MASS becomes a critical part in our discussion. Just like time, it has different definitions.
- MASS, Definition 1. In general, we think of mass as how much matter a body has. (the total number of neutrons, electrons, and protons found inside the object) To determine the weight of something in everyday terms, we multiply the mass of an object by the effect of the earth’s gravitational pull. IT IS OF UTMOST IMPORTANCE TO REALIZE THAT MASS IS INDEPENDENT OF LOCATION IN SPACE. The same volume of mass on the earth has less weight on the moon as a direct result of the force of gravity the moon exerts in comparison to the force of gravity the earth exerts.
- MASS, Definition 2. This definition of mass is what was used in physics. For a long time, relativistic mass was the standard physics definition. It says that mass depends on a body’s motion in relation to the motion of an observer. If you measure your weight while in motion, you will always weigh the same amount. Yet, if someone were to measure your weight while observing you in motion, your weight would actually increase as the amount of activity or motion increases. This is because mass was defined as the amount of force required to cause a body to accelerate. You were combining the physical characteristics of an object with the energy used to put it in motion.
- MASS, Definition 3. Today, in physics, mass and energy are nearly always viewed simply as energy. It is important for you to understand that there is a direct relationship between mass and energy. Hopefully, I have not yet lost you.
- ENERGY is an object’s ability to perform “WORK”. The two most commonly recognized forms of energy are potential (a body at rest) and kinetic (a body in motion). The basic law of conservation of energy says that energy can neither be created nor destroyed. It can merely change forms. If you have a car parked in your driveway, it is an object at rest. It has the potential of motion, but it just sits there. Once you get in and start the motor and engage the drivetrain, you change its potential to actual motion or kinetic energy. If you wish to convert the kinetic energy or motion of the vehicle back to a state of rest such as stopping at a stop sign, you apply your brakes causing friction to occur. Friction creates heat which is also a form of energy which dissipates the kinetic energy of the car’s motion bringing it back to its potential state. The long way around saying that you can start, drive, and stop a car.
- LIGHT is energy. It has two unique concepts of existence. It can be defined by its wave characteristics or diffraction. It can also be defined by the pieces of energy in motion known as discrete particles. These two separate but constant features are known as duality. IT IS CRITICAL TO UNDERSTAND THAT BOTH OF THESE CHARACTERISTICS ARE PRESENT AT THE SAME TIME. IT IS NOT AN EITHER/OR EVENT. (Both waves and particles occur at the same time) The very same beam of light that displays energy (particles) can also be diffracted by a prism (wave). The scientific terms for such are
photon for particle and electromagnetic radiation for wave.
Light is very important to us. It is a primary factor in how we gather information. If we “see” something, it is because we have registered a transmission of electromagnetic radiation. (Life sure gets complex when looked at from a physics viewpoint) Before we get further into this part of light, maybe we should first look at photons. A photon is defined as emitted energy from an atom.
Electrons orbit the nucleus of an atom. Unlike our solar system, these orbital paths that electrons follow are uniform or equidistant from the core. These paths are called ORBITALS. Each individual orbital can have only one electron’s worth of energy contained in it. This is known as a discrete or limited amount.
Atoms can absorb and shed energy (change the location of electrons surrounding the nucleus) by external stimulation. Fire (heat), compression, freezing, etc… are forms of force that alter the energy of an atom. The amount of energy that an atom takes on or gives off does not change its physical characteristic. A hot piece of lead is still lead. A cold piece of gold is still gold. They simply have reacted to external stimulation.
Generally speaking, the period that an individual atom is excited is very short, measured in millionths and billionths of a second. What happens when an electron absorbs energy is that an electron close to the nucleus of the atom will jump to an orbital path further away from the nucleus. At the end of the period of excitement, the electron will fall back to its normal position in relation to the nucleus. Depending upon the level of stimulation, a packet of energy called a photon is released from the atom. This is the difference in the amount of energy between the peaks and valleys (highs and lows) of energy levels when electrons shift their orbital patterns. This packet of energy or photon has wave or frequency value which may be seen as a form of light if it falls into the visible spectrum.
In other words, light waves are a form of energy created by excited atoms that oscillate or change back and forth between states. If this change falls in a very narrow frequency range between ultra-violet (very high) and infra-red (very low), we will be able to see them as some variety of “color”. In reality, this oscillation is a change between an electric field and a magnetic field which is where the name electromagnetic radiation comes from. Radiation in this sense refers to “the giving off” or shedding of energy. As energy levels can vary that excite atoms, so do the frequencies of the oscillation or photons vary. What we commonly refer to as “light” is only a small band of the electromagnetic radiation available.
At the very high end of this spectrum are the gamma rays. This is the very high energy stuff that can penetrate nearly every substance. These gamma rays can cause very significant damage to our biological system. This is the damaging part of nuclear radiation which is a whole other topic. As noted earlier, each frequency of electromagnetic radiation has a specific value. Within the visible spectrum, each frequency is seen by us as a specific color.
