raapjeqp

2022-10-22

Dark energy physically can be interpreted as either a fluid with positive mass but pressure the negative of its density (pressure has units of energy/volume, and energy is mass), or a property of space. If it's a fluid, it should add to the mass of black holes like any form of energy (no hair), and the black hole should grow? However, if dark energy is a property of space, then this won't happen. Is my reasoning correct that we can differentiate (in theory) by looking at black hole's growth rate?

fjaldangi

Beginner2022-10-23Added 9 answers

No, it is not possible to differentiate between these two interpretations because they're ultimately physically equivalent.

First, we should separate the discussions in cases where the total energy is conserved and where it isn't. The existence of conserved energy in general relativity (which must be ADM-like) actually requires vanishing or negative cosmological constant – the spacetime is Minkowski or AdS. In de Sitter space, there's no nonzero gauge-invariant definition of energy that would be conserved because the de Sitter space has no asymptotic region at infinity.

In the de Sitter space, masses of objects may therefore change in various general ways and by measuring them, you can't deduce pretty much anything.

In Minkowski or AdS space, the energy is conserved. Let's consider an anti de Sitter space with a negative cosmological constant. This means $\rho <0$, a negative energy density, with a positive pressure $p=-\rho >0$. The energy of the mass that ends up as the black hole is conserved, it's the total energy in the spacetime, assuming that everything collapses. However, the value of this total mass/energy is given before the black hole is formed – it stays the same by the conservation law – which means that we can't deduce anything new if we measure the same value at the end.

What you really want to do is to "attribute" or "divide" the total mass/energy of the black hole into different regions – either the generic black hole interior or the singularity. But this "attribution" or "localization" of matter is exactly what is impossible according to general relativity. The conserved total mass/energy cannot be written as an integral of a well-defined energy density. Such a thing may only be written in the "Newtonian" limit of weak gravitational fields and the existence of black hole is exactly the opposite situation in which the "weak fields" condition is dramatically violated.

So no, your verbal descriptions of the situations are just heuristic and to see what actually happens, you need to discuss things quantitatively, using the right concepts suggested by general relativity and using the right equations. The (Einstein's) equations say a very clear thing about the impact of cosmological constant in the absence of black holes much like in their presence and any idea about "two possibilities" (the cosmological constant is a property of space or a form of energy) is a mere illusion, an artifact of non-quantitative thinking about the problem.

First, we should separate the discussions in cases where the total energy is conserved and where it isn't. The existence of conserved energy in general relativity (which must be ADM-like) actually requires vanishing or negative cosmological constant – the spacetime is Minkowski or AdS. In de Sitter space, there's no nonzero gauge-invariant definition of energy that would be conserved because the de Sitter space has no asymptotic region at infinity.

In the de Sitter space, masses of objects may therefore change in various general ways and by measuring them, you can't deduce pretty much anything.

In Minkowski or AdS space, the energy is conserved. Let's consider an anti de Sitter space with a negative cosmological constant. This means $\rho <0$, a negative energy density, with a positive pressure $p=-\rho >0$. The energy of the mass that ends up as the black hole is conserved, it's the total energy in the spacetime, assuming that everything collapses. However, the value of this total mass/energy is given before the black hole is formed – it stays the same by the conservation law – which means that we can't deduce anything new if we measure the same value at the end.

What you really want to do is to "attribute" or "divide" the total mass/energy of the black hole into different regions – either the generic black hole interior or the singularity. But this "attribution" or "localization" of matter is exactly what is impossible according to general relativity. The conserved total mass/energy cannot be written as an integral of a well-defined energy density. Such a thing may only be written in the "Newtonian" limit of weak gravitational fields and the existence of black hole is exactly the opposite situation in which the "weak fields" condition is dramatically violated.

So no, your verbal descriptions of the situations are just heuristic and to see what actually happens, you need to discuss things quantitatively, using the right concepts suggested by general relativity and using the right equations. The (Einstein's) equations say a very clear thing about the impact of cosmological constant in the absence of black holes much like in their presence and any idea about "two possibilities" (the cosmological constant is a property of space or a form of energy) is a mere illusion, an artifact of non-quantitative thinking about the problem.

What makes the planets rotate around the sun.

What is the distance from the Earth's center to a point in space where the gravitational acceleration due to the Earth is 1/24 of its value at the Earth's surface.

What is proper motion.

Why is the image formed in a pinhole camera is inverted?

