# Questions about Negative Pressure and Vacuum Energy

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• darkdark10
In summary, the conversation discusses the concept of dark energy and its properties, specifically its negative pressure. However, there is a problem with the assertion of negative pressure and it is argued that it is illogical to assume negative kinetic energy while maintaining a positive mass density. The discussion also touches on the equation dU=-PdV which is used to calculate the change in internal energy, but it is questioned whether this equation holds for a system where energy is not conserved. Furthermore, it is debated whether vacuum energy with positive energy density can produce negative pressure.
darkdark10
Currently, dark energy is described as a being that exerts a negative pressure while having a positive energy density.

$${\rho _\Lambda } + 3{P_\Lambda } = {\rho _\Lambda } + 3( - {\rho _\Lambda }) = - 2{\rho _\Lambda }$$

However, there seems to be a problem with the negative pressure assertion.

In the book(An Introduction to Modern Astrophysics P1161)
Note that the effect of the pressure P is to slow down the expansion (assuming P>0). If this seems counterintuitive, recall that because the pressure is the same everywhere in the universe, both inside and outside the shell, there is no pressure gradient to exert a net force on the expanding sphere. The answer lies in the motion of the particles that creates the fluid’s pressure. The equivalent mass of the particle’s kinetic energy creates a gravitational attraction that slows down the expansion just as their actual mass does.
Pressure P = equivalent energy density of kinetic energy
P/c^2 = equivalent mass density of kinetic energy

1. Sign of the negative pressure
In general relativity, pressure is equivalent energy density of the kinetic energy.
In the acceleration equation, 3P(c=1) has the idea of an equivalent mass density corresponding to the kinetic energy of the particle. So, assuming that the pressure P term has a negative energy density is same assuming that it has negative kinetic energy. In order to have negative kinetic energy, it must have negative inertial mass or imaginary velocity. But, because they assumed a positive inertial mass, it is a logical contradiction.
$$K = \frac{1}{2}m{v^2} < 0$$
m<0 or v=Vi : negative mass or imaginary speed.
Negative mass contradicts the assumption of positive energy density, and energy density with imaginary speed is far from physical reality.

2. Size of the negative pressure
In the ideal gas state equation, we obtain,
$$P = \frac{1}{3}(\frac{{{v^2}}}{{{c^2}}})\rho = \omega \rho$$
In the case of matter, v << c, So $$P = \frac{1}{3}{(\frac{v}{c})^2}\rho \simeq 0$$
In the case of radiation, v=c, So $$P = \frac{1}{3}{(\frac{v}{c})^2}\rho = \frac{1}{3}\rho$$
However, in the case of dark energy $$P = \frac{1}{3}{(\frac{v}{c})^2}\rho = - \rho$$
$$v = (\sqrt 3 c)i$$
We need energy density with imaginary super-luminous speed.

Ideal gas state equation applies to "massless ~ no upper limit'' particles, and the velocity ranges from "0 ~ c'' are all included. If we seriously consider the physical presence of P= -ρ = - 3((1/3)ρ), we will find that there is a serious problem.

It is very strange. So, let's look again at the logic we're convinced of negative pressure.
dU=dQ-dW, if dQ=0, dU=-dW=-PdV
dU=-PdV : if dU=ρdV, P=-ρ

I thought about whether the logic using dU=-PdV is correct.
1) The argument regarding negative pressure is an inverted explanation
Pressure P = equivalent energy density of kinetic energy
Since pressure is a property of an object, pressure exists first, and because of this pressure, changes in internal energy according to volume change appear.
That is, since pressure is positive, if dV>0, then dU<0. Since the pressure is positive, if dV<0, then dU>0.
By the way, we use the logic "if dV>0, dU>0, then P<0''. Can we be sure that this logic is correct?

2) ρΛ+ 3PΛ = ρΛ + 3(-ρΛ) =-2ρΛ
Mass density ρ and pressure P are properties of the object to be analyzed. Both mass density ρ and pressure P are sources of gravity.
It means that even if the region maintains a constant size without expanding or contracting, gravitational force is applied as much as ρΛ+ 3PΛ =-2ρΛ. In other words, it suggests that the object (or energy density) has a gravity with a negative mass density of -2ρΛ. This is different from a vacuum with a positive energy density +ρΛ, which we think of.

3) dU = - PdV is the expression obtained when the law of conservation of energy is established
However, in the case of vacuum energy and the cosmological constant, energy conservation does not hold. As the universe expands, the total energy in the system increases. Therefore, we cannot guarantee that dU = - PdV holds.

I am not sure if this equation(dU=-PdV) holds even for negative pressure. However, although this equation holds even in the case of negative pressure, its interpretation is as follows.

This equation holds true when substances in radius r_1 expand from r_1 to r_2 (r_2 > r_1), and have the same uniform density in r_1 and r_2. In other words, it is argued that a negative pressure is required to create a uniform density effect only with the material present in radius r_1. But, vacuum energy is a form in which energy is newly generated by an increased volume. It is also energy that can be assumed to have an initial speed of 0 ~ c.

Q1. Is it possible to have negative kinetic energy while having positive mass density?

Q2. Does the dU=-PdV equation hold for a system in which energy is not conserved?

Q3. Does vacuum energy with positive energy density ρ produce negative pressure?

