 Welcome to video eight of this series on the theory of relativity. Here we want to investigate how we can form a yardstick for space-time measurements. At the end of the previous video, we had seen that the methods of differential geometry appeared ideal for expressing Einstein's idea that gravity represents curvature of space-time. But differential geometry is built upon the assumption that between any two points there exists an invariant distance and that in any coordinate system we can express that distance in terms of coordinate differences. However, special relativity has shown us that two reference frames in general won't agree on space or time measurements. The key insight is primarily due to Herman Minkowski. A translation of his 1908 presentation titled Space and Time appears in the Dover publication book of reprints that we've referred to several times in this series. In his introduction, he states, the views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth, space by itself and time by itself are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality. Well, let's see what he meant by that. Recall video two where we developed the equations of special relativity that relate time and space measurements and two inertial reference frames that are moving relative to each other with velocity v. Here's Einstein's version. We prefer to use units in which the speed of light is one so we can scratch the c-squared terms out of his equations. One system uses Latin letters t, x, y, and z for time and space, and the other uses Greek letters, tau, psi, eta, and zeta. Our problem is that the Greek time and space coordinates, tau and psi, are a mixture of the Latin time and space coordinates, t and x. To see what Minkowski did, consider an analogy. Suppose centered on some particular point you set up a system of coordinates. For a second point, you can then determine the coordinates x and y in your system that uniquely specify the location of that point. Suppose someone else chooses a system rotated with respect to yours. The position of the second point can also be expressed in their system, say by coordinates psi and eta, but clearly your x and y will in general not be the same as their psi and eta. The coordinates of a point are specific to a particular coordinate system. However, the distance between the points, let's call it s, has a physical significance independent of any coordinate system. Anybody can take a tape measure and determine that distance. Specifically, in the x, y system, you can draw a triangle with sides x and y, and the hypotenuse will be s. You can figure out s from x and y using the Pythagorean theorem, s squared equals x squared plus y squared. In the psi, eta system, you can also draw a triangle with s as the hypotenuse, but with sides psi and eta. The Pythagorean theorem will again give you s, but now in terms of psi and eta. s squared equals x squared plus y squared is the equation of a circle of radius s, and so is s squared equals psi squared plus eta squared. Any point on the circle in one coordinate system will also be on the circle in the other, although with different coordinates. It doesn't matter what angle theta a circle is rotated by, it always looks the same. We say circles are invariant under rotation. The equation of a circle of radius s doesn't change regardless of how we rotate the coordinates. With the little trigonometry, we can write each Greek coordinate as a combination of the two Latin coordinates with factors of the cosine and sine of the rotation angle theta. If you sum the squares of the right side expressions, you find the reason it reduces to x squared plus y squared is because for any angle theta, cosine squared theta plus sine squared theta equals one. Greek and Latin coordinates are related in Einstein's special relativity equations look a lot like the rotated coordinate expressions with beta, the dilation factor, and beta times v taking the place of cosine and sine of theta, and the plus sine replaced by a minus sine, similar but different. The sum of squares of the Greek coordinates does not equal the sum of squares of the Latin coordinates. In this case, the key fact is that beta squared minus beta v squared equals one. Because of this, it's the difference of squares that is the same for the two systems. We get s squared equals t squared minus x squared equals tau squared minus psi squared. The graphical representation we developed in video two shows the similarity and difference between rotation and the special relativity transformation. Here are the blue lines that note the Latin coordinates and the red lines, Greek coordinates. The green lines correspond to the speed of light. The relation between Latin and Greek coordinates looks like a rotation except the two axes are rotated in different directions. It's more of a squeezing and stretching operation. Still, we've found a coordinate relationship that looks the same in all reference frames. For rotation, s squared equals x squared plus y squared is a circle of radius s. What does s squared equals t squared minus x squared represent? If you choose a value of s, you get a plot like the black curve shown here. This is a hyperbola. This particular hyperbola has s squared equals five. Since three squared minus two squared, nine minus four equals five, the point t equals three, x equals two, or minus two, should be on the hyperbola. In the blue Latin system, if we go three units to the right and two units up or down, we indeed fall on the hyperbola. If we do the same thing in the red system, three units to the right and two units up or down, we again fall on the hyperbola. In special relativity, it's hyperbolas that are invariant. So, s squared equals t squared minus x squared. But what does s represent? Let's plot a patch of the tx coordinates in blue and the tau xi coordinates in red. The green line shows the speed of light and the black curve is a piece of our hyperbola. If the two events happen at the same place in the blue system, then x equals zero and we simply have s equals t. In the blue system, moving along the x equals zero line, it takes one, two, three, four units of time to reach the hyperbola. In the red system, this would look like five units of time. Likewise, if the