# Einstein's General Theory of Relativity | Lecture 2 | Summary and Q&A

## Summary

This video discusses the concept of dark energy and its effects on atoms and other bound structures. The concept of dark energy is introduced as a small repulsive force that adds to the attractive force between particles. The video explains that this force is very small and negligible at small distances but becomes significant at cosmological scales. The video also mentions the theory of the "big rip", which suggests that the dark energy could potentially tear atoms apart if it increases with time. However, the video argues that there is no reason to believe that dark energy increases with time and that this theory violates fundamental principles of physics.

## Questions & Answers

### Q: What is dark energy and what are its effects on atoms?

Dark energy is a small repulsive force that adds to the attractive force between particles. At small distances, the force is negligible and has no significant effect on atoms. However, at cosmological scales, dark energy becomes significant and could potentially tear atoms apart.

### Q: How does the force of dark energy compare to other forces in the solar system?

Dark energy is very small compared to other forces in the solar system, such as gravity. Its effects are only noticeable at cosmological distances and do not have a significant impact on objects smaller than the entire universe.

### Q: Is dark energy a constant or does it change with time?

Most current thinking suggests that dark energy is a constant, as indicated by the term "cosmological constant". However, it does not violate any fundamental principles for dark energy to decrease with time. On the other hand, an increase in dark energy with time would violate these principles.

### Q: Can dark energy be harnessed for energy purposes?

Dark energy has an extremely small density, making it impractical to harness for energy purposes. To extract a significant amount of energy from dark energy, one would need a volume of dark energy equivalent to the size of the moon's orbit or larger. Therefore, there is currently no feasible way to tap into the energy of dark energy.

### Q: Is the gravitational field within the Earth uniform?

The gravitational field within the Earth is not uniform, but it can be approximated as such. Assuming a uniform mass density within the Earth, the acceleration experienced by a test mass would be proportional to the distance from the center of the Earth.

### Q: Does the distribution of mass outside a spherical shell have any effect on the gravitational field inside the shell?

No, the distribution of mass outside a spherical shell has no effect on the gravitational field inside the shell. According to Gauss's law, the gravitational field inside a shell depends only on the mass enclosed within the shell, not the distribution of mass outside it.

### Q: How does the one over r-squared law relate to Gauss's law?

Gauss's law allows us to derive the one over r-squared law for the gravitational field. Assuming spherical symmetry and using Gauss's law, we can derive the relationship between the mass within a certain radius and the gravitational field at that radius.

### Q: What is Gauss's theorem and how is it related to Gauss's law?

Gauss's theorem relates the divergence of a vector field to the flux of the field through a closed surface. It states that the integral of the divergence over a volume is equal to the integral of the perpendicular component of the field over the surface. Gauss's law is a specific case of Gauss's theorem and applies to gravitational fields, relating the divergence of the acceleration to the mass density.

### Q: Can Gauss's law be used to calculate the gravitational field within the Earth?

Yes, Gauss's law can be used to calculate the gravitational field within the Earth. By assuming a uniform mass density within the Earth and using Gauss's law, we can derive the acceleration experienced by a test mass inside the Earth. This acceleration can be represented as a linearly increasing force towards the center of the Earth, similar to a harmonic oscillator.

### Q: How does the distribution of mass within a sphere affect the gravitational field inside?

The distribution of mass within a sphere determines the gravitational field inside the sphere. Assuming spherical symmetry, the gravitational field inside a sphere only depends on the mass within the radius of the sphere. The distribution of mass outside the sphere does not have any effect on the gravitational field inside.