the orbital period observed from}\ \ r = R[/math] [math]\dfrac{R^3 \Big(1 - \dfrac{2GM}{c^2 R}\Big)}{T_R^2} = \dfrac{GM}{4\pi^2}[/math] [math]\text{For a lightlike trajectory:}\ \ \ T_R = \dfrac{2\pi R}{c}\ \ \ ;\ \ \ \dfrac{T_R^2}{R^2} = \dfrac{4\pi^2}{c^2}[/math] [math]R^3 - \dfrac{2GM}{c^2} R^2 = \dfrac{GM}{4\pi^2} T_R^2[/math] [math]R = \dfrac{2GM}{c^2} + \dfrac{GM}{4\pi^2} \dfrac{T_R^2}{R^2}[/math] [math]= \dfrac{2GM}{c^2} + \dfrac{GM}{c^2}[/math] [math]= \dfrac{3GM}{c^2}[/math] [math]\text{For a timelike trajectory:}\ \ \ T_R = \dfrac{2\pi R}{c \beta}\ \ \ ;\ \ \ \dfrac{T_R^2}{R^2} = \dfrac{4\pi^2}{c^2 \beta^2}[/math] [math]R = \dfrac{2GM}{c^2} + \dfrac{GM}{4\pi^2} \dfrac{T_R^2}{R^2}[/math] [math]= \dfrac{2GM}{c^2} + \dfrac{GM}{c^2 \beta^2}[/math] [math]= \Big(2 + \dfrac{1}{\beta^2}\Big) \dfrac{GM}{c^2}[/math] [math]\dfrac{1}{\beta^2} = \dfrac{c^2 R}{GM} - 2 = \Big(\dfrac{GM}{c^2 R}\Big)^{-1} \Big(1 - \dfrac{2GM}{c^2 R}\Big)[/math] [math]\beta^2 = \dfrac{GM}{c^2 R} \Big(1 - \dfrac{2GM}{c^2 R}\Big)^{-1}[/math] Finally, the time dilation of the orbit is obtained compared to time at infinity. This comes from the [math]tt[/math]-component of the rotating metric. And because the [math]t[/math]-coordinate was unchanged by the coordinate transformation to the rotating metric, the rotating metric can be directly compared to the non-rotating metric. [math]\text{Also:}\ \ \ 1 - \dfrac{2GM}{c^2 R} - \dfrac{\omega^2 R^2}{c^2} = 1 - \dfrac{2GM}{c^2 R} - \dfrac{4\pi^2 R^2}{c^2 T_\infty^2}[/math] [math]= 1 - \dfrac{2GM}{c^2 R} - \dfrac{GM}{c^2 R}[/math] [math]= 1 - \dfrac{3GM}{c^2 R}[/math] [math]\text{Therefore, the time dilation for an object in a circular orbit:}\ \ \ \dfrac{\Delta t_R}{\Delta t_\infty } = \sqrt{1 - \dfrac{3GM}{c^2 R}}[/math] From this, and comparing to the [math]tt[/math]-component of the non-rotating metric, gravitation contributes [math]\dfrac{2GM}{c^2}[/math] to the time dilation, and speed contributes [math]\dfrac{GM}{c^2}[/math] to the time dilation. But, we also obtained [math]\beta^2 = \dfrac{GM}{c^2 R} \Big(1 - \dfrac{2GM}{c^2 R}\Big)^{-1}[/math]. That is because speed [math]c \beta[/math] was specified in terms of time at [math]r = R[/math] rather than time at [math]r = \infty[/math], as required by the invariance of the local speed of light, and as observed from [math]r = \infty[/math], it is slower by the time dilation factor [math]\sqrt{1 - \dfrac{2GM}{c^2}}[/math]. Therefore, the contribution of [math]\beta^2[/math], which is not time dilated as it is a ratio, to the time dilation of the object is reduced by factor [math]1 - \dfrac{2GM}{c^2 R}[/math], resulting in a contribution of [math]\dfrac{GM}{c^2}[/math], in agreement with the time dilation obtained for the object in a circular orbit.

