# Not even a point on the HARDEST physics olympiad? IPhO Solution Friedmann equation

TLDRIn this intriguing video, the host explores the acceleration of the universe by applying the first law of thermodynamics to the cosmos. They simplify the process by considering it as an adiabatic system, where heat transfer is zero. The host differentiates the volume of the universe with respect to time, using the scale factor 'a(t)' and derives the expression for the rate of volume change. They cleverly utilize Einstein's mass-energy equivalence, E = mc^2, to relate mass, density, and volume changes over time. The video concludes with a fascinating derivation of the second Friedmann equation, which describes the expansion rate of the universe, leaving viewers with a deeper understanding of cosmological dynamics.

###### Takeaways

- π The video discusses the acceleration equation of the universe by examining a question from the International Physics Olympiad.
- π§ The first law of thermodynamics is applied to the entire universe, simplifying to an equation where the change in energy is equal to the work done plus the energy input, with the latter being zero for an adiabatic process.
- π The volume of the universe is expressed in terms of its radius, which is time-dependent due to the universe's expansion.
- π The differentiation of volume with respect to time is performed to find the rate of change of the universe's volume.
- π The time derivative of the scale factor (a(t)) is crucial in determining the universe's acceleration.
- π Einstein's mass-energy equivalence formula, E = mc^2, is used to relate mass, energy, and the speed of light.
- π The change in mass is considered as a function of time, with mass being density times volume, both of which change with the expansion of the universe.
- βοΈ The equation is simplified by dividing by c^2 to make the terms more manageable.
- π The final equation derived is an expression of a (the scale factor) in terms of the time derivative of the density and pressure, which is a key part of the Friedmann equations.
- π The video concludes with an invitation to watch another video on Pascal's infinite resistors, suggesting a series of educational content.

###### Q & A

### What is the first law of thermodynamics as applied to the universe in the script?

-The first law of thermodynamics, as applied to the universe, states that the change in energy of the system is equal to the work done on the system plus the energy added to the system. In an adiabatic process, the term for heat transfer (delta Q) is zero, simplifying the equation to delta E + P*V = 0.

### Why is the process considered adiabatic in the context of the universe's expansion?

-The process is considered adiabatic because there is no heat transfer into or out of the system; the universe is expanding without exchanging heat with its surroundings.

### How is the volume of the universe expressed in terms of its radius?

-The volume of the universe is expressed as a function of its radius, which is time-dependent due to the universe's expansion. It is represented as V = (4/3) * pi * (r(t))^3, where r(t) is the radius of the universe as a function of time.

### What is the expression for the time derivative of the volume with respect to time?

-The time derivative of the volume with respect to time, denoted as dV/dt, is derived from the volume expression and is equal to (4/3) * pi * r^2 * (da/dt), where r is the constant radius of a spherical piece of the universe, and a(t) is the scale factor as a function of time.

### What does the term 'a dot' represent in the script?

-'a dot' represents the time derivative of the scale factor (a(t)), which indicates how the universe is accelerating or decelerating over time.

### How is the mass of the universe related to its density and volume?

-The mass of the universe is equal to its density (rho) multiplied by its volume (V). As both density and volume are functions of time due to the universe's expansion, the mass is also time-dependent.

### What is the significance of the equation E = mc^2 in the context of the universe's energy change?

-The equation E = mc^2, where E is energy, m is mass, and c is the speed of light, is used to relate the change in energy of the universe to its mass. Since mass is time-dependent, the change in energy is also tied to the dynamics of the universe's expansion.

### What is the product rule used for in the derivation of the energy change equation?

-The product rule is used to differentiate the product of two functions of time, in this case, the mass of the universe, which is the product of density and volume, both of which are time-dependent.

### How does the script simplify the equation involving the time derivative of density, volume, and pressure?

-The script simplifies the equation by dividing both sides by c^2 and then substituting the expression for dV/dt, leading to an equation that relates the time derivative of density, the scale factor, and pressure to the acceleration of the universe.

### What is the final equation derived in the script, and what does it represent?

-The final equation derived is an expression for the acceleration of the universe in terms of the density, pressure, and the scale factor. It represents the second of the Friedman equations, which describes how the universe expands over time.

### What is the significance of the video mentioning Pascal's infinite resistors?

-The mention of Pascal's infinite resistors is a suggestion for a related topic that viewers might find interesting to explore next, indicating that the principles of physics can be applied in various contexts, including both cosmology and electrical theory.

###### Outlines

##### π Understanding the Universe's Acceleration

The paragraph explores the acceleration equation of the universe by applying the first law of thermodynamics, which states that the change in energy within a system is equal to the work done plus the energy added. In the context of the universe, which is expanding, this law simplifies to an equation involving the time derivative of volume. The paragraph carefully explains how to differentiate the volume of the universe over time, assuming it to be spherical, and introduces the concept of the scale factor 'a(t)', which changes as the universe expands. The relationship between the original volume and the scale factor's derivative is established, showing how this expression reveals the universe's acceleration. This forms the foundation for applying further equations to understand cosmic expansion.

##### π Deriving the Second Friedman Equation

This paragraph continues from the previous discussion, diving deeper into the derivation of the second Friedman equation. By substituting the previously derived expression for the rate of change of volume (V dot) into the thermodynamic equation, the author simplifies the equation further. The paragraph explains how terms can be canceled out, leading to an equation that relates the density of the universe, pressure, and the scale factor's rate of change. The process highlights the importance of understanding these relationships in cosmology, as the equation derived is one of the key Friedman equations that describe the dynamics of the universe's expansion. The paragraph concludes with a reference to the significance of the result and suggests further exploration of related topics, such as Pascal's infinite resistors.

###### Mindmap

###### Keywords

##### π‘First Law of Thermodynamics

##### π‘Adiabatic Process

##### π‘Volume of the Universe

##### π‘Scale Factor (a(t))

##### π‘Friedmann Equations

##### π‘Energy Density (Ο)

##### π‘Pressure (P)

##### π‘Mass-Energy Equivalence (E = mcΒ²)

##### π‘Differentiation

##### π‘Chain Rule

###### Highlights

The acceleration equation of the universe is derived using the laws of thermodynamics.

The first law of thermodynamics is simplified for an adiabatic process in the universe, where delta Q equals zero.

Differentiation of volume with respect to time is essential for understanding the universe's expansion.

The volume of the universe is expressed in terms of its radius, which is time-dependent.

The time derivative of the volume, v dot, is calculated using the chain rule.

The scale factor 'a' of the universe is used to describe its expansion over time.

The derivative of the scale factor, a dot, indicates the rate of the universe's acceleration.

Einstein's equation E=mc^2 is applied to the changing energy in the universe.

The mass of the universe is considered as a function of time due to its expansion.

The product rule is used to differentiate the mass-energy equation in the context of the expanding universe.

The density of the universe and its time derivative are key components in the energy change equation.

The equation is simplified by dividing by c^2 to focus on the relevant terms.

The final rearranged equation relates the density, pressure, and acceleration of the universe.

The derived equation is a form of the second Friedman equation, crucial for cosmology.

The video concludes with an invitation to watch another educational video on a different topic.

###### Transcripts

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