Chapter 13 the Theories of Special and General Relativity Special

Ron Ferril SBCC Physics 101 Chapter 13 2017Jul23A Page 1 of 14

Chapter 13 The Theories of Special and General Relativity

Special Relativity The Theory of Special Relativity, often called the Special Theory of Relativity or just “special relativity” for a shorter name, is a replacement of Galilean relativity and is necessary for describing dynamics involving high speeds. Galilean relativity is very useful as long as the speeds of bodies are fairly small. For example, the speeds of a supersonic jet, a rifle bullet, and a rocket (such as the Apollo vehicle that went to and from the Moon at about 25,000 miles per hour) are all handled well by Galilean relativity. However, special relativity is required in explanations of the dynamics at the higher speeds of high-energy subatomic particles in cosmic rays and in large particle accelerators.

Both Galilean relativity and special relativity involve “frames of reference” which are also called “reference frames.” A frame of reference is the location of an observer of physical processes. For a more mathematical view, a frame of reference can be viewed as a coordinate system for specifying the locations of bodies or physical events from the viewpoint of the observer. An observer may specify positions of bodies and events as distances relative to his or her own position. The positions can also be specified by a combination of distance from the observer and angles from the direction the observer is facing. The distances and angles can be called “coordinates.”

An “inertial frame of reference” or “inertial reference frame” is a frame of reference that does not accelerate. If an observer is accelerated, you can expect the observer would experience inertial forces. Thus, an observer does not experience inertial forces in an inertial frame of reference. The observer can consider himself or herself to be at rest in his or her own inertial frame.

For example, suppose two people on a moving train are throwing a ball back and forth in a game of “catch” and this game is observed by people on the train and people on the ground outside the train. Each observer of the game can specify the position of the ball relative to himself or herself at any time.

Thus, a man throwing the ball forward, in the direction the train is moving, to be caught by a woman can specify the position of the ball as the distance from him. The woman can specify the position of the ball by the distance of the ball from her. An observer on the ground can specify the position of the ball by the distance of the ball past a reference point on the ground. Since we can expect the speeds (magnitudes of velocities) involved to be much smaller than a rifle bullet, we should be able to use either Galilean relativity (which was introduced in a previous chapter) or special relativity which is being introduced in this chapter. The Bohr correspondence principle assures us that the newer Theory of Special Relativity should give either the same or almost the same results as Galilean relativity for such speeds. The Bohr correspondence principle allows small differences in the results because of tiny corrections special relativity may introduce.

To make our example of the game of catch on a train easier to visualize, let us assign some numerical values to the example. Suppose the train is moving as 40 miles per hour relative to the ground, and the man throws the ball forward, in the direction the train is moving, at a speed of 30 miles per hour relative to himself. The man does not need to notice any movement of the train. By Galilean relativity, an observer on the ground sees the train moving at 40 miles per hour and the ball moving at 70 miles per hour (40+30=70) relative to this observer on the ground. Special relativity presents these speed values as very good approximations and the differences in results between the two types of relativity are so small that only the most sensitive laboratory devices (devices called “interferometers”) can detect them.

To make the differences introduced by special relativity more plain, we can consider a similar example involving a high speed. We will still have a train moving at 40 miles per hour, but suppose the man and woman on the moving train of the previous example replace their ball with light. The man can shine a flashlight forward at the woman, and instruments can measure the speed of the light. Earlier chapters told us that the speed of light is maximum in vacuum. Thus, let us assume the light shines forward through a vacuum. This is surely a high enough speed that we will need to use special relativity. The man and woman on the train can measure the speed and confirm that it travels at the speed of light in vacuum and can denote this speed by c. However, observers on the ground can also measure the speed of the light. By Galilean relativity, we would expect the speed of the light to be the sum of the speed of the train and the speed measured by the man and woman. If the speed of the train is v then we speak of light relative to observers on the ground would be expected to be v+c. However, a major claim of special relativity is that observers on the ground would measure the same speed c for the light in vacuum as the man and woman who are both on the train measure.

