We don't often see an entire sequence of own goals being celebrated with such exuberance.
Oooh, Clinger is slinging mud. I like it when people who have no counterargument do that. It just makes them look stupid and vindictive and bitter. And yawn, Dopa is boring his imaginary audience to death again.
It isn't a question of believing Einstein.
It isn't about
believing Einstein. It's about
understanding Einstein,
working through the mathematical consequences of his theory of general relativity, and
comparing those mathematical consequences to experiment.
Farsight has done none of those things.
Throughout this thread,
Farsight has been denying what Einstein called "the general postulate of relativity". In what follows, I'll quote from Einstein's 1916 paper on "The Foundation of the General Theory of Relativity", highlighting some of Einstein's words that
Farsight's been ignoring or denying.
Albert Einstein said:
The general laws of nature are to be expressed by equations which hold good for all systems of co-ordinates, that is, are co-variant with respect to any substitutions whatever (generally covariant).
It is clear that a physical theory which satisfies this postulate will also be suitable for the general postulate of relativity.
...snip...
Having seen in the foregoing that the general postulate of relativity leads to the requirement that the equations of physics shall be covariant in the face of any substitution of the co-ordinates x1, . . . x4, we have to consider how such generally covariant equations can be found. We now turn to this purely mathematical task, and we shall find that in its solution a fundamental role is played by the invariant ds given in equation (3), which, borrowing from Gauss's theory of surfaces, we have called the "linear element."
Einstein's equation (3) is
Albert Einstein said:
[latex]
\[
ds^2 = \sum_{\tau, \sigma} g_{\sigma\tau} dx_{\sigma} dx_{\tau}
\hbox{\ \ \ \ (3)}
\]
[/latex]
where the
gστ will be functions of the
xσ.
In section 8, Einstein calls that invariant expression the "covariant fundamental tensor". Nowadays we call it the metric tensor. (If we're unusually precise with our language, we call it the pseudo-metric tensor field.) In section 4, Einstein says "we shall hold fast to the view" that the components of this fundamental metric tensor "describe the gravitational field".
In section 6, Einstein stated a general law that explains how the coordinate-dependent components of any covariant tensor (such as the fundamental metric tensor
gστ) are transformed when we change from one (unprimed) coordinate system to another (primed) coordinate system:
Albert Einstein said:
[latex]
\[
A^\prime_{\sigma\tau}
= \frac{\partial x_{\mu}}{\partial x^\prime_{\sigma}}
\frac{\partial x_{\nu}}{\partial x^\prime_{\tau}}
A_{\mu\nu}
\hbox{\ \ \ \ (11)}
\]
[/latex]
In that equation, Einstein's using the
Einstein summation convention, which he introduced in this paper. That notation says there's an implied sum over all indices that appear twice within an expression.
Let's work through an example. In Schwarzschild coordinates t, r, θ, ϕ with MTW notational conventions, the components of the fundamental metric tensor are
[latex]
\[
\begin{align*}
g_{00} &= g_{tt} = - (1 - \beta^2) \\
g_{11} &= g_{rr} = (1 - \beta^2)^{-1} \\
g_{22} &= g_{\theta\theta} = r^2 \\
g_{33} &= g_{\phi\phi} = r^2 \sin^2 \theta
\end{align*}
\]
[/latex]
where
[latex]
\[
\beta = \sqrt{\frac{2m}{r}}
\]
[/latex]
and the other 12 components of the fundamental metric tensor are zero.
According to Einstein, as quoted above, we can transform those Schwarzschild coordinates to a primed coordinate system as follows.
