Special Relativity | Lecture 2

TL;DR

Lorentz transformations relate two frames moving along one axis: x' equals (x minus vt) divided by the square root of 1 minus v squared, with t' following the same form. Setting the speed of light to one simplifies the equations, and factors of c can always be restored for dimensional consistency. Perpendicular coordinates stay unchanged.

Transcript

Stanford University. All right. We were last time we worked out basic Lorent transformations relating two frames of reference. Uh whether or not I said it, let me say it now. We were mainly dealing with problems in which all of the motion and in particular the relative motion of different observers is along one particular axis. We didn't uh try to ... Read More

Key Insights

• Lorentz transformations relate a stationary frame and a moving frame along one axis: x prime equals x minus vt divided by the square root of 1 minus v squared, and t prime equals t minus vx over the same square root.
• The speed of light being the same for all reference frames is Einstein's basic hypothesis; with correctly chosen units such as years and light years, all observers measure the speed of light as one.
• Restoring the speed of light into the equations is always unique: you insert appropriate factors of c so the equations are dimensionally consistent, with v squared over c squared replacing v squared.
• The relationship between frames is truly reciprocal; the inverse transformation is identical except the velocity changes sign, because the two observers move in opposite directions relative to each other.
• Perpendicular directions do not change under a velocity change along one axis; if relative velocity is along x, then y prime equals y and z prime equals z remain unchanged.
• An event is something happening at a point of space and a point of time, like a flash bulb going off; the stationary observer assigns coordinates x and t while the moving observer assigns x prime and t prime to the same event.
• You can read the relative velocity directly from the transformation: x prime equals zero, the origin of the moving frame, corresponds to x equals vt, which shows the relative velocity is v.
• Relativistic velocity addition is solved by chaining transformations across three frames: a car moving at velocity u inside a train moving at velocity v is described to the stationary observer by substituting the prime coordinates in terms of unprimed ones.

Questions & Answers

Q: What are the Lorentz transformation equations for two frames moving along one axis?

For a moving frame with velocity v along the x axis, x prime equals x minus vt divided by the square root of 1 minus v squared, and t prime equals t minus vx divided by the same square root of 1 minus v squared. Newton would recognize x minus vt, but not the square root denominator. Perpendicular coordinates stay the same, so y prime equals y and z prime equals z.

Q: Why is the speed of light set equal to one in these equations?

Setting the speed of light to one is a choice of units that simplifies the equations. If you work in years and light years, or seconds and light seconds, or any correctly chosen units, the speed of light becomes one. It follows from Einstein's basic hypothesis that all reference frames measure the speed of light exactly the same, and choosing these units makes that shared value equal to one.

Q: How do you put the speed of light back into the equations?

There is always a unique way to restore the speed of light c. You modify the equations by inserting appropriate factors of c so that the equations become dimensionally consistent. For example, the denominator square root of 1 minus v squared becomes 1 minus v squared over c squared. If the velocity is small compared with the speed of light, this correction is terribly tiny, so it is often dropped.

Q: How do you invert the Lorentz transformations to get x and t?

Inverting is just a matter of solving for x and t in terms of x prime and t prime, with nothing sophisticated involved. The result is x equals x prime plus v t prime divided by the same square root of 1 minus v squared, and t equals t prime plus v x prime over 1 minus v squared. The only asymmetry is that the sign of the velocity changes, because the relative velocities point in opposite directions.

Q: How can you read the relative velocity between two frames from the transformation?

You look at the x equation. The moving, or prime, observer's origin sits at x prime equals zero, which is the position of the origin of coordinates inside the train. Setting x prime equals zero corresponds to x equals vt. You do not need the denominator. Reading x equals vt directly tells you that the relative velocity between the two observers is v.

Q: What is an event in special relativity?

An event is something that takes place at a point of space and a point of time, in other words at some point of spacetime. A useful picture is a flash bulb exploding or going off somewhere. The stationary observer ascribes coordinates x and t to that event using his meter sticks and clock, while the moving observer ascribes coordinates x prime and t prime to the same event. It does not matter whether it happens inside or outside the train.

Q: Why do perpendicular coordinates stay unchanged under a Lorentz transformation?

