Energy density Definition and 202 Threads

I am reading Peskin-Schroeder's QFT text, and there on pg. 98 Equation (4.56) they derive the expression for the vacuum energy density (relative to the zero of energy set by ##H_0|0\rangle = 0##): $$ \frac{E_0}{\rm{Volume}} = \frac{i\,\sum\text{(all disconnected pieces)}}{(2\pi)^4\,\delta^4(0)}\...

Given the metric $$c^2 d\tau^2 = B(r) c^2 dt^2 - A(r) dr^2 - C(r) r^2 d\Omega^2$$ how would the Einstein field equations be spelled out algebraicly for the energy density and radial and tangent pressures in terms of the unknown functions A, B, and C, while also including a cosmological constant...

\frac{1}{R}\frac{{{d^2}R}}{{d{t^2}}} = - \frac{{4\pi G}}{3}\left[ {{\rho _m} + {\rho _{rad}} + {\rho _\Lambda } + \frac{{3({P_m} + {P_{rad}} + {P_\Lambda })}}{{{c^2}}}} \right] In the Friedmann equation, ρ is the mass density. === https://en.wikipedia.org/wiki/Friedmann_equations They were...

The starting point is the identity $$\left(\frac{\partial u}{\partial T}\right)_n = T\left(\frac{\partial s}{\partial T}\right)_n.$$ I then try to proceed as follows: Integrating both with respect to ##T## after dividing through by ##T##, we find $$ \int_0^T \left(\frac{\partial s}{\partial...

I need the theoretical energy density of inflation for my story. I seem to recall it as an enormous 1095 ergs per cubic centimeter.

Is there a way to independently determine the proportion of dark energy density to total energy density of the universe apart from using 1 -(Ωmatter+Ωdark matter )?

I am confused about how the electric field changes in this problem - is E' = E/Ke=E/2? Is E = V/d a correct usage? When I solve it this way, the answer is incorrect: change in energy density = (1/2)ε(E'2- E2) = (1/2)ε(E2/4 - E2) = (1/2)ε(-3/4)(V/2d)2. What am I doing wrong? Thanks.

I developed three arguments to answer this question. Argument no 2 seems to be wrong, but I cant figure out why. I know one/more of my arguments are flawed. Please be kind to help me figure this out. Argument 1) Since they have same charge on them, the ##E## between them must be same. The one...

The calculation of the vacuum energy density gives us a discrepancy with reality. There should be a mass equivalent of about $10^{96}$ kilograms. I'm wondering if the assumed point-like "structure" of particles could be the cause of this wrong value. Since string theory doesn't assume a...

Hi. I'm not sure where to put this question, it concerns particles, mass-energy equivalence and various things. Classical electromagnetism seems to be as sensible a place as any. There is energy stored in an E field. Energy density (at position r, time t) = \frac{1}{2}...

I have doubts about the wording of the exercise: (1) energy density is ##u=\varepsilon_0 (cB)^2## but since the question asks for mean energy density should I perhaps average over ##cos^2 (\omega t)## (there due to the ##B^2##) and thus use ##<u>=\frac{1}{2}\varepsilon_0 (cB)^2##? (2) it seems...

Hi, The problem I am working on requires me to work out the the pressure on the outer conductor of a coaxial cable due to the current on the inner one. This cable carries a dc current of 5000 Amps on the inner wire of radius 2 cm. The outer cylindrical wire of radius 5cm carries the return...

Can the energy-momentum tensor of matter and energy be cast in terms of energy density of matter and energy, similar to how the energy-momentum tensor of vacuum energy can be cast in terms of the energy density of vacuum energy?

According to the wiki entry on Planck units, https://en.wikipedia.org/wiki/Planck_units, the energy density of the universe, 1.8 × 10−123, is 1/16th the cosmological constant, 2.9 × 10−122. Is there a theoretical reason for this precise relationship?

In Principles of Lasers by Svelto, while deriving the Planck radiation formula, equation 2.2.3 says $$I_{\nu} = \frac {c_0} {4n} \rho_\nu$$ where ##I_\nu## is the spectral intensity at some hole in the cavity wall (energy per time per area per frequency), ##c_0## is the speed of light in...

