Cosmology | Lecture 2

TL;DR

The equations governing how the universe expands or contracts are the Friedmann-Robertson-Walker equations, and they can be derived from Newton's equations before invoking general relativity. In a homogeneous, isotropic universe, Hubble's Law holds: velocity is linearly proportional to distance, with the Hubble constant fixed across space but varying with time.

Transcript

this program is brought to you by Stanford University okay let's uh let's go back to where we were the last time we were talking about the metric of space today what we're going to do is a little bit of geometry and then we're going to get really into the Dynamics of cosmology the Dynamics actually means how does the scale factor we discussed the s... Read More

Key Insights

• The Friedmann-Robertson-Walker (FRW) equations govern how the scale factor changes with time, determining how the universe grows, shrinks, or contracts. Susskind derives them from Newton's equations first, then later connects them to general relativity.
• Hubble's Law states that velocity is linearly proportional to distance, and it always holds whenever the universe is homogeneous and isotropic. The proportionality constant is the Hubble constant.
• The Hubble constant is constant with respect to position in space (homogeneity) but not with respect to time. Different regions expanding differently would mean space is not homogeneous.
• Three-dimensional space can be flat while spacetime itself is curved. A time-dependent metric describing flat spatial slices embedded in four-dimensional spacetime produces a curved spacetime even when the space is flat.
• Meter sticks and other bound objects do not expand with the universe because the electric and magnetic forces holding them together are many orders of magnitude stronger than the minute forces due to expansion.
• Galaxies far enough apart that they feel no strong gravitational binding will separate as the universe expands. Only objects large and diffuse enough to escape binding forces expand with the ambient expansion.
• The scale factor 'a' codifies the relationship between coordinates and actual distances measured in meters. Each galaxy sits at a fixed coordinate value that never changes; only the meaning of the coordinates changes with time.
• The spacetime metric takes the form ds squared equals dt squared minus a(t) squared times (dx squared plus dy squared plus dz squared). The coefficients depend only on time, not position, reflecting the homogeneity of the universe.

Questions & Answers

Q: What are the Friedmann-Robertson-Walker equations?

The Friedmann-Robertson-Walker (FRW) equations, really the Friedmann equations, are the evolution equations that govern how the universe expands and, if it eventually contracts, how it contracts. They describe how the scale factor changes with time, meaning how the universe grows or shrinks. Susskind notes they can be derived from Newton's equations to get started, and only later fitted together with the general theory of relativity.

Q: Why is Hubble's Law always true in a homogeneous, isotropic universe?

If the universe is homogeneous and isotropic and changes with time, then Hubble's Law, that velocity is proportional to distance, will always be true. The proportionality is linear with distance at a fixed time. The constant of proportionality, the Hubble constant, is constant with respect to position in space, which reflects homogeneity, but it generally depends on time. If different regions of space expanded differently, space would not be homogeneous.

Q: Can space be flat while spacetime is curved?

Yes. Three-dimensional space can be flat while spacetime itself is curved if you have a time-dependent metric describing flat spatial slices embedded in four-dimensional spacetime. The space can be pictured as an infinite flat rubber sheet, but if giants far away pull on it to expand or contract it over time, the spacetime itself becomes curved even though each spatial slice remains flat.

Q: Why don't meter sticks expand with the universe?

Meter sticks do not expand because the forces that hold them together, strong electric and magnetic and other forces, are far stronger, by many orders of magnitude, than the forces due to expansion. The coordinates expand, but bound objects stay held together. A tiny repulsive expansion force would only shift the equilibrium position between atoms by an immeasurably small amount, not tear the object apart, much like a small extra pull on a stiff spring.

Q: What is the scale factor and what does it represent?

The scale factor 'a' appears in front of the spatial metric and represents the distance measured in meters between neighboring grid points separated by one coordinate unit. When the universe expands, the distance between neighboring grid points grows with time, so 'a' becomes a function of time. Each galaxy stays at a fixed coordinate value; what changes with time is the relationship between coordinates and actual distances, which the scale factor codifies.

Q: How is the spacetime metric written when the universe expands?

The spacetime metric becomes ds squared equals dt squared minus a(t) squared times (dx squared plus dy squared plus dz squared). The coefficients do not depend on position in space, only on time, which expresses homogeneity. In the metric tensor, the time-time component is one, all off-diagonal terms are zero, and the three spatial diagonal terms are each minus a squared. It is nearly as simple as flat spacetime except the relationship between space and time is time dependent.

Q: Does expansion by itself create a force between objects?

Expansion by itself does not create a force. If there were no forces between objects, they would simply expand along with the universe. However, some kinds of expansion, namely accelerated expansion, can be modeled in terms of a force between points, such as between atoms. A plain force would produce acceleration on its own rather than the linear growth of ordinary expansion, so the two are distinguished carefully.

Q: How does special relativity relate to this expanding metric?

In ordinary special relativity the proper time between neighboring spacetime points is dt squared minus the spatial distance squared, and introducing the speed of light gives the spatial terms divided by c squared. Reading off that form, the constant 'a' corresponds to one over the speed of light. The expanding-universe case generalizes this by letting 'a' change with time, introducing time dependence into the spacetime metric while keeping it independent of spatial position.