Perpendicular directions do not change under a change of velocity along a given axis. If the relative velocities of the two frames are along the x axis, then the coordinates perpendicular to the tracks, meaning the vertical direction and the direction out of the board, are unaffected. This means y prime equals y and z prime equals z. Only the coordinates along the direction of relative motion, x and t, get transformed.

Q: How do you find the velocity of a car moving inside a moving train relative to the station?

You introduce three coordinate sets: the stationary x and t, the passenger's x prime and t prime, and the car driver's double-prime coordinates. The car moves at velocity u relative to the train, giving a Lorentz transformation between double-prime and prime frames using u. Since you already know x prime and t prime in terms of x and t, you substitute them in to connect the car's coordinates to the station, yielding the combined velocity W in terms of u and v.

Summary

This video discusses basic Lorentz transformations in frames of reference and the concept of proper time or interval. It explains how the coordinates of one observer relate to the coordinates of another observer and introduces the notion of four-vectors, including four-velocity.

Questions & Answers

Q: What is the main concept discussed in this video?

The main concept discussed in this video is Lorentz transformations and the notion of proper time or interval.

Q: How are Lorentz transformations related to frames of reference?

Lorentz transformations relate the coordinates of one observer to the coordinates of another observer in different frames of reference. They account for the relative motion of observers and ensure that all reference frames see the speed of light the same way.

Q: What are the Lorentz transformations based on?

The Lorentz transformations are based on the hypothesis of Einstein that all reference frames see the speed of light exactly the same.

Q: How are the coordinates of one observer related to the coordinates of another observer?

The coordinates of one observer (stationary observer) can be related to the coordinates of another observer (moving observer) through Lorentz transformations. The coordinates depend on the relative velocity between the two observers.

Q: What if the motion of observers is not only along one axis?

In that case, Lorentz transformations can still be used to relate the coordinates of the stationary observer to the coordinates of the moving observer in multiple dimensions. The transformations for the additional axes are simpler and do not involve changes in velocity.

Q: What is the proper time or interval between two points in space-time?

The proper time or interval between two points in space-time is an invariant quantity that is the same in all reference frames. It is given by T^2 - X^2, where T is the time coordinate and X is the spatial coordinate.

Q: How are Lorentz transformations represented mathematically?

Lorentz transformations can be represented mathematically using matrices. For example, for the transformation in the x-direction, the matrix would be [[1, -V], [-V, 1]].

Q: What is a four-vector?

A four-vector is a four-dimensional object that includes both the space and time components. It is represented by coordinates X^(mu), where mu runs from 0 to 3 and corresponds to T, X, Y, and Z, respectively.

Q: How does four velocity differ from ordinary velocity?

Four velocity, represented by U^(mu), is a four-dimensional quantity that incorporates both space and time components. It is obtained by dividing the change in coordinates (Delta X^(mu)) by the invariant distance (Delta Tau) between two points in space-time.

Q: How is four velocity related to ordinary velocity?

Four velocity is related to ordinary velocity by dividing the space component of four velocity by the time component. This gives the velocity in three-dimensional space.

Takeaways

In this video, we learned about Lorentz transformations and their application in relating coordinates between different frames of reference. We also explored the concept of proper time or interval, which is an invariant quantity in all reference frames. Additionally, we discussed four-vectors and four velocity, which extend the notion of velocity to incorporate the space and time components in a four-dimensional framework. These concepts are crucial in understanding the motion and dynamics of particles in the context of relativity theory.

Summary & Key Takeaways

• The lecture reviews Lorentz transformations relating two one-dimensional frames: stationary observers at a station and observers on a moving train. The relative motion is along a single railroad-track axis, and the transformations are built on Einstein's hypothesis that all frames measure the same speed of light.

• With suitable units like light years or light seconds, the speed of light equals one, simplifying the equations. The speed of light c can always be restored uniquely by adding factors that keep the equations dimensionally consistent, appearing as v squared over c squared in the denominators.

• A third observer, a kid in a kitty car moving at velocity u inside the train, sets up a velocity-addition problem. Using three coordinate sets and chaining the transformations by substitution, the stationary observer's velocity W for the car is found in terms of u and v.