Improved Argument
The Energy ##E## of a fundamental string due to its length ##L## goes like
$$E\sim TL$$
where string tension ##T## is given by
$$T \sim \frac{1}{l_P^2}$$
(Using natural units ##\hbar=c=1## with Planck length ##l_P##)
(e.g. see...
Isn't normal ordering just a "trick" to define the Hamiltonian so that there is no zero point energy? I understand that experiments show that there is such a thing as zero point energy in systems such as cooled atoms for example.
see https://physics.stackexchange.com/a/433824/22307
I understand that strings have a size of roughly the Planck length ##l_P## of ##10^{-35}## m.
If that is the case then one would expect that their mass would be roughly the Planck mass which is an enormous ##10^{19}## GeV.
(Strings that have small spins, like standard model particles, are...
As I understand it energy is conserved in a metric with a time-like Killing vector.
It is well-known that FRW metrics do not have a time-like Killing vector but my above calculations confirm that they do have a conformal time-like Killing vector ##K^\mu=(1,0,0,0)## such that
$$L_K g_{\mu\nu} =...
I calculate ##P \cdot U## at the origin and then calculate its covariant derivative as the particle moves forward in time. By integrating I find a value for ##P \cdot U## at the later time. Does it make sense to do this? Have I unambiguously found an expression for the energy of the particle...
I can't give any reference. I'm just trying to understand this stuff myself.
My question to a GR expert is this: Can an observer fixed at the origin of a curved spacetime calculate the energy of a particle at a distant spacetime position?
It is normalised as the observer stays fixed at the origin of the coordinate system ##(\eta,x,y,z)=(0,0,0,0)##
I try to calculate the energy of the comoving particle at arbitrary time ##\eta## relative to the observer who stays fixed at the origin where ##\eta=0##.
Assume that we have a flat FRW metric expressed in conformal time ##\eta## so that the line element is
$$ds^2=a^2(\eta)(d\eta^2-dx^2-dy^2-dz^2)\tag{1}$$
where ##a=1## at the present time ##\eta=0## and the speed of light ##c=1##.
This metric has the following non-zero Christoffel symbols...
According to Sean Carroll's The Cosmological constant(https://arxiv.org/pdf/astro-ph/0004075.pdf) (Eqn.20) cosmological observations imply that the magnitude of the vacuum energy density in natural units is given by
$$|\rho^{(obs)}_\Lambda|\le (10^{-12}\ \rm{GeV})^4.$$
Does this imply that the...
Does the Friedmann vacuum equation have a linear solution rather than an exponential one?
Using natural units one can write Friedmann's equation for the vacuum as
$$
\begin{eqnarray*}
\left(\frac{\dot a}{a}\right)^2 &=& \frac{8\pi G}{3}\rho_{vac}\\\tag{1}
&=& L^2 \left(\frac{\rho_0}{L^4}\right)...
I was wondering if the [Feynman-Heaviside formula](http://www.feynmanlectures.caltech.edu/II_21.html) for the electric field of a moving charge could be used to write down the force/reaction force between charges ##q_1## and ##q_2## in a Machian purely relational way.
The retarded electric...
Consider an electric dipole consisting of charges ##q## and ##-q##, both of mass ##m##, separated by a distance ##d##.
If the dipole is given an acceleration ##a## perpendicular to its moment the total electric force on it, due to each charge acting on the other, is given approximately by...
Can liquid ammonia NH3 be electolyzed to produce H2 and N2?
I have heard of ammonia cracking using high temperature and a catalyst but I was wondering if electrolysis would be an easier way to produce hydrogen from ammonia. The H2 could be fed into a PEM fuel cell to produce electricity to run...
The Compton wavelength of a particle is given by
$$\lambda=\frac{h}{mc}.$$
One can construct an expression for the energy density ##\rho## of a particle of mass ##m## given by
$$\rho = \frac{mc^2}{\lambda^3}=\frac{m^4 c^5}{h^3}.$$
What is the physical significance of the mass scale ##m## in the...
Could one make a negatively-charged insulator with the extra electrons trapped all the way through its volume by building it up layer by layer with electrons "sprayed" onto each layer as it was constructed?
I guess the electrons would be trapped in empty atomic orbitals within the material - is...
Could one insert the target into the accelerator ring so that any beam particles that don't collide with target nuclei are collected and sent round the accelerator again until they do collide?
The CNO cycle (see https://en.wikipedia.org/wiki/CNO_cycle) is a catalytic fusion reaction that produces energy in stars larger than the sun. It converts four protons into a helium-4 nucleus using a cycle of carbon, nitrogen and oxygen isotopes as catalysts and releases 26.7 MeV of energy mostly...
What do people think of Fulvio Melia's argument for the necessity of "zero active mass" in FRW cosmologies? (i.e. the overall equation of state must be ##\rho+3p=0## at all times)
Here is a link to an interesting lecture video:
Here is a recent paper:
https://arxiv.org/abs/1807.07587
I start by outlining the little I know about the basics of quantum field theory.
The simplest relativistic field theory is described by the Klein-Gordon equation of motion for a scalar field ##\large \phi(\vec{x},t)##:
$$\large \frac{\partial^2\phi}{\partial t^2}-\nabla^2\phi+m^2\phi=0.$$
We...
If the uncertainty in the age of the Universe is ##\Delta t## then the Uncertainty Principle implies that it has an uncertainty in its energy ##\Delta E## given by
$$\Delta E \ \Delta t \sim h.\tag{1}$$
If this energy fluctuation excites the zero-point electromagnetic field of the vacuum then a...
The Dynamical Casimir Effect is the production of real photons from the vacuum in a system where one has moving mirrors (see https://www.technologyreview.com/s/424111/first-observation-of-the-dynamical-casimir-effect/). The frequency of the photons is related to the ratio of the velocity v of...
So where does the Unruh effect energy come from? If the matter in the Earth is stable then it must come from the vacuum itself when it is subjected to a gravitational field. In that case is the law of conservation of energy violated?
According to the Unruh effect an observer who is has an acceleration ##g## will observe the temperature of the vacuum to be
$$T=\frac{\hbar g}{2 \pi c k_B}.$$
According to the equivalence principle the observer should measure the same Unruh temperature if he is sitting on a planet whose surface...
Ok I was wrong. I accept that antiparticles have positive energy. However my initial question still stands. Why are the negative energy modes ignored in the vacuum energy density calculation?
According to [Dark Energy and the Accelerating Universe](https://ned.ipac.caltech.edu/level5/March08/Frieman/Frieman5.html) quantum field theory says that the energy density of the vacuum, ##\rho_{vac}##, should be given by
$$\rho_{vac}=\frac{1}{2}\sum_{\rm...
Heisenberg's uncertainty relation says:
$$\Delta x \Delta p \ge \hbar$$
If we assume a massless quantum object then we have the relationship ##\Delta E = c\Delta p## so that the above uncertainty relationship becomes
$$\Delta E \ge \frac{\hbar c}{\Delta x}.\tag{1}$$
I understand that if we have...