Similar experiments were actually done with light in partial vacuums inside pipes and the motion of the train replaced with the motion of a point on the Earth's surface as the Earth rotated and revolved around the Sun. One such experiment, the Michelson-Morley experiment, used a “Michelson interferometer” in an attempt to detect a medium in which light was thought to travel. One reason for suspecting the existence of such a medium is that all other waves known at the time traveled in media. Thus, it was thought that light may travel in a special type of media that existed even in vacuum. The experiment did not detect any such medium. However, this “null” result of the experiment caused much thought about how light travels. Several questions were answered by new fundamental physical laws introduced in special relativity.

When physicists develop a new fundamental theory of physics, they find there are several ways they can choose and express the basic postulates—the fundamental laws—of the theory. When Einstein was developing the Theory of Special Relativity, he chose the following two postulates as being the foundation for his new theory. In these postulates, the term “uniformly moving” means moving without acceleration.

The Relativity Principle: All laws of nature are the same in all uniformly moving frames of reference.

Law of Invariance of the Speed of Light in Vacuum: The speed of light in vacuum remains the same constant for all frames of reference regardless of the motions of the source of the light or observers.

Galilean relativity and special relativity can both assert that the same physical laws apply in all uniformly moving frames, but Galilean relativity is incompatible with the invariance of the speed of light in vacuum. The different results between Galilean and special relativity become tiny when all the speeds involved are much smaller than c, but the differences are significant when speeds closer to c are involved. The consequences of the invariance of the speed of light in vacuum are severe. The concept of simultaneous events (events that occur at the same time) becomes complicated because two events can be simultaneous when viewed by observers in one frame of reference, but not simultaneous when viewed from another frame of reference which is moving relative to the first frame of reference. Also, measurements of lengths, distances and time may be different for different observers.

Consider motion in one direction to avoid confusion from multiple directions. Suppose observers in two frames of reference measure the speed of a body or wave. Let the measured speed of the body for the first frame be vb1 and the speed be vb2 relative to the second frame. Let the speed of the second frame relative to the first frame be v21 and in the same direction as the velocity of the body or wave. Let the speed of the first frame to the second be v12. Both Galilean and special relativity assure us that v21 =

-v21. Galilean relativity has vb2=vb1-v21 and vb1=vb2+v21. However, special relativity has

2 vb2 = [vb1 - v21] / [1 - vb1v21 /c ]

2 vb1 = [vb2 + v21] / [1 + vb2v21 /c ]

2 2 When vb1 is much smaller than c, 1+vb1v21/c and 1-vb2v21/c are both approximately 1 and the equations from special relativity give approximately the same values for vb1 and vb2 as the equations from Galilean relativity. (Thus, the Bohr Correspondence Principle is demonstrated.)

Time Dilation and Length Contraction Measurements of length and time can be different for different frames of reference if the frames move relative to each other. Consider again the example of the train. Suppose identical clocks, very precise clocks, are placed on the train and on the ground. If the train moves forward at a fast speed v relative to the ground, and there are good precise clocks on the train and mounted on the ground, then observers on the ground see the clocks on the train are slow and can tell the passengers on the train that they measure the rate of clocks on the train as being slow. The equation for the time ttrain measured by clocks on the train is related to the time tground measured by clocks on the ground. If the clocks on the ground advance to a time reading tground then observers on the ground see the clocks on the train require a time ttrain to advance to the same time reading.

2 2 ttrain = tground / √ [1 – v /c ] = γ tground

2 2 where γ = 1 / √ [1 – v /c ] = the Lorentz factor. Notice ttrain > tground and the clocks on the train require a greater time ttrain, as measured by observers on the ground, to advance to a reading tground. This slowing of clocks in a moving frame is called “time dilation.” The equation ttrain=γtground is valid if the observers on the ground make the measurements of both times ttrain and tground and the time ttrain is the time required for the clocks on the train to catch up to the clocks on the ground. It is very important to notice that observers on the ground use the clocks on the ground to measure the slower running of the moving clocks. The observers on the ground see the time dilation, but observers moving with the train do not see any time dilation.