In ΛCDM, H(t0) = 70km/s/Mpc, Ωd(t0) = 0.3, Ωr(t0) = 0 and ΩΛ(t0) =0.7, so that Ω(t0) = Ωd(t0) + Ωr(t0) + ΩΛ(t0) = 1and the universe is spatially flat. I want to know the t and z when the matter density equal to the vacuum energy density. By total energy density equation, I think Ωd(t) +...

Hi, In Problem 9.12 of Griffiths Introduction to Electrodynamics, 4th edition (Problem 9.11 3rd edition), in the problem, he says that one can calculate the average energy density and Poynting vector as using the formula I don't really understand how to do...

1. If today vacuum and matter contribute 71 % and 29 % to the total energy density of the universe, at what redshift z were they contributing equally? 2. If today vacuum, matter, and radiation contribute 71 %, 29 %, and 0.01% to the total energy density of the universe, at what redshift z were...

There are some universe models where ##\Lambda < 0##. In this case, the energy density of the dark-energy becomes negative. At this point, does it make sense to talk about "negative dark energy density"? Or is it possible to think of this energy as curvature on space-time? Such that, ##\Lambda <...

I am reading an article, which talks about graduated dark energy (gDE) model. In this model, it's assumed that the inertial mass density exhibits power-law dependence to its energy density $$\rho_{inert} = \gamma\rho_0(\frac{\rho}{\rho_0})^{\lambda}$$ Where ##\gamma## and ##\lambda## are real...

The energy density of an EM wave is given as (1/2) ϵ E^2 + (1/(2μ)) B^2. This is derived from the energy density of the electric and magnetic fields of capacitors and inductors, respectively. But why should the energy density of the fields of capacitors and inductors be the same as that of...

How many nanoseconds after the Big Bang was its energy density that achievable in modern particle accelerators?

In Special Relativity I'm given the energy-momentum tensor for a perfect fluid:$$ T^{\mu\nu}=\left(\rho+p\right)U^\mu U^\nu+p\eta^{\mu\nu} $$where ##\rho## is the energy density, ##p## is the pressure, ##U^\mu=\partial x^\mu/\partial\tau## is the four-velocity of the fluid. In the...

If we start with the Lagrangian \begin{equation} \begin{split} \mathcal{L} = & \frac{1}{2}(\partial_\mu \phi)^2 + \frac{1}{2}\mu^2 \phi^2 - \frac{1}{4}\lambda^2 \phi^4\\ \end{split} \end{equation} and give the scalar field a VEV so that we can define the field ##\eta##, where $$\eta = \phi...

V(ρ) = V_o*ln(ρ/0.0018)/ln(45/180) (Attached picture is where the unit vector of r is really ρ.) In cylindrical coordinates ∇V = ρ*dV/dρ + 0 + 0 ∇V =derivative[V_o*ln(ρ/0.0018)/1.386]dρ ∇V = V_o*0.0018/(1.386*ρ) E = V_o*0.0012987/ρ Work = 0.5∫∫∫εE•E dv Bounds: 0.0018 to 0.00045 m D = εE =...

Hi all, Just had a look at the 2016 paper by Wang, Zhu, and Unruh, "How the huge energy of quantum vacuum gravitates to drive the slow accelerating expansion of the Universe," Qingdi Wang, Zhen Zhu, and William G. Unruh, Phys. Rev. D 95, 103504 – Published 11 May 2017 The paper states...

I understand that the energy of an electric field arises from the work put into gathering the electrons together to create the field. Bringing electrons close together requires energy because they naturally want to repel. This potential energy is stored in the field itself and the field has an...

Recent observations report w < -1.3 for z > 1.5. What was the dark energy density compared to matter density during that time? Was the universe briefly accelerating?

Let ##(x_1,y_1)## and ##(x_2,y_2)## be the point where the rods intersect the ##x,y## plane. I know that on any given point there will be the superpositions of ##E_1=\frac{2\lambda}{4\pi \epsilon_0}\frac{1}{(x-x_1)^2+(y-y_1)^2}\hat{r}_1## and ##E_2=\frac{2\lambda}{4\pi...