That's a reasonable assessment. However, I see it as a way to get around the unprovability of physics to utilise the provability of mathematics. It's not perfect, the weakness is the limitation of the mathematics to describe physical reality. You might say that is the point you are making to me, but this is as much a question about physical reality as it is about mathematics. For example, can quantum theory be described with the same mathematics as general relativity? Nevertheless, it is my answer to the question of "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" that by producing a mathematical description of physical reality, it is only natural that the mathematical properties of descriptions translate to observed properties of physical reality. The problem with this is that, as suggested above, you do not know the ontology of reality. You have suggested that observed reality is a projection of a hidden reality that your theory uncovers, and that this hidden reality is the true ontology. But I would say that physical reality itself does not have mathematical properties or logic, and that strictly speaking, these can only be applied to mathematical descriptions. Physical reality is what it is, nothing more, nothing less. And because what you are describing is hidden, that description doesn't correspond to observed reality. In other words, you are overstating what you know about physical reality. That's not to say that mathematical descriptions have to correspond directly to observed reality. The power of mathematics is its ability to transform descriptions into some other form where obscure details can be revealed. Even the notion of "mathematical properties" of descriptions may involve substantial mathematical processing of a description to reveal. Because of this, the mathematics associated with the descriptions of physical reality can't really be said to be the ontology of reality. For example, suppose one has a waveform over some region of spacetime. A Fourier transform of that waveform creates a description in the domain of wavenumber and frequency. Does that mean the ontology of physical reality is the domain of wavenumber and frequency? Can we even say that the ontology of physical reality is the domain of spacetime? What about a fractional Fourier transform? To treat the observed physical reality as if it were a projection of a higher hidden reality is not necessarily wrong if this reveals a mathematical property of the observed physical reality, and one is not regarding the higher hidden reality as the true ontology. For example, one might choose to embed four-dimensional curved spacetime into a higher-dimensional flat spacetime (I believe it requires ten dimensions to embed an arbitrarily curved four-dimensional spacetime, though I've not seen a proof of this). Why would one do this? Because dealing with flat spaces is less problematic than dealing with curved spaces. Maybe. The mathematics is about extracting knowledge about the observable reality, not about describing an ontology. That's straightforward: When you're doing mathematics, you're doing mathematics, and when you're performing experiments, making observations, taking measurements, you're doing physics. But how do you know that its not just "when holding a hammer everything around looks like a nail"? By how effective the mathematical descriptions are at describing reality. On the other hand, a differentiable manifold might not be effective at describing reality at extremely small scale. That is, the best mathematical tools are those that match the physical reality being described. This is not entirely unlike me choosing to consider mathematical descriptions instead of physical reality itself as the notion to be studied. And we encounter more-or-less the same problem when we choose the mathematics to start from. However, I feel that choosing a mathematical notion that looks like physical reality is a good place to start. If you had said that RG augments a differentiable manifold, I could maybe accept this. But I don't think there is enough detail in RG to actually replace a differentiable manifold in real-world problems. Do you deny that an experimental physicist can set up a coordinate grid in the laboratory? Surely, gravitational lensing and the photon sphere are based on the same principle? So why would gravitational lensing involve [math]2\kappa^2[/math], and the photon sphere involve only [math]\kappa^2[/math]? Also, for gravitational lensing, you said [math]\beta^2 = 1[/math], whereas for the photon sphere, you said [math]\beta^2 = \dfrac{1}{3}[/math]? One could extricate density from the energy-momentum tensor so that: [math]T_{\mu\nu} = \rho c^2 U_{\mu\nu}[/math] where [math]\rho[/math] is the mass part of the energy-momentum tensor, and [math]U_{\mu\nu}[/math] is the kinetic part of the energy-momentum tensor. Whereas [math]\rho[/math] would determine the overall strength of the gravitation, [math]U_{\mu\nu}[/math] would be about the directional aspect of the gravitation, which seems to be less important to you. According to the Wikipedia article, "De Sitter space": [math]\Lambda = \dfrac{3}{\alpha^2}[/math] where [math]\alpha[/math] is the cosmological horizon.

What do I think? I think this belongs in the Trash Can. There is nothing in the above that is worthy of discussion. It seems likely to me that you are just taking the piss.