Consider another measurement in which the observers on the ground merely measure a time interval between two events using the clock on the ground and also using the clock on the train. Notice that the observers on the ground are using the clocks here. The clock on the ground measures time interval

Δtground and the clock on the train measures Δttrain. Then

2 2 Δttrain = Δtground √ [1 – v /c ] = Δtground / γ

Notice the difference between this equation for Δttrain and Δttground and the previous equation for ttrain and ttground. Both equations were for observers on the ground. The equation for ttrain and ttground had ttrain being the time for the clock on the train to catch up as measured by observers on the ground. For example, if the clock on the ground require 10 seconds to the clock to advance 10 seconds, the clock on the moving train require more than 10 seconds to advance its reading 10 seconds. On the other hand, the equation for Δttrain and Δttground had Δttrain as the time the clock on the train advances while the clock on the ground advances Δttground. This may seem confusing because I have given more than one equation for time dilation, so I give you this rule: A helpful rule for uniformly moving frames of reference: The observers in a uniformly moving frame of reference consider themselves to be at rest and see their clocks running normally, but they see the clocks in other uniformly moving frames as running slower than their own clocks as measured using their own clocks.

(That rule works unless the two observers are in very different gravitational fields because gravitational fields also cause clocks to run slower, but that is a topic that is beyond the Theory of Special Relativity and is explained by the Theory of General Relativity.) Thus, if the observers on the ground see their own clock advance 10 seconds in a time interval, they see the clock on the train advances less than 10 seconds in that time interval and requires more time to advance 10 seconds. During the 10 seconds the clock on the ground measures, the time the clocks on the train advance is Δttrain < 10 seconds and the time they need to read 10 seconds is ttrain > 10 seconds.

There seems to be a problem here which you may have noticed. The observers on the ground consider their clocks good and themselves at rest, and they see the train moving and the clocks on the train running slow. However, the passengers on the train can consider themselves at rest and the ground moving. The passengers see their clocks running fine and not slow, but see the the clocks on the ground running slow. Thus, people on the ground can yell at the passengers on the train and say the clocks on the train are running slow, and the passengers can yell at people on the ground and say the clocks on the ground are running slow. Which observers, the ones on the train or the ones on the ground, are correct? Einstein claimed that both sets of observers are correct. The correct values for time depend on the frame of reference and every observer was able to correctly state the values for their own frames.

Suppose the people want to stop the train to resolve the matter of who's clocks are slow. The idea is that people in one frame of reference will hand their clocks to observers in the other frame of reference so they can see who's clocks are running slow. Stopping the train involves an acceleration backwards (which is a deceleration forward). Thus far we have seen time dilation in a frame moving uniformly. Here “uniformly” means without acceleration. The acceleration of the train changes matters. The observers on the train see the clocks on the ground running fast during the acceleration (or deceleration) of the train. There would be no inconsistencies when the people compare clocks at the stopped train, but there could still be an argument regarding who had the slow clocks while the train was in motion.

Suppose lengths are measured by the observers rather than times, and all observers have identical measuring tools before the train begins moving. When the train is moving, all the observers, both on the moving train and on the ground, would agree on measurements of the heights of objects because the height is measured in a direction perpendicular to the direction the train is moving. All observers in two frames of reference observe the same distances along directions perpendicular to the velocity of one frame relative to the other. Next consider measurements of length along the direction the train is moving. The observers in different frames of reference would not obtain the same length values. When the observers on the ground investigate by observing the measuring tools used on the train, the observers would see the lengths of the measuring tools used on the train have contracted so they are now shorter, in the direction the train is moving, than the tools used by observers on the ground. The length Ltrain of the tools on the train as seen by the observers on the ground are related to the length

Lground of the tools used by the observers on the ground by

2 2 Ltrain = Lground √ [1 – v /c ] = Lground / γ This contraction of the length (in the direction the train is moving) is called the Lorentz contraction. The passengers would look at their own tools and not see them as contracted, but would see the tools used by observers on the ground as being contracted. When the train stops, the tools on the train are identical to the tools used by observers on the ground.