Summary

This video discusses the dynamics of cosmology, focusing on the evolution equations that govern how the universe expands or contracts over time. The speaker explores the concept of flat space and how it can be curved when the space itself is expanding or contracting. The video also delves into the measurement of distances in expanding space and the relationship between the expansion and the forces holding objects together. The speaker then introduces the metric of space-time and discusses its time dependence. Finally, the video touches upon Newtonian cosmology and the behavior of particles in a gravitational field.

Questions & Answers

Q: What are the dynamics of cosmology?

The dynamics of cosmology refers to how the scale factor, which represents the distance between neighboring points in space, changes over time. It encompasses understanding how the universe grows, shrinks, or evolves and the equations that govern these changes.

Q: What are the Friedman-Robertson-Walker equations?

The Friedman-Robertson-Walker (FRW) equations are the evolution equations that determine how the scale factor changes with time in an expanding or contracting universe. They describe the dynamics of cosmology from a cosmological perspective.

Q: How is the expansion of the universe related to the Hubble constant?

The expansion of the universe is related to the Hubble constant. Hubble's law states that the velocity of a galaxy is proportional to its distance from us. The Hubble constant represents the proportionality constant in this relationship. However, the Hubble constant is not constant with respect to position and time, as it varies with time in an expanding universe.

Q: Can space be flat while space-time is curved?

Yes, it is possible for three-dimensional space to be flat while space-time is curved. This occurs when the metric describing the flat spatial slices in four-dimensional space-time is time-dependent and the space itself is expanding or contracting. In this scenario, the space-time itself is curved.

Q: Do measuring rods expand in an expanding universe?

No, measuring rods do not expand in an expanding universe. The forces holding meter sticks together, such as electrical and magnetic forces, are much stronger than the forces due to expansion. Therefore, meter sticks are not affected by the ambient expansion of the universe. However, large and diffuse objects like galaxies, which do not experience strong gravitational forces, will expand with the expansion of the universe.

Q: How can the expansion of the universe be modeled as a force between points?

The expansion of the universe cannot be directly modeled as a force between points. While expansion by itself does not create a force, some types of expansion, such as accelerated expansion, can be modeled as a force between points. However, the forces due to expansion are much smaller than the interatomic forces holding objects together and cannot be measured in terms of the slight modification of the size of a meter stick.

Q: Is the Hubble constant constant with respect to position in space?

Yes, the Hubble constant is constant with respect to position in space. It does not vary from point to point, which is a property of homogeneity. However, the Hubble constant is not constant with respect to position and time, as it varies with time in most scenarios.

Q: How does the metric of space change with the expansion of the universe?

The metric of space changes with the expansion of the universe by a factor called the scale factor. This scale factor represents the distance between neighboring points at a fixed instant of time. As time progresses, the scale factor determines how the actual distance between these points grows or shrinks. In an expanding universe, the scale factor increases with time, leading to larger separations between objects.

Q: What is the metric of space-time in a homogeneous and isotropic universe?

In a homogeneous and isotropic universe, the metric of space-time is time-independent and described by the Friedmann-Lemaître-Robertson-Walker (FLRW) metric. The FLRW metric has a fixed coefficient in the time-time component, while the space components are negative and proportional to the square of a function called the scale factor, which represents the distance between neighboring points.

Q: How does energy conservation relate to the motion of particles in a gravitational field?

Energy conservation is an important concept when studying the motion of particles in a gravitational field. The total energy, including kinetic and potential energy, of a particle remains constant over time. This means that the sum of the kinetic and potential energy, which depend on the mass, velocity, and distance, remains the same for a given particle. This allows us to understand how particles behave under the influence of gravity and determine their escape velocities or bound orbits.

Takeaways

In this video, the speaker delves into the dynamics of cosmology, focusing on the scale factor and how it changes with time. They discuss the expansion of the universe, its relationship with the Hubble constant, and the behavior of objects in an expanding space. The video also touches upon the metric of space-time, proper time, and the energy conservation concept in gravitational fields. Overall, this exploration provides valuable insights into the foundations and principles behind the study of cosmology.

Summary & Key Takeaways

• The lecture's goal is the dynamics of cosmology: how the scale factor changes with time and how the universe expands or contracts. These evolution equations are the Friedmann-Robertson-Walker equations, derived tonight from Newton's equations rather than general relativity, with the relativity connection shown later.

• Assuming a homogeneous and isotropic universe, Hubble's Law (velocity linearly proportional to distance) always holds. The Hubble constant is constant across space but generally varies with time. Flat three-dimensional space can still sit inside a curved, time-dependent four-dimensional spacetime like an expanding rubber sheet.

• Meter sticks stay fixed because binding forces vastly exceed expansion forces, while unbound galaxies separate. Expanding coordinates are embedded with the galaxies, and the scale factor 'a', now a function of time, relates coordinate separations to real distances in the spacetime metric ds squared equals dt squared minus a(t) squared times the spatial terms.