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comment Sensor Temperature with time constant
Migrating per OP request.
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comment Does the 1-D poisson's equation have monotonic potentials if $\rho=\rho(\phi(z))$?
@Anode: no. You evaluated at $\phi = 0$, which is nonsense. Integrating a derivative in $z$ you should evaluate at $z = z_0$ and $z = 0$. Incidentally it also makes your choice of upper bound of integration "$\phi'$" look very bizarre. (Also, at this level it is a problem with calculus, and has relatively little to do with physics.)
Feb
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comment Does the 1-D poisson's equation have monotonic potentials if $\rho=\rho(\phi(z))$?
Let me expand on Qmechanic's comment: $$\int_0^{z_0} \frac{d}{dz}\left( \frac{d\phi}{dz}\right)^2 dz \neq \left(\frac{d\phi}{dz}\right)^2$$ Instead $$\int_0^{z_0} \frac{d}{dz}\left( \frac{d\phi}{dz}\right)^2 dz = \left(\frac{d\phi}{dz}\right)^2 \Big|_0^{z_0}$$The "wrong" expression looks "manifestly positive". The correct one doesn't.
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reviewed Reject What software programs are used to draw physics diagrams, and what are their relative merits?
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comment How is it possible for astronomers to see something 13B light years away?
let us continue this discussion in chat
Dec
5
comment How is it possible for astronomers to see something 13B light years away?
@Alfe: you should really ask a new question :-). But the answer to your questions are, in order: (1) Yes. (2) The problem with explaining ly is that in GR, while there is a notion of absolute time difference (by maximal proper time) between two causally related events, there is no notion of "absolute spatial distance". Spatial distance is only defined relative to fixing what it means by "now" (remember, time is not absolute!) That things are simpler in special relativity is really a happy accident. (3) No problem. (4) No. (5) What about it?
Dec
5
comment How is it possible for astronomers to see something 13B light years away?
@Alfe: the "contradiction" is only in your understanding of general relativity. General covariance does not mean what you think it means, and in particular it does not mean that by attaching a local coordinate system you can pretend you live in Minkowski space and do computations as if you live in Minkowski space.
Dec
5
comment How is it possible for astronomers to see something 13B light years away?
@Alfe: you assumed that the universe is static. Going back to basic noneuclidean geometry in two dimensions, let us consider the following scenario: start with two points, draw a straight (geodesic) line between them. Start the two points moving with the same speed in the direction perpendicular to the line. In the flat Euclidean geometry the distance between the two points remain forever the same. In a hyperbolic (negatively curved) geometry, the distance between the two points get larger and larger. The latter is basically what happens in an expanding universe.