Light travels at a very high speed

Due to rectilinear propagation of light

Light can reflect from a surface

Screen of the pinhole camera is invertedIn order to calculate the cross-section of an interaction process the following formula is often used for first approximations:

$\sigma =\frac{2\pi}{\hslash \phantom{\rule{thinmathspace}{0ex}}{v}_{i}}{\left|{M}_{fi}\right|}^{2}\varrho \left({E}_{f}\right)\phantom{\rule{thinmathspace}{0ex}}V$

${M}_{fi}=\u27e8{\psi}_{f}|{H}_{int}|{\psi}_{i}\u27e9$

${M}_{fi}=\u27e8{\psi}_{f}|{H}_{int}|{\psi}_{i}\u27e9$

$\varrho \left({E}_{f}\right)=\frac{\mathrm{d}n\left({E}_{f}\right)}{\mathrm{d}{E}_{f}}=\frac{4\pi {{p}_{f}}^{2}}{{\left(2\pi \hslash \right)}^{3}}\frac{V}{{v}_{f}}$

The derivation of this equation in the context of the non relativistic Schrödinger equation. Use this formula in the relativistic limit: ${v}_{i},{v}_{f}\to c\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{1em}{0ex}}{p}_{f}\approx {E}_{f}/c$

Very often books simply use this equation with matrix element derived from some relativistic theory, e.g. coupling factors and propagators from the Dirac equation or Electroweak interaction. How is this justified?Why does the azimuthal angle, $\varphi $, remain unchanged between reference frames in special relativity?

I think this comes from the aberration formula, showing dependence only on the polar angle, $\theta $.

The aberration formula is:

$\mathrm{tan}\theta =\frac{{u}_{\perp}}{{u}_{||}}=\frac{{u}^{\prime}\mathrm{sin}{\theta}^{\prime}}{\gamma ({u}^{\prime}\mathrm{cos}{\theta}^{\prime}+v)}$

where ' indicates a property of the moving frame.For $n$ dimensions, $n+1$ reference points are sufficient to fully define a reference frame. I just want the above line explanation.

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If time in systems moving with different speed goes differently, does speed of entropy change differ in these systems?

If a rocket ship is traveling at .99c for 1 year, and is streaming a video at 30 frames/sec to earth, how would the earth feed be affected? Would it show the video at a much slower rate, would it remain constant, or would it be sped up?

Suppose we want to construct a wave function for 2 free (relativistic) fermions. As we are dealing with fermions the total wave function has to be antisymmetric under interchange of the coordinates,Suppose we want to construct a wave function for 2 free (relativistic) fermions. As we are dealing with fermions the total wave function has to be antisymmetric under interchange of the coordinates,

$\mathrm{\Psi}({x}_{1},{x}_{2})=-\mathrm{\Psi}({x}_{2},{x}_{1})$

If we assume that we can factorize the wave function in terms of single particle wave functions we can write

$\mathrm{\Psi}({x}_{1},{x}_{2})={\psi}_{1}({x}_{1}){\psi}_{2}({x}_{2})-{\psi}_{1}({x}_{1}){\psi}_{2}({x}_{2})$

which fulfills the anti-symmetry requirement. The plane wave single particle states are given by,

${\psi}_{\mathbf{k},{m}_{s}}(x)={u}_{\mathbf{k},{m}_{s}}(s)\varphi (\mathbf{k}\cdot \mathbf{r})$

So expect the total wavefunction to be

$\begin{array}{rl}\mathrm{\Psi}({x}_{1},{x}_{2})& ={u}_{{\mathbf{k}}_{1},{m}_{{s}_{1}}}({s}_{1})\varphi ({\mathbf{k}}_{1}\cdot {\mathbf{r}}_{1}){u}_{{\mathbf{k}}_{2},{m}_{{s}_{2}}}({s}_{2})\varphi ({\mathbf{k}}_{2}\cdot {\mathbf{r}}_{2})-{u}_{{\mathbf{k}}_{1},{m}_{{s}_{1}}}({s}_{2})\varphi ({\mathbf{k}}_{1}\cdot {\mathbf{r}}_{2}){u}_{{\mathbf{k}}_{2},{m}_{{s}_{2}}}({s}_{1})\varphi ({\mathbf{k}}_{2}\cdot {\mathbf{r}}_{1})\\ & ={u}_{{\mathbf{k}}_{1},{m}_{{s}_{1}}}({s}_{1}){u}_{{\mathbf{k}}_{2},{m}_{{s}_{2}}}({s}_{2})\varphi ({\mathbf{k}}_{1}\cdot {\mathbf{r}}_{1})\varphi ({\mathbf{k}}_{2}\cdot {\mathbf{r}}_{2})-{u}_{{\mathbf{k}}_{1},{m}_{{s}_{1}}}({s}_{2}){u}_{{\mathbf{k}}_{2},{m}_{{s}_{2}}}({s}_{1})\varphi ({\mathbf{k}}_{1}\cdot {\mathbf{r}}_{2})\varphi ({\mathbf{k}}_{2}\cdot {\mathbf{r}}_{1})\end{array}$

However

$u({\mathbf{k}}_{1},{m}_{{s}_{1}})u({\mathbf{k}}_{2},{m}_{{s}_{2}})\varphi ({\mathbf{k}}_{1}\cdot {\mathbf{r}}_{1})\varphi ({\mathbf{k}}_{2}\cdot {\mathbf{r}}_{2})-u({\mathbf{k}}_{2},{m}_{{s}_{2}})u({\mathbf{k}}_{1},{m}_{{s}_{1}})\varphi ({\mathbf{k}}_{1}\cdot {\mathbf{r}}_{2})\varphi ({\mathbf{k}}_{2}\cdot {\mathbf{r}}_{1})$

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$d{s}^{2}={g}_{\mu \nu}d{x}^{\mu}d{x}^{\nu},$

How to calculate $ds$?