Suppose the experiment with lengths is done another way with the train. Instead of tools, suppose the passengers and observers on the ground acquire identical circular metal plates before the train starts moving. When the train is moving, the observers in either frame of reference will see the plates in the other frame have contracted in one direction so they are no longer circular but have oval shapes. When the train stops, all the plates are again identical.

Another helpful rule for uniformly moving frames of reference: The observers in a uniformly moving frame of reference consider themselves to be at rest and see their tools for measuring lengths operating normally, but they see the length-measuring tools in other moving frames as giving shorter length values for lengths in the direction those frames are moving.

Rectangular Coordinate Systems in Four Dimensions We often use a rectangular coordinate system with coordinates x, y and z. For example, x and y can be coordinates for two perpendicular but horizontal directions and z can be a coordinate for a vertical direction. Special relativity introduces a fourth rectangular coordinate as being a time coordinate. However time does not have the same units as x, y and z. This problem is quickly solved by taking the fourth coordinate to be the product of time t and the speed c of light in vacuum so the fourth coordinate is ct. The coordinates of any point in this four-dimensional “spacetime” can be written as an ordered quadruplet (ct,x,y,z). Consider two such coordinate systems with the ct, x, y, z coordinate system being considered to be at rest and another coordinate system moving along the x-axis with coordinates (ct’,x’,y’,z’) for the same point and the x’-axis parallel to the x-axis. Then the following “Lorentz transformation” equations give the relations between the coordinates. ct’ = γ[ct – (v/c)x] x’ = γ[x – (v/c)ct] = γ[x – vt] y’ = y z’ = z Notice that these coordinate relations show that the coordinates in directions perpendicular to the velocity of the moving reference frame are not affected by the motion of the frame of reference. Galilean relativity would have the following relations for the coordinates. ct’ = ct x’ = x – vt y’ = y z’ = z For fairly small speed v, the Lorentz transformation and the Galilean transformation give similar results. As speeds get larger, the two types of transformation give very different results. However, both

Lorentz and Galilean transformations show that the coordinates are unaffected for directions perpendicular to the velocity of the moving frame.

Momentum, Mass and Energy Since observations of length, time and velocity yield different values for different uniformly moving frames of reference, we can expect that measurements of linear momentum will differ between frames. We consider linear momentum to be the product of mass m and velocity v. Thus, linear momentum p is a vector. We expect the velocity v to depend on the frame of reference but, in order for everything to be consistent, the only way we can maintain p=mv is for the mass to also change. Thus, physicists can adopt either of two views. Either • the mass changes with velocity of the frame of reference and p=mv remains valid, or

• the mass remains constant and p=mv must be replaced with a new expression.

With either viewpoint, the linear momentum is p=γmv and this expression can be considered as either • having the old definition of linear momentum (p=mv) and a mass γm, called the “relativistic mass,” that depends on speed, or

• having the mass m independent of speed (and sometimes called the “rest mass” m0 in many old books) and the new expression for linear momentum is p=γmv. The second viewpoint, with mass independent of speed and the expression for linear momentum changing from p=mv to p=γmv, has become more popular among physicists. Newton's second law

Fnet=KΔp/Δt (but not Fnet=ma) holds for either viewpoint.

Choosing the viewpoint of mass changing with speed causes a problem with gravitational mass mgrav and inertial mass minert. The gravitational mass of a body is defined by the force on the body in a specified gravitational field. Newton's Law of Gravity gives us the gravitational force Fgrav on a body of mass mgrav as

2 Fgrav = G M mgrav / r

2 mgrav = Fgrav r / G M

However inertial mass is defined by a form of Newton's Second Law of Motion that is often written for constant mass.