We can considered the vacuum energy density with P=0. However, in order to have a uniform energy density even when space expands, does it have to suddenly have negative pressure? Shouldn't it be newly created with P=0, and filling the larger volume?

Q4. Can't there be a vacuum energy density with P=0 ~ (1/3)ρ?

darkdark10 said:
there seems to be a problem with the negative pressure assertion.
No, there isn't. The vacuum has to be Lorentz invariant. That requires that the vacuum have the equation of state ##p = - \rho##.

Your objections are based on misconceptions:

darkdark10 said:
In general relativity, pressure is equivalent energy density of the kinetic energy.
No, it isn't.

darkdark10 said:
We need energy density with imaginary super-luminous speed.
No, we don't. The energy density of the vacuum is not moving. I don't know where you are getting your "ideal gas state equation" in terms of ##v / c##, but it doesn't apply to the vacuum.

darkdark10 said:
Both mass density ρ and pressure P are sources of gravity.
Yes, but that doesn't mean energy density and pressure are the same thing, which is what you're basing your reasoning on.

darkdark10 said:
in the case of vacuum energy and the cosmological constant, energy conservation does not hold.
This is correct for the meaning you are assigning to "energy conservation". It's also not a problem, since GR does not require that "energy conservation" in your sense must hold. The only "energy conservation" that must hold in GR is that the covariant divergence of the stress-energy tensor is zero. The stress-energy tensor of the vacuum is ##\Lamba g_{\mu \nu}##, with ##\Lambda## a constant, and the covariant divergence of that is zero.

darkdark10 said:
Q1. Is it possible to have negative kinetic energy while having positive mass density?
No, but that has nothing to do with vacuum energy. See above.

darkdark10 said:
Q2. Does the dU=-PdV equation hold for a system in which energy is not conserved?
Not necessarily, but that's not a problem. See above.

darkdark10 said:
Q3. Does vacuum energy with positive energy density ρ produce negative pressure?
Yes. See above for the equation of state of vacuum energy.

darkdark10 said:
Q4. Can't there be a vacuum energy density with P=0 ~ (1/3)ρ?
No.

In the book(An Introduction to Modern Astrophysics - Bradley W. Carroll and Dale A.Ostile)

Pressure P = equivalent energy density of kinetic energy
P/c^2 = equivalent mass density of kinetic energy

Isn't the pressure P in the acceleration equation the equivalent energy density of kinetic energy?

Last edited:
darkdark10 said:
In the book(An Introduction to Modern Astrophysics - Bradley W. Carroll and Dale A.Ostile)

View attachment 302768
Pressure P = equivalent energy density of kinetic energy
P/c^2 = equivalent mass density of kinetic energy

Isn't the pressure P in the acceleration equation the equivalent energy density of kinetic energy?
Dark energy is not caused by a song on the Mason Williams Phonograph Record.

berkeman
darkdark10 said:
Isn't the pressure P in the acceleration equation the equivalent energy density of kinetic energy?

darkdark10 said:
Pressure P = equivalent energy density of kinetic energy
P/c^2 = equivalent mass density of kinetic energy
No. I'm not familiar with the textbook you referenced, but I would say the passage you quoted is badly worded. The "equivalent mass of the particle's kinetic energy" goes in the mass density (or energy density, depending on which units you use) ##\rho##, not the pressure ##P##. The pressure ##P## is the force per unit area exerted by one parcel of fluid on another (assuming we are talking about a perfect fluid). Pressure does appear in the stress-energy tensor and in the "source of gravity" term in the Friedmann equation, but it is not the same as "equivalent energy density of kinetic energy"--kinetic energy goes in ##\rho##, not ##P##, as above.

## 1. What is negative pressure and how does it relate to vacuum energy?

Negative pressure is a concept in physics that describes a situation where the pressure exerted by a substance or field is less than the surrounding pressure. Vacuum energy, on the other hand, is the energy associated with the vacuum of space. In some theories, negative pressure is thought to be responsible for the expansion of the universe, and it is believed that vacuum energy plays a role in this process.

## 2. How is negative pressure different from positive pressure?

Negative pressure and positive pressure are opposite concepts. Positive pressure describes a situation where the pressure exerted by a substance or field is greater than the surrounding pressure. This can be seen in everyday situations, such as blowing up a balloon or pumping air into a tire. Negative pressure, on the other hand, is less common and is often associated with more extreme environments, such as outer space.

## 3. Can negative pressure be created artificially?

Yes, negative pressure can be created artificially in controlled environments. This is often done in laboratory settings, where researchers use specialized equipment to create low-pressure conditions. Negative pressure can also be created in certain industrial processes, such as in the production of vacuum-sealed products.

## 4. What are the potential applications of negative pressure and vacuum energy?

Negative pressure and vacuum energy have several potential applications in various fields of science and technology. In cosmology, these concepts are thought to play a role in the expansion of the universe. In materials science, negative pressure can be used to create new materials with unique properties. In medicine, negative pressure is utilized in wound healing and in some medical procedures, such as removing air from the lungs in patients with collapsed lungs.

## 5. Is there a relationship between negative pressure and dark energy?

There is currently no definitive answer to this question. Some theories suggest that negative pressure and vacuum energy may be related to dark energy, which is thought to be responsible for the accelerating expansion of the universe. However, more research is needed to fully understand the relationship between these concepts.

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