The energy density of an electromagnetic field with a linear dielectric is often expressed as . It is also known that energy can be found by . Using the latter, the energy density is found to be , as is well known. If you integrate the latter only over free charge and ignore bound charge, you...

Hi everyone! I'm currently strudying some astrophysical equation of states, some stuff about Fermi's gas and I'm kinda confused about the relation between the energy density and the mass density, $$ \frac{\epsilon}{c^2}=\rho. $$ I don't get why they do not use whole $$...

<< Mentor Note -- thread moved to the Sci-Fi writing forum after starting in the technical forums >> Hey folks, I'm interested in the feasibility of providing high DC voltages (~kV) in a physically small and low mass package (~grams). The power source does not actually need to be very energy...

I don't know GR so while answering the question if you prefer not to use that, I would be happy. In the Friedmann Equations, is energy density has an effect on curvature or vice versa? Or they are separate things and they don't affect each other? For example can we have an energy density...

Excuse me for bad wording in the title, but there is only so much you can do with the character limit. So, has there ever been a proposal for a mechanism through which negative energy density could be created? Or the only possibility considered so far was that it would have been created in the...

Vacuum or dark energy have energy densities. (Markus, a science advisor at Physics Forums in 2003, estimated that dark energy has an energy density of about 0.5 Joule per cubic km.) I assume that the structure of space-time has an energy density, that it was measured and that it can be...

There is a Baez essay about the vaccum energy density, where he says: So did they do what I might naively think of doing, namely just plotting the redshift data over a few years? Or is it a more subtle method that directly measures the time derivative of the expansion? And if I understand...

Dark energy density at this time is a constant and our universe is expanding (accelerating). This is expected to continue indefinitely. What would happen, if for some unknown reason, dark energy density started to decrease? If over time, in billion of years or longer, dark energy density...

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...

Homework Statement A charged isolated metal sphere of diameter d has a potential V relative to V = 0 at infinity. Calculate the energy density in the electric field near the surface of the sphere. State your answer in terms of the given variables, using ε0 if necessary.Homework Equations Since...

How can I find the relation between the radiance and the energy density of a black body? According to Planck's law, the energy density inside a blackbody cavity for modes with frequency ##\nu \in [\nu, \nu + \mathrm{d}\nu]## is given by $$ \rho(\nu, T)\mathrm{d}\nu =...

I am a bit confused about the energy density in an EM wave. why do we take the Peak value of E vector while calculating the energy density? Like if the E field is ##E_0 Sin(kx-wt)## what is the energy density of the EM wave(Magnetic + Electric)? is it A) ##\frac {e_0E_0^2}{2} ## or B) ##...

I have the problem of making "at home" or almost, a measure of a laser's pulse energy for unit area of the target: those kinds lasts for ~ tens of milliseconds, up to some hundreds of ms and I should be able to verify that this energy "density" doesn't go beyond 40 J/cm^2 for a single pulse...

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...

<Moderator's note: Two threads on the same topic merged in order to have arguments and sources at one place.> Can antimatter-matter be used as a fuel for a rocket? There are various problems for anti-matter to be used as a fuel as it produces a lot of gamma rays. Gamma rays are not healthy to...

why sp2 hyb. have more e. density comp with sp3 in nmr calculation stuff ?

If mass of a particle is less than the vacuum energy density.. what would happen.. is this possible (also for some dark matter species)? How about photons.. are they more or less than the vacuum energy density? And what exactly is the value of vacuum energy density?

If i have Energy Density (U) -> U.Area= F but F.Area = pressure (p) but p must be U . I'm confused! In which cases we can say that energy density is pressure?

Hi folks, here's a thought/conceptual question I've been wondering about. What is the maximum theoretical specific energy (IE Joules/kg or equivalent) for energy stored in the electric field of a capacitor? I know the energy stored in a capacitor is given by U=C V^2/2, and the mass of the system...

Homework Statement Taken from Purcell Problem 1.33 Consider the electric field of two protons a distance b apart. The potential energy of the system ought to be given by U=∫E2dv. Let E1 be the field of one particle alone and E2 that of the other. Evaluate ε0∫E1⋅E2dv. Set one of the protons...

is there any way to derive p=rho/3 from 4 vector or stress-energy tensor?