F = minert a

minert = F / a where F is the magnitude of force and a is the magnitude of acceleration. In Newton's time, the gravitational mass and inertial mass were assumed to be proportional to each other. With modern units in which gravitational mass and inertial mass are measured in the same units, the gravitational mass and inertial mass would be equal. miner=mgrav. This would not hold if we allow the inertial mass to vary with speed except in the limit as speed goes to zero. As speed increases toward c, the relativisitc inertial mass increases toward infinity. Thus, the trend among new physics books is to refer to the rest mass as the mass and replace p=mv with p=γmv. Then mass is a measure of inertia at (or near) rest though it must be noted that inertia increases as the speed approaches c.

Mass is treated as an energy, called mass energy, in special relativity. Actually, the mass energy E for a mass m is E=mc2. A question that may arise after a discussion of “relativistic mass” γm and “rest mass” 2 m0 is which mass is to be used in E=mc ? The mass used depends on which energy you are trying to 2 calculate. The “rest energy” E0=m0 c of a particle or body is the energy due just to its rest mass m0 and this energy can be converted to other forms of energy. Converting mass energy to other forms of energy 2 involves converting matter to those forms of energy. The “total energy” of a particle or body Etot=γmc is the total energy of the particle including both mass energy and kinetic energy of the particle. The old 2 formula for kinetic energy being ½mv is no longer valid. The new formula for kinetic energy EK is

2 2 EK = Etot – mc = Etot – m0c where m is the rest mass m0 of the body.

Examples of matter or mass energy being converted to other forms of energy include the following. • Matter-antimatter annihilations which are subatomic processes in which a particle and its “antiparticle” are replaced by electromagnetic energy in the form of two gamma rays. The two particles are gone but energy is conserved. • Fusion of a proton and neutron to form a deuteron which has less mass than the sum of the masses of the separate proton and neutron. Some of the mass energy is converted to binding energy which is potential energy due to the strong force. • Nuclear power plants in which part of the mass of uranium is converted to heat.

• Nuclear fusion in the Sun which converts mass energy into electromagnetic energy and heat.

The Speed of Information and Mass An important consequence of special relativity is that the speed c of light in vacuum is the maximum speed for information. Information cannot travel faster than the speed of light in vacuum. If any particle or wave travels faster than c, then it cannot carry information to us. There are phenomena that travel faster than c, but such phenomena don't carry information. For example, there are waves that can travel faster than c.

If a body has nonzero mass, then its inertia and linear momentum both increase with increasing speed. As the speed approaches c, the inertia and the linear momentum approach infinity. Anything with mass and moving must travel at speeds slower than c with respect to the space or spacetime in which it travels. Neither information nor mass can travel faster than the speed of light in vacuum. Thus, the space travel presented in science fiction, where a person travels across the galaxy, sends messages back to Earth and returns to Earth in his or her lifetime, does not seem possible.

The General Theory of Relativity The Theory of General Relativity, also called the General Theory of Relativity or just “general relativity” for a short name, is a theory of gravity. Newton's Law of Gravity was a postulate, a fundamental physical law, that did not give any reason why gravity should exist. Einstein developed general relativity to explain gravity in a way consistent with special relativity. The whole development of general relativity starts with Einstein's Equivalence Principle. Equivalence Principle: Physical processes in an accelerated frame are identical to physical processes in a (Newtonian) gravitational field.

Remember the inertial forces discussed early in this course. Inertial forces come from an interpretation of Newton's Second Law of Motion.

Fnet = K Δp / Δt The left side is the total or net force. However, the equation can be re-arranged.

Fnet – K Δp / Δt = 0

Then the term -Δp/Δt is the inertial force. Thus, if I sit in a car and the car accelerates, the total (real) force on the car would be Fnet but I would feel -KΔp/Δt as though it was a force. If you are in a closed room and think you feel an inertial force, you don't know if what you feel is really an inertial force due to the room being accelerated or the (real) force of gravity pulling you. This is a major point of the Equivalence Principle: inertial forces and gravitational forces seem to be identical.