As light flows through space, it encounters all kinds of matter. Some of these items reflect light and others diffuse or bend light. The moon, planets, and any non-illuminating item around us such as trees, grass, houses, people, etc… all reflect light at various wave lengths or frequencies. This bending of light is called refraction. That is why we see these things in color assuming our eyes are working properly. Even though these items are not a source of light in and of themselves, they are interacting with light. Sometimes as light is traveling around these different masses, it is bent in such a fashion as to display a specific color. When you block certain frequencies of light, then specific colors are seen. Green leaves for instance are a filter taking out all the color spectrum except green. A rainbow occurs when light from the sun bends around molecules of moisture suspended in the air. Prisms also provide this same type of effect.
- THE SPEED OF LIGHT is a simple way to talk about the speed of electromagnetic radiation. This is the speed that we receive information. You cannot know that something happens until you “see” evidence of it. This evidence comes to our eyes either by reflection or direct emission of light from a source. Radio signals are converted to sound that we “hear”, but do start life as testimony to motion as a form of light. So, until the light reaches you with evidence of motion, you do not know that the motion occurred. (radio signals, flash of light, etc…are all forms of electromagnetic radiation) All light forms travel at 186,000 miles per second (in a vacuum).
Picture a line of cars at a stop sign. As the signal turns green, each car begins to move after a delay waiting for the car in front of it to begin to move. The longer the line, the greater the delay. Even if all the cars were able to move the very instant that they saw the light (not taking into account the delay for the reaction time of other drivers in front), the last car in the que would still have to wait until the signal light change is seen. Granted, this occurs almost instantaneously, but there is still a minimum delay time that it takes the electromagnetic radiation (green light) wave to reach from the light to the eye of the last car driver. This delay time is what relativity is all about.
- REFERENCE FRAMES were the key to Einstein’s approach to relativity. The theory is that the observation point for the event occurring is the point of reference. As you sit and read this, you feel as though you are stationary. Yet from the vantage point of space, you would see the world you sit on is rotating on its axis in an orbit around the sun.
There is no place in the universe that is absolutely still. Everything is in motion, all the time. There is no absolute point of reference. Therefore, there can be no single place on which to base observations. In other words, all observations are relative relationships. You are running toward me and I am stationary. There are two points of observation. You see me coming closer as you run toward me and I see you coming closer as you run toward me. We both observe the event of your motion from different perspectives or reference frames.
- THE LORENTZ TRANSFORMATIONS are mathematical equations that allow us to transform from one coordinate system to another. Why would we want to do this? Because special relativity deals with frames of reference. When you analyze properties from one frame to another, it is necessary to first transform from one coordinate system to another. Thus, we can utilize the Lorentz Transforms to convert length and time from one frame of reference to another. For example, if you are flying in an airplane and I am standing still on the ground, you could apply the transformations to transform my frame of reference into your frame of reference and I could do the same for you in my frame of reference. The previous statements imply that lengths and times are not the same for objects that are in motion with respect to each other. As unbelievable as this may seem, it is a result of SR. Einstein utilized the transformations because they provide a method of translating the properties from one frame of reference to another when the speed of light is held constant in both. I will not delve into the problem of the mathematics of the Lorentz Transforms, but it is crucial to understand their place in this whole theory.
Postulates
Postulate #1 says that the laws of physics hold true for all frames of reference. If you run head-long into a brick wall, the point of impact, aside from hurting a lot, is also the point of energy conservation for the crash. (conversion of kinetic to static energy) The point of energy conservation for your crash into the wall is the same whether observed from where I stand or you crash.
Because the laws of physics hold true in all points of reference, we will be able to predict events and outcomes. The laws therefore allow us to grasp an understanding of how and why our environment reacts the way it does.
Postulate #2 seems like a redundant statement of postulate #1, but it is significant. THE SPEED OF LIGHT IS MEASURED AS A CONSTANT IN ALL FRAMES OF REFERENCE. In other words, if the laws of physics hold true, then you can say that light (electromagnetic radiation) must travel at the same speed regardless of the frame of reference chosen. This is not quite as simple as it seems. A few examples, if I might.
In general terms, we view speed of objects in motion as compounding. If you drive towards me at 50 miles per hour and I drive towards you at 50 miles per hour, we are moving a combined speed of 100 miles per hour.
If you are driving at 50 miles per hour and throw a rock at me as you pass by and the rock initially travels at 20 miles per hour (not getting into the factors of friction and gravity), the combined velocity of the rock as projecting at me would be 70 miles per hour. Yet from your perspective, the rock is moving at 20 miles per hour as you are stationary from your point of view.
Assuming I have not lost you on the concept of your being stationary even though you are in the moving vehicle, we now look at the problem of the speed of light.