[math]x \overline{x} x[/math] [math]x \underline{x} x[/math] [math]x \buildrel \rm abc \over {xyz} x[/math] [math]x \buildrel \rm {abc} \over {xyz} x[/math] [math]x \buildrel \rm abc \under {xyz} x[/math] [math]x \buildrel \rm {abc} \under {xyz} x[/math]

I deliberately chose not to use the term "differentiable manifold". That term gives the subject a topological viewpoint. No, this is about analysis. It is an extension of the mathematics that is taught at school, so it is at the foundations of mathematics. You might argue that pedagogical foundations are different to logical foundations, but is it really? I'm starting from the Cartesian [math]n[/math]-th power of the real number line (or of the complex number plane) [math]\mathbb{R}^n[/math] (or [math]\mathbb{C}^n[/math]), defining functions over that domain similar to the way a school student would recognise [math]f(x)[/math], except that I'm considering [math]f(x^1,...,x^n)[/math], a function of [math]n[/math] variables instead of one, identifying each of the [math]x^p[/math] with each of the real number lines of [math]\mathbb{R}^n[/math] (or each of the complex number planes of [math]\mathbb{C}^n[/math]) using superscript indices [math]p = 1\ \text{to}\ n[/math]. And just as function [math]f(x)[/math] returns a single number from the domain of [math]\mathbb{R}[/math] (or [math]\mathbb{C}[/math]), a function [math]T^{p_1,...,p_r}_{q_1,...,q_s}(x^1,...,x^n)[/math] can return [math]n^{r+s}[/math] values from the domain of [math]\mathbb{R}^{n^{r+s}}[/math] (or [math]\mathbb{C}^{n^{r+s}}[/math]). With the introduction of the notion of limits, calculus can be performed. But note that by considering the [math]n[/math] partial derivatives of [math]f(x^1,...,x^n)[/math]: [math]\dfrac{\partial f(x^1,...,x^n)}{\partial x^p}[/math] for [math]p = 1\ \text{to}\ n[/math] one has naturally introduced a function that returns a vector. So really, I could've introduced limits (calculus) at the [math]f(x^1,...,x^n)[/math] stage and allow functions that return higher level objects to define themselves naturally. So, by using the term "differentiable manifold", you've made what I said above seem more advanced and esoteric than it really is. I don't think there really is "one step before that assumption". You have introduced a "Topological Requirement" in your logical flow diagram, suggesting to me that you believe Topology is a foundational subject in mathematics, and that because your topology has less structure than a differentiable manifold, that your theory is closer to first principles than the way I described the foundations of general relativity. But I personally regard mathematics as the manipulation of symbols that are defined by symbolic expressions called axioms, and that a theorem can only be considered to be rigorously proven if it can be done mechanically. This is an ideal that I have yet to achieve, though I have learnt a lot from my attempts. Thus, to me, mathematics is like a tree where the theories that are higher up rely on theories that are below it but not on the theories that are above it. But because of the symbolic nature of mathematics, the foundations of mathematics (the trunk of the tree) is based on the logic of symbol manipulation rather than what the symbols may represent. Please explain how you get from RC to a "differentiable manifold". The way I see it, the foundations of reality are not purely mathematical. I ask what can be assumed to be true without any prior knowledge of how reality behaves? The fundamental answer to this question is that reality can be mathematically described and that any knowledge gained about the properties of mathematical descriptions translate to the properties of reality. And because reality looks like a "differentiable manifold", this can form the basis of a mathematical description of reality. I don't think I do that. Please explain why you think I do that. Doesn't that disagree with the photon sphere [math]\dfrac{3}{2}\kappa^2[/math] ? What is [math]\rho/\rho_{\max}[/math] referring to? Also, GR has the energy-momentum tensor [math]T_{\mu\nu}[/math] (also known as the stress-energy tensor), which has ten independent components. Why are you ignoring nine of the independent components? In what way is [math]r_s/r[/math] representative of GR beyond the Schwarzschild metric?

I see RG as a somewhat abstract theory that seems to coincidently exploit a particular simplicity of the Schwarzschild metric in coordinates for which the radial coordinate maintains a Euclidean spherical surface area (there is a tendency for particular formulae to agree with Newtonian theory as a result). It's not clear to me if general relativity was used during the derivation of RG, even though you do use the Schwarzschild radius. I also note that according to the virial theorem under Newtonian gravitation: 2<T> = –<V> where <T> is the time average of the total kinetic energy of all the particles, and <V> is the time average of the total potential energy of all the particles. Apart from the minus sign, it seems to correspond with your [math]2\beta^2 = \kappa^2[/math] noting that [math]\beta^2[/math] is kinetic and [math]\kappa^2[/math] is gravitational. I would like a detailed explanation of: Gravitational Lensing [math]\alpha = 2\kappa^2[/math] Overall, it seems to me that your theory would be too simple to handle more difficult problems in general relativity. However, I am still unable to figure out the rationale behind your formulae. I feel you need to ground your theory on a more well-known basis instead of what seems to me to be a little too esoteric.