One consequence of the Equivalence Principle is that gravity must be able to bend light from its straight path. Although Einstein's general relativity mostly agrees with Newton's Law of Gravity, this bending of light is one significant deviation from Newton's concept. Newton’s Law of Gravity did not mention forces on particles that have zero mass. Photons of light have zero mass (or zero rest mass since they can be considered, in one viewpoint, as having nonzero relativistic mass). With zero mass, Newton's Law of Gravity assigns zero gravitational force to light. However, the Equivalence Principle shows that light should bend even with zero mass.

General relativity involves the following important generalizations. • General relativity and special relativity both involve a four-dimensional “spacetime” instead of just a three-dimensional space. Three-dimensional space has width, depth and height. Four- dimensional spacetime adds time as the fourth dimension. Actually time alone doesn't have correct units for a fourth dimension. Thus, Einstein had the fourth dimension be the product of time t and the speed of light in vacuum c. Thus, you can write the coordinates of spacetime as width x, height y, depth z and the time coordinate ct. • Curved spacetime is a major feature of general relativity even though our experiences make spacetime appear flat. To get a feel of what curved spacetime is like, consider drawing a large triangle on the surface of the Earth. The surface of the Earth is curved. The old version of the Pythagorean Theorem you learned in school no longer applies to a right triangle on this curved surface. That is a key point. Euclidean geometry applies to flat space rather than curved spacetime. Curved spacetime is handled by differential geometry (a branch of matematics). • A generalization for curved spacetime of the old Pythagorean Theorem for right triangles is at the heart of general relativity. The main fundamental physical law of modern general relativity is one equation, the Einstein Field Equation, that allows determining the details of the revised Pythagorean Theorem for a given physical situation. (After the Einstein Field Equation is learned, much of the rest of general relativity is mostly advanced mathematics. Thus, a course in general relativity may be described by some people as a little physics and much mathematics. However, the boundary between physics and mathematics is vague.)

Newton described gravity as a force and objects thrown into the air follow a curved path because of this force. The curved path is evidence of the gravitational force. Einstein described gravity as curved spacetime and could describe the path of the thrown object as a nearly straight path in a curved spaccetime. The nearly straight path is called a “geodesic.” With the nearly straight path, one viewpoint is that there is no force but just a curved spacetime. In a curved spacetime, light goes along a straight path but bends because the spacetime is curved. In another viewpoint which has good agreement with Newton's Law of Gravity, the gravitational force is considered to pull on energy since mass (or the product of mass and c2) is a form of energy (mass energy). Thus, general relativity results in an inverse- square law like Newton's equation for gravity with the square r2 of distance r between bodies in the denominator but with mass replaced by energy (and the universal gravitational constant G replaced by another constant). (Newton did not know about the equivalence E=mc2 of mass and energy, and this equivalence is a major feature of modern relativity.)

Time and Gravitational Red Shift Strong gravitational fields slow clocks much like high speeds do, but an observer in a strong gravitational field can agree with an observer far away from the field because both observers agree that the clocks in the strong gravitational field are running slower than clocks away from the field. Light sources in strong gravitational fields tend to emit light that appears to observers away from the gravitational field to have lower frequency because the clocks run slower in the field. This is the cause of the “gravitational red shift” seen for light coming from regions of large gravitational field. By the Equivalence Principle, clocks in accelerated frames also run slower. Some books present the example of clocks on a turntable that is rapidly turning. The clocks run slow where the centrifugal force (an inertial force) is high. This effect on time can be viewed as being the cause of the precession of planetary orbits (for Mercury) near the Sun.

The Pythagorean Theorem and General Relativity This section is optional (extra reading material) in that I don't expect students of this course to do calculations in general relativity. However, students are typically interested in the mathematics involved. The equations presented in this section are actually from a branch of mathematics called “differential geometry.” These equations should give a student a feel for what “curved” spacetime

© Copyright 2015, 2016 and 2017 by Ron Ferril. All rights reserved. Ron Ferril SBCC Physics 101 Chapter 13 2017Jul23A Page 13 of 14 means. Curved means the Pythagorean equation has more terms in it than would be present for “flat” spacetime.