events happen at the same place in the red system, then xi equals zero and s equals tau. Moving along the xi equals zero red line, we travel one, two, three, four units of time to reach the hyperbola. In the blue system, this would look like five units of time. Minkowski called s the proper time between two events. It's the elapsed time as seen in a frame where the two events occur at the same place. The invariant hyperbola has the significance that any clock leaving the origin at any velocity will reach the hyperbola in the same s units of time as measured by a clock in that system. Minkowski's proper time idea also gives us another way to understand the twin paradox as what we'll call the principle of extremal aging. Suppose in some reference frame we put a clock at the origin and let it sit there for two units of time. This defines these two blue events and the clock naturally moves between the two. Since x doesn't change, the proper time is just t equals two. Now suppose we choose another event at a different position in space at time t equals one. We move a clock from the first event to this intermediate event. The proper time it will record is s one equals the square root of one minus x squared. From there we change directions and move it back to the final event. The proper time of that interval will also be square root of one minus x squared because in each case the clock moved to distance x. Since one minus something positive is less than one, the two proper times s one and s two are each less than one. So the sum is less than two, which is the proper time on the blue path that corresponds to inertial or natural motion. We have the result that the natural inertial motion between two events is that which results in the longest elapsed time as seen by a clock that moves between the two events. This is just a twin paradox. There the earth twin was the one who moved naturally, that is without changing velocity, between the two events and ended up older than his space twin sister who experienced acceleration. Now for the flat spacetime of special relativity, Minkowski's invariant proper time is an interesting way to look at things, but we're looking for a yardstick for the curved spacetime of general relativity. Ah, but now back to the happiest thought of Einstein's life. If a person falls freely, he will not feel his own weight. So even in the gravitational field, proper time will provide a yardstick for a small freely falling reference frame. Any other reference frame is just a transformation of the freely falling frame so proper time must be the yardstick we're looking for. The only difference is the small caveat. As we've seen that means that we have to limit ourselves to small differential displacements. So we use an expression like ds squared equals dt squared minus dx squared and then we can add up the little increments. But the final result follows the same principle. Consider earth's surface. Gravity pulls things down. Suppose a clock starts at t is equal to zero and we let it rest on the surface to say t equals 30 seconds. This clock will feel its own weight so 30 seconds can't be the proper time between these two events. After proper time, we need to throw a clock from the first event so it freely falls to the second event. At every instant of time, it's moving according to special relativity. By the principle of extremal aging, the proper time s that it shows will be longer than the time t on the clock that felt its own weight. This then is the yardstick of space time, flat or curved. No matter what the gravitational field, you could be near a black hole or in between two rapidly orbiting neutron stars. Remember, throw a clock from one event so that it falls freely to the second event. The elapsed time recorded by that clock is the proper time between the two events and all reference frames will be able to calculate this time using their metric tensors. We've seen that light plays a central role in relativity so we need to look at this special case. In special relativity, the speed of light, which is one in our units, is represented by the green line on this graph. Light moves such that in say three units of time, it covers three units of distance. In general then, dx will be plus or minus dt, plus or minus because light can move either up or down or left or right and so on. The resulting proper time is therefore zero. The proper time between two events connected by array of light is always zero. If you could move at the speed of light, apparently you could go anywhere in the universe in zero time, at least as measured by your own clock. We now finally have the complete general theory of relativity, at least for empty space outside of matter. And we can write it all down on one page. You blanket all of space-time with four coordinates, the first three for space and the fourth for time in any way you want. The catch is you cannot assume these coordinates have any specific physical meaning. We'll have to let the theory tell us what they mean. We might say that no prior geometry can be assumed. Then the generalized Pythagorean theorem allows us to convert small coordinate differences between events into the corresponding invariant proper time. The change in a coordinate x over the change in proper time s, we interpret as a space-time velocity u, the tangent vector of differential geometry. Particles move along geodesics, so the geodesic equations provide the equations of motion by telling us how to change velocity or space-time direction as we move along the path. The draw-square test for space-time curvature generates the Riemann tensor. Summing various terms of the Riemann tensor produces the Rieschi tensor that measures how the volume of a dust cloud changes as it falls in a gravitational field. Einstein's equation of gravity are that the dust cloud volume doesn't change, at least for a short time, so the Rieschi tensor vanishes. Finally, this gives us the equations we need to solve to determine the metric coefficients of the generalized Pythagorean theorem. And from there we can calculate how any object, and even light, will move in a gravitational field. We're now in a position in future videos to work out in detail the incredible predictions of Einstein's crowning masterpiece.