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Ok, here's how I personally see it: We start from an n-tuple of real-numbered (or perhaps complex-numbered) variables, typically [math](x^1,...,x^n)[/math]. This is a coordinate system. Over this coordinate system are various functions of the coordinate system variables. These functions can be scalar fields, components of vector fields, or object fields in general. Also, we can set up another n-tuple of variables, say [math](y^1,...,y^n)[/math] such that each of the [math]y^p[/math] are differentiable functions of all of the [math]x^q[/math] and this system of functions can be inverted so that each of the [math]x^r[/math] are differentiable functions of all of the [math]y^s[/math]. The functions of one set of coordinate variables in terms of another set of coordinate variables is a coordinate transformation. Under a coordinate transformation, the object field functions of the [math](x^1,...,x^n)[/math] variables are transformed to the corresponding object field functions of the [math](y^1,...,y^n)[/math] by the composition of functions, for example: [math]f(x^1,...,x^n) = f(x^1(y^1,...,y^n),...,x^n(y^1,...,y^n)) = \bar{f}(y^1,...,y^n)[/math] as well as some coordinate transformation law in accordance with the mathematical properties of the object field. Implicit in the notion of a coordinate transformation is that [math](y^1(x^1,...,x^n),...,y^n(x^1,...,x^n))[/math] is the same location as [math](x^1,...,x^n)[/math] even though [math](y^1,...,y^n)[/math] may be numerically different to [math](x^1,...,x^n)[/math]. Also implicit in the notion of a coordinate transformation is that an object field in one coordinate system is the same object field in the other coordinate system, even though they may numerically differ. The idea is that I can use the above to create a mathematical description of physical reality, and as a mathematical description, it has mathematical properties that can be determined mathematically and can be correlated back to the physical reality. While one might wish to create a mathematical description of the actual physical reality, I'm more interested in mathematical descriptions of hypothetical arbitrary realities, logically deriving "laws of physics". It is only natural that because physical reality looks like a space over which a coordinate system can be applied, that the above is the appropriate formalism to mathematically describe physical reality. At this point, there is no metric. Because physical reality has no coordinate system, and because the above coordinate systems are arbitrary, the laws of physics must be independent of any coordinate system and therefore be expressed as tensor equations. However, the partial derivative of a tensor is not a tensor. Thus, one must introduce a connection object field, also not a tensor, that converts a partial derivative to a covariant derivative, which is a tensor. The coordinate transformation law of the connection object field compensates for the non-tensor coordinate transformation law of the partial derivative. If one attempts to coordinate-transform the connection object field to zero, so that the covariant derivative is equal to the partial derivative, then this is only possible if a particular expression in terms of the connection object field is zero. This particular expression defines the Riemann curvature tensor field. What we have now is the existence of non-equivalent mathematical descriptions, identifiable by the connection object field as well as the Riemann curvature tensor field. If one makes a couple of assumptions (which I won't discuss because they go beyond standard general relativity), the metric tensor can be introduced such that the connection object field can be expressed in terms of the metric tensor and its first-order partial derivatives. The metric tensor introduces the notion of magnitude to the coordinate system. It also identifies non-equivalent mathematical descriptions. It is in this sense that one can say that the metric "implicitly defines the coordinate system and the 'underlying 4D manifold' itself, providing structure". That is, although the metric is expressed in terms of the coordinate system, in doing so it implicitly defines the coordinate system, as well as the "underlying 4D manifold" due to its identification of non-equivalent mathematical descriptions.

Two things seem to enforce causality within relativity: (1): The 3+1 metric signature ensures that the solution of the electromagnetic wave equation allows only spacetime regions on and inside the lightcone to be affected by a past location. Unlike a 2+2 or 4+0 metric signature, a 3+1 metric signature distinguishes space and time, with space always being the 3 and time always being the 1, in accordance with the nature of lightcones. (2): Future-directed timelike or lightlike energy-momentum vectors always sum to future-directed timelike or lightlike energy-momentum vectors. The presence of any energy-momentum vectors not so restricted will allow arbitrarily directed energy-momentum vectors to exist. In addition to the above, transition probabilities are not time-reversible with the resulting cause and effect being statistically valid only in the future direction.