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It’s Relatively Simple… Nick Bremer, Erik Cox, Scott McKinney, Mike Miller, Logan Petersen, AJ Schmucker, Nick Thull
IN REVIEW • Einstein’s Postulates • Lorentz Frames • Minkowski Inner Product
Einstein’s Postulates There exists a Lorentz frame for Spacetime. A Lorentz transformation of a Lorentz frame gives a Lorentz frame.
1st Lorentz Postulate • For stationary events, Physical Clock Time and Coordinate Time should agree. • That is, we assume that stationary standard clocks measure coordinate time.
2nd Lorentz Postulate • The velocity of light called c = 1. • Light always moves in straight lines with unit velocity in a vacuum. • , time and spatial position • Note: Think of the light pulse as a moving particle.
Minkowski Space(Geometry of Spacetime) • The symmetric, non-degenerate bilinear form of the inner product has the properties • <x,y>=<y,x> • <x1 + x2, y> = <x1, y> + <x2, y> • <cx,y> = c<x,y> • The inner product does not have to be positive definite, which means the product of it with itself could be negative. • Non-degenerate meaning only the zero vector is orthogonal to all other vectors • Spacetime has it’s own geometry described by the Minkowski Inner Product.
Minkowski Inner Product • Defined on R4: • u = (u0,u1,u2,u3) • v = (v0,v1,v2,v3) • <u,v>:=u0v0- u1v1- u2v2- u3v3 • <•,•> also called the Lorentz Metric, the Minkowski metric, and the Metric Tensor • M = R4 with Minkowski Inner Product • “•” represents the usual inner product (dot product) in R3. • In this case you have an inner product that allows negative length.
Vector Position Functions andWorldlines • In Newtonian physics/calculus, moving particles are described by functions t r(t) • r(t) = ( x(t) , y(t) , z(t) ) • This curve gives the “history” of the particle. Z r(t) X Y
Vector position functions andWorldlines cont… • View this from R4 perspective t ( t, r(t) ) • In the above ‘t’ represents time and ‘r(t)’ represents the position. • This can be thought of as a “curve in R4”, called the Worldline of the particle. • A worldline (at +b) is given a non-Euclidian “geometry” in M by the Minkowski Metric.
Light source Mirror The Einstein – Langevin Clock • Time is measured where the period between light emission and return is regarded as one unit. Where L = length of tube, and c = speed of light.
The Relativistic Time Dilation Factor Let t’ = time of ½ pendulum Light Source L = ct’ L Mirror Consider a spaceship with an Einstein-Langevin Clock onboard. We will look at what happens after time t.
Z Z’ “e” Y O Y’ O’ X X’ Time Relative to a Stationary Lorentz Frame Ship has moved D = vt. Light Source ct L Mirror vt We’ll let t be the measure of ½ of the light pendulum according to the clock of the stationary Lorentz frame. Observe: (ct)2 = L2 + (vt)2
Relativistic Dilation Factor • Solve for (ct)2 = L2 + (vt)2 And recall: L = ct’ We call the relativistic dilation factor. This shows up in many equations in relativity theory. Too make things easier, we often set c = 1.
Recall: where represents the elapsed time of the moving object in proper time and t is the elapsed time of the moving object in coordinate time (using as a conversion factor). What Is Proper Time? Proper time is the elapsed time measured by a moving object. However, this formula only applies for constant velocities.
Average Velocity for small Solution: Take small time intervals on the worldline, each with approximately constant velocity.
As , summing the intervals together yields the following: With constant velocity, we can use: Proper Time!
Proper Time where t-a is the elapsed time measured on the stationary clock and the resulting integral is the elapsed time measured on the moving clock. Note: Note: T is Tau
Recall: Spacetime = E = the set of all possible events e where E is modeled by Worldline: where and Parameterize using coordinate time: Differentiate: Parameterize the Worldline Using Coordinate Time
Fact About Proper Time Proper time represents a non-Euclidian arc-length. Proof:
x = x(t), y = y(t) z(t) = ( x(t),y(t) ) t = b t = a Notice the similarities between the common arc-length formula and the expression above: This shows that Proper time is a non-euclidean arc-length.