[latex]
\[
\begin{align*}
t^\prime &= t - \int_{r}^{\infty} \frac{\beta}{1 - \beta^2} dr \\
r^\prime &= r \\
\theta^\prime &= \theta \\
\phi^\prime &= \phi \\
\end{align*}
\]
[/latex]
To apply Einstein's equation (11), we'll have to know the partial derivatives of the unprimed (Schwarzschild) coordinates with respect to the primed coordinates. To compute those partial derivatives, we'll need the inverse coordinate transformation (from primed coordinates to Schwarzschild):
[latex]
\[
\begin{align*}
t &= t^\prime + \int_{r}^{\infty} \frac{\beta}{1 - \beta^2} dr \\
r &= r^\prime \\
\theta &= \theta^\prime \\
\phi &= \phi^\prime \\
\end{align*}
\]
[/latex]
Of the 16 partial derivatives needed to apply Einstein's equation (11), all but these 5 are zero:
[latex]
\[
\begin{align*}
\frac{\partial x_0}{\partial x_0^\prime}
&= \frac{\partial t}{\partial t^\prime}
= 1 \\
\frac{\partial x_0}{\partial x_1^\prime}
&= \frac{\partial t}{\partial r^\prime}
= - \frac{\beta}{1 - \beta^2} \\
\frac{\partial x_1}{\partial x_1^\prime}
&= \frac{\partial r}{\partial r^\prime}
= 1 \\
\frac{\partial x_2}{\partial x_2^\prime}
&= \frac{\partial \theta}{\partial \theta^\prime}
= 1 \\
\frac{\partial x_3}{\partial x_3^\prime}
&= \frac{\partial \phi}{\partial \phi^\prime}
= 1
\end{align*}
\]
[/latex]
With those partial derivatives in hand, it's easy to calculate the components of the fundamental metric tensor in the primed coordinate system:
[latex]
\[
\begin{align*}
g^\prime_{00} &=
\frac{\partial x_{\mu}}{\partial t^\prime}
\frac{\partial x_{\nu}}{\partial t^\prime}
g_{\mu\nu}
= g_{00}
= - (1 - \beta^2) \\
g^\prime_{01} &=
\frac{\partial x_{\mu}}{\partial t^\prime}
\frac{\partial x_{\nu}}{\partial r^\prime}
g_{\mu\nu}
=
\frac{\partial t}{\partial t^\prime}
\frac{\partial t}{\partial r^\prime}
g_{00}
= - \frac{\beta}{1 - \beta^2} g_{00}
= \beta \\
g^\prime_{11} &=
\frac{\partial x_{\mu}}{\partial r^\prime}
\frac{\partial x_{\nu}}{\partial r^\prime}
g_{\mu\nu}
=
\frac{\partial t}{\partial r^\prime}
\frac{\partial t}{\partial r^\prime}
g_{00}
+
\frac{\partial r}{\partial r^\prime}
\frac{\partial r}{\partial r^\prime}
g_{11}
= - \frac{\beta^2}{1 - \beta^2} + \frac{1}{1 - \beta^2}
= 1 \\
g^\prime_{22} &= g_{22} = r^2 \\
g^\prime_{33} &= g_{33} = r^2 \sin^2 \theta
\end{align*}
\]
[/latex]
We've seen those components before, in another thread. They're the components of the
Painlevé-Gullstrand metric form.
The primed coordinates above are
the Painlevé-Gullstrand coordinates that prove Farsight wrong.
According to Einstein's general postulate of relativity, Painlevé-Gullstrand coordinates are just as good as Schwarzschild coordinates. According to Einstein, we have just as much right to describe spacetime around a black hole using Painlevé-Gullstrand coordinates as using Schwarzschild coordinates. According to Einstein, every experimental prediction that can be made using Schwarzschild coordinates can also be made using Painlevé-Gullstrand coordinates, and vice versa. According to Einstein, all of those experimental predictions will agree.
Farsight disagrees with Einstein on all of those points.
Other words of Einstein don't "directly refute some of my claims". Don't believe Clinger when he makes that claim, he's just trying to distract you from the scientific evidence.
Farsight will probably object to the mathematics in this post, but the math is Einstein's. I'm just working through a concrete example of Einstein's math.
I know what the theory predicts.
Bare assertion, contradicted by evidence.
I'm no crank. Cranks are the guys who say "Einstein was wrong". Now take a look at which side of the fence you're on. LOL, the irony!
Irony is right. Whenever
Farsight objects to Painlevé-Gullstrand, Lemaître, Eddington-Finkelstein, or Kruskal-Szekeres coordinates, he's saying Einstein was wrong about what Einstein called the general postulate of relativity.
No problem with that. There's empirical evidence that the speed of light varies with gravitational potential, which means the maths of Kruskal-Szekeres coordinates is flawed. It isn't wrong as such, it's flawed in that it presents you with a description that does not match the behaviour of the universe.
Farsight has been unable to cite any examples of experiments that conflict with predictions made using Kruskal-Szekeres coordinates. He has repeatedly referred to experiments for which Kruskal-Szekeres coordinates predict exactly the same results as are predicted by Schwarzschild coordinates:
Yes. I've given it repeatedly. The speed of light varies with gravitational potential just like Einstein said.
Just like Einstein said. Just like Kruskal-Szekeres coordinates say.
But people who are convinced that it's absolutely constant absolutely refuse to accept it.
Farsight absolutely refuses to accept what Einstein wrote.
It's doubtful whether
Farsight even understands what Einstein wrote. Einstein wrote math, which is a language
Farsight does not understand.
But I'm with Einstein on this.
Bare assertion, contradicted by evidence.
You can't move through spacetime. No kidding. You can move through space and plot a worldline in spacetime, but you can't move through spacetime.
Einstein defined what it means to move through spacetime in his section 9: "The Equation of the Geodetic Line. The Motion of a Particle."
It doesn't disprove relativity at all. It's important because people say "relativity tells us X" when they're contradicting what Einstein actually said.
Within this thread, no one has contradicted Einstein more often or more fundamentally than has
Farsight.
I'm no crank. And I know my stuff.
Bare assertions, contradicted by evidence.