The Pythagorean Theorem in flat two-dimensional space is (Δs)2 = (Δx)2 + (Δy)2 where Δs is the length of hypotenuse of a right triangle, and Δx and Δy are the lengths of the legs of the right triangle. In three-dimensional space, the Pythagorean equation is (Δs)2 = (Δx)2 + (Δy)2 + (Δz)2 Here the Δx, Δy and Δz are coordinate intervals of a line segment of length Δs. Rather than write the coordinates as x, y and z, a popular notation uses the superscripted variables Δx1, Δx2 and Δx3. With this notation, the Pythagorean equation for flat space becomes (Δs)2 = (Δx1)2 + (Δx2)2 + (Δx3)2

In the four-dimensional spacetime of either special relativity or general relativity, time is treated like a displacement coordinate. This may seem absurd because time has units like seconds and hours but displacement has units like meters and miles. Einstein used the product of time and the speed of light in vacuum to form his new coordinate Δx0=cΔt. Thus, we can use the coordinate Δx1 = x to represent width, Δx2 = y to represent height, Δx3 = z to represent depth and Δx0=cΔt to represent intervals of the new time coordinate. In the “flat” spacetime of special relativity, the four-dimensional Pythagorean Theorem becomes (Δs)2 = (Δx1)2 + (Δx2)2 + (Δx3)2 - (Δx0)2 where Δs is the length of a short line segment or hypotenuse in the four-dimensional spacetime. The minus sign shows time as a special coordinate. This four-dimensional Pythagorean equation can be written in another notation to prepare for the changes introduced by general relativity.

2 0 2 1 2 2 2 3 2 (Δs) = g00(Δx ) + g11(Δx ) + g22(Δx ) + g33(Δx ) where g00 = -1 and g11 = g22 = g33 = 1 for the “flat” spacetime of special relativity. In this form, the four coordinates all seem to be treated the same because there is no minus sign except inside g00.

General relativity allows for “curved” spacetime in which the Pythagorean Theorem can have more terms and the absolute values of g00, g11, g22 and g33 are no longer required to be one. The more general equation that replaces the simple Pythagorean equation when spacetime is curved has the form

2 0 2 1 2 2 2 3 2 (Δs) = g00(Δx ) + g11(Δx ) + g22(Δx ) + g33(Δx ) +

0 1 0 2 0 3 2g01Δx Δx + 2g02Δx Δx + 2g03Δx Δx +

0 1 0 1 0 1 2g12Δx Δx + 2g13Δx Δx + 2g23Δx Δx All these “g” values are components of a mathematical object called the “metric tensor.” This tensor is said to be symmetric because its elements are symmetric under interchange of indexes as in g01 = g10, or g21 = g12, for example.

Example Calculations

Example 01. Suppose a simple pendulum with period T0 is placed aboard a spacecraft and the length

L0 of the spacecraft is measured while it is at rest before it leaves Earth, but then the spacecraft leaves Earth and travels at half the speed of light in vacuum. We can calculate the length contraction and time dilation. Observers at “rest” on Earth will see a new length L for the spacecraft and period T for the pendulum.

2 2 2 L = L0 √ [1 – v /c ] = L0 / γ = L0 √ [1 – (½) ] = L0 √ [3] / 2

2 2 T = T0 / √ [1 – v /c ] = γ T0 = (2√ [3] / 3) T0

Example 02. Suppose a body has a mass m = 1.0 x 10-9 kg and a velocity of half the speed of light in vacuum (c/2) to the right. We can calculate the magnitude of the linear momentum of the body. p = γmv = mv/√[1–v2/c2] = [1.0x10-9 kg x 1.50x108 m/s]/[√[3]/2] to the right = 0.17 kg m/s to the right Thus, the linear momentum is 0.17 kg m/s to the right.