For any curve, we can make a change of parameter using a function to get By Chain Rule Since Proper Time is similar to arc-length, can the worldline be parameterized by proper time? Yes, provided the quantity called proper time is “independent of parameter”. Change parameter of the worldline: Differentiate z with respect to s:
Make a substitution: Pick b so t(b) = a (i.e. if s = b, then t = a) Thus showing proper time is independent of parameter.
4-dimensional, relativistic analog of traditional three- velocity, represented by: or Four-velocity What is four-velocity?
Four-velocity and Three-velocity Note: shown later. Factor out How is four-velocity related to traditional three-velocity? Recall that we can parameterize worldlines by proper time:
Four-velocity is a timelike unit vector Proof: Show Important Fact About Four-velocity Recall: Differentiate using Fund. Thm. Calculus: Using Chain Rule and inverses, we get:
Recall: Thus showing and proving four-velocity is a timelike unit vector.
Conservation of Momentum Recall the Classical Law of Conservation of Momentum It can be shown, however, that this Law is NOT Lorentz invariant for inelastic collisions. Due to the principle of covariance, the laws of physics should hold for any Lorentz Frame. Something must not be quite right!
So what do we do? Question: If the Classical Law isn’t quite right, how do we fix it? Answer: Would a 4-dimensional analog work?
What if we use 4-velocity[ ]in place of standard velocity? Recall: and (space component of ) Note that for small , Gearing up
Relativistic Momentum Our first analog of the Classical Law will look like: We’ll call this equation *. Let’s consider the time and space components of *.
The time component of U(T) is simply: Which gives: As you may have noticed, this seems trivial since for small all it says is that . However, we will see that this will be a new law analogous to the Classical Law of Conservation of Energy. Time Component
a.k.a. (space component of ) Space Component Now let’s examine the space part of *. Recall that:
Notice that when is small, and we get the Classical Law of Conservation of Momentum. Space Component Now we can write the space component of * as:
Philosophy It behooves us to stop and think here for a moment. It seems that the space component of * is very close to the Classical Law of Conservation of Momentum. Does this make sense? It does. When moving from 3 dimensions to 4, we added time. If we look at the space part of our 4-dimensional analog, it seems reasonable to see things that were developed in 3-dimensional space.
is very close to the Classical Law of Conservation of Momentum. If we could somehow redefine mass as some sort of rest mass times , then this equation would match the Classical Law. So now what?
Mass Can Mass Change? If so, what happens if Mass is not constant?
How to Define Rest Mass • Fix a Lorentz coordinatization • Arbitrarily choose a particle at rest • Define the rest mass of that particle to be one
Equals the post collision mass and velocities Than by solving for m1 and than looking as v1 approaches zero In other words, The initial mass and velocity
Rest mass is than defined as just the mass of an object having velocity zero
Relativistic Momentum also called 4- Momentum or energy momentum p = (Rest mass)(4-velocity)
The break down of p The time component in P is related to the Newtonian concept of Kinetic energy. While the space component in P is related to the Classical Law of Conservation of Momentum.
Substitution and Distribution Recall,
Inspired by the tendency of Newtonian concepts to appear in spatial components of relativistic momentum. We determine relativistic mass to equal M=(rest mass)(time dilation factor)
Hence, p can now be defined as a vector with relation to M.
= speed of the object = mass of the object Background on Kinetic Energy • Kinetic energy is the energy of motion • Kinetic energy is a scalar quantity; it does not have a direction. • The Kinetic energy of an object is completely described by magnitude alone.
Newton’s 2nd Law of Motion Explains how an object will change velocity if it is pushed or pulled upon Setup for Conversion to Newton Newtonian Perspective versus the classical definition of momentum