# Einstein's General Theory of Relativity | Lecture 3 | Summary and Q&A

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February 5, 2009
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Stanford
Einstein's General Theory of Relativity | Lecture 3

## Summary

In this video, the speaker discusses the behavior of a light ray in a gravitational field or an accelerating elevator. They explain how the color or frequency of a light beam changes when fired vertically in an elevator. The effects of gravity and relativity, as well as the principle of equivalence, are explored. The speaker also touches on the topic of coordinate transformations and how they affect the measurements and observations of light.

### Q: What effect does gravity have on the behavior of a light ray traveling vertically in an elevator?

When a light ray is beamed vertically upward in an elevator, the frequency or color of the light beam changes due to the effects of gravity. Unlike a bullet, which decelerates under the influence of gravity, a light beam experiences a change in frequency or color as it travels vertically. This effect is more pronounced when considering gravity and relativity, particularly for objects moving close to the speed of light.

### Q: How does the principle of equivalence relate to the behavior of a light ray in a gravitational field?

The principle of equivalence, as proposed by Einstein, suggests that the effects of gravity can be equivalent to the effects of an accelerating reference frame. When considering the behavior of a light ray in a gravitational field, Einstein advises not to ask about the effect of gravity on a vertical light beam, but instead to consider the effect of the beam when detected at another location, such as the ceiling or the floor of the elevator. This principle helps explain the changing frequency or color of a light beam in a gravitational field.

### Q: What happens to the frequency or color of a light beam when fired vertically from the top of the elevator and detected at the bottom?

When a light beam is fired downward from the top of an elevator and detected at the bottom, its frequency or color undergoes a blueshift. This means that the light beam appears slightly bluer than it would if detected at the top of its arc. The effect is similar to the Doppler shift experienced by a light beam when intercepted by something moving relative to the source. The acceleration of the elevator causes the detector to move faster upwards, resulting in a blueshift.

### Q: Does the same effect occur when detecting a light beam fired from the floor and detected at the ceiling?

Yes, the same effect occurs when a light beam is fired from the floor of an elevator and detected at the ceiling. In this case, the light beam appears slightly redshifted, meaning it appears slightly redder than it would if detected at the floor. This is the opposite effect to the blueshift observed when the light beam is fired from the top and detected at the bottom. Similar redshift effects can also be observed when detecting light beams emitted from the surface of stars.

### Q: Does the size of a star affect the redshift or blueshift of its emitted light?

Yes, the size of a star can affect the redshift or blueshift of its emitted light. The relevant factor is the escape velocity, with larger and more compact stars, such as black holes, producing more significant effects. For example, a light beam emitted near the surface of a black hole would appear deeply infrared by the time it reaches an observer. The effect of redshift or blueshift depends on the strength and nature of the gravitational field.

### Q: How do the Doppler shifts of a light beam traveling in an elevator cancel each other out?

The Doppler shifts of a light beam traveling in an elevator cancel each other out due to the equivalence of the elevator's acceleration and gravity. When a light beam is fired downward, its frequency or color undergoes a blueshift due to the elevator's acceleration. However, the genuine blueshift caused by the gravitational field cancels out this acceleration-induced shift. The cancellation occurs because the detector is moving faster upwards than the emitter due to acceleration, resulting in a net cancellation of the Doppler shifts.

### Q: Can the effects of gravity be completely removed by choosing an accelerated reference frame?

No, the effects of gravity cannot be completely removed by choosing an accelerated reference frame. While an accelerated reference frame can cancel out the gravitational field up to a certain point, tidal forces and variations in the strength of gravity prevent complete removal. For example, in a freely falling elevator, the effects of gravity are mostly removed, but residual effects known as tidal forces remain due to the variation in gravity's strength. Completely removing the effects of gravity globally is obstructed by the non-uniformity of the gravitational field caused by mass.

### Q: Is it possible to remove the effects of gravity through coordinate transformation?

No, removing the effects of gravity through coordinate transformation is obstructed by the non-uniformity of the gravitational field caused by mass. While coordinate transformation can remove the effects of gravity within a small volume and for a short duration, it cannot remove the overall effect of gravity globally. The curvature introduced by coordinate transformation to remove gravity is closely related to the obstruction encountered in removing the curvature of a curved space. This limitation is an essential feature of general relativity and the behavior of gravity in curved space.

### Q: What is the transformation rule for changing coordinates from x to y for a scalar quantity?

For a scalar quantity, such as temperature, the transformation rule when changing coordinates from x to y is straightforward. The value of the scalar remains the same, regardless of which coordinate system is chosen. Therefore, we can express the scalar phi in both sets of coordinates as phi(x) = phi(y) if x and y represent the same point. Scalars are invariant under coordinate transformations.

### Q: How do the components of a vector transform when changing from one coordinate system to another?

The components of a vector transform when changing from one coordinate system to another by a specific transformation rule. The transformation involves the derivative of the new coordinates (y) with respect to the old coordinates (x) multiplied by the components of the vector expressed in the old coordinate system (x). The resulting components are the vector expressed in the new coordinate system (y). This transformation rule allows for comparing and relating vectors in different coordinate systems.

## Takeaways

In this video, the behavior of light rays in gravitational fields and accelerating elevators is explored. The speaker emphasizes the importance of considering coordinate transformations and the effects they have on measurements and observations. The principle of equivalence is discussed in the context of understanding gravity and acceleration. The transformation rules for scalars and vectors are presented, highlighting their dependence on coordinates and the relationship between different coordinate systems. Understanding these concepts is crucial for grasping the behavior of gravity in curved spaces and tackling the mathematical aspects of general relativity.