In the process of trying to establish a new land speed record, you jump in a vehicle traveling 500 miles per hour. Because the evening is about to fall, you turn on your headlights. What is the speed, from my vantage point of your vehicle’s headlight beams? Using the previous logic, you would add 500 mph to 669,600,000 miles per hour and come up with a total of 669,600,500 and you would be wrong. See, it is getting a little more complex.
You define speed as the distance traveled in a given (finite) amount of time. When you were driving 500 miles per hour, you were moving a relative speed equal to covering the finite distance of 500 miles in the finite time of 1 hour. So, how come if you launch your light beams from your moving car, the speed of the light does not increase by 500 miles per hour? Obviously something is skewed with this logic. As it turns out, not one but 2 things are slightly out of wack. KEY ELEMENT HERE: SPEED IS DISTANCE TRAVELED BY TIME. Light will still only travel 669,600,000 miles during the hour that you have the headlights on. You are simply changing the point at which the light is generated from. Therefore, because the point of reference for the initiation of the light source is a reference point, light does its thing. Just because you relocate the source of the light in reference to SOME OTHER POINT OUTSIDE OF THE REFERENCE FRAME WHERE THE LIGHT IS GENERATED, you do not change how far the light can travel in one hour (comparing apples and oranges).
Just for good measure, what screws up just about everybody with this concept is that the distance you use in your measurement is not the same distance that light uses in its measurement. When an object (mass) is in motion, its physical length (one of the 3 spatial factors of mass) literally shrinks (compresses) in the direction it is moving. (we will get into this in a little while when we do a comparison of the Doppler effect in sound to the compression of mass). Light on the other hand is the far end of the finite scale of motion. It is an absolute. Key here being it is the measured END of a finite scale. (I won’t bore you with the mathematics of this calculation, but it is a true statement). When a mass or object, if you prefer, reaches the speed of light, its measured length is 0. It becomes a 2 dimensional object. This is how an observer standing OUTSIDE the motion would view the occurrence. As you watch the velocity increase, the object becomes shorter and shorter until it simply disappears. (Star Wars/Star Trek jumps to light speed/warp speed respectively) A flash and then nothing, so to speak. Yet, if you were inside the vehicle as it approached the speed of light, you would not notice the compression. This is because from your point of reference, nothing has relatively changed. If you were to take a tape measure and measure the length of pencil you were holding while parked at a stop sign and it measured 6”, it would still measure 6” long as you approached the speed of light because the ruler would also be compressing in the same relative manner. Again note: only length is affected, width and height are not changed. This phenomenon is known in the scientific community as “length contraction.” There is general consensus that this has now been successfully proven. BE VERY, VERY CAREFUL NOT TO THINK THAT TRAVEL BEYOND THE SPEED OF LIGHT IS NOT POSSIBLE. THIS CONCEPT OF LENGTH CONTRACTION MERELY POINTS TO A CHANGE IN PHYSICAL CHARACTERISTICS, NOT THE
CESSATION OF MOTION.
In reality, the speed of light also applies to everyday measurement of distance over time. The only thing is that at our relatively slow speeds, there is so little noticeable effect that for all intent and purpose the change is nil. For instance if you were to measure the amount of compression of a person walking at 3 miles per hour compared to the speed of light, the total compression would be less than 1 trillionth of an inch. Kind of hard to comprehend that change. So, as a matter of form, general physics just ignores that type of distortion which is why you normally compound forces like the earlier examples of collisions and throwing rocks, even though it is technically an incorrect action.
Now the fun sets in. This is where Einstein made use of the Lorentz Transforms mentioned earlier. The Lorentz Transforms allow us to calculate the length contraction. How much contraction occurs is dependent on how fast an object is traveling with respect to the observer. Just to put some numbers to this, assume that a 12-inch football flies past you and it is moving at a rate of 60% the speed of light. You would measure the football to be 9.6 inches long. So at 60% the speed of light, you measure the football to be 80% of its original length (original 12 inch measurement was made at rest with respect to you). Keep in mind that all measurements are in the direction of the motion - The diameter of the ball is not changed by the ball's forward motion. Here are
a few more items to keep in mind:
If you ran beside the football at the same speed, 60% the speed of light, you would always measure the length to be 12 inches. This is no different than you standing still and measuring the football while holding it. If a lady running with the football measured a ruler that you are holding, she would measure you and your ruler to be length contracted as well. Remember, she has equal right to view you as being in motion with respect to her. This is the tricky one. If you are standing still and she is moving in relationship to you from your perspective, you must realize that you are in motion when compared to her perspective and the same contraction of length is occurring to you from her vantage point as you perceive in her. Motion is relative to who views it. There is no correlation between activity and motion only static and dynamic positions.
We should now have digested the two concepts of the absolute speed of light and that any mass in motion compresses its length and the faster it moves, the more compression that occurs.
TIME DILATION is just like length contraction only with the 4th dimensional element of time. Time slows down as motion increases but it only becomes apparent at speeds close to light. ONCE AGAIN, THE STOPPAGE OF TIME IS ONLY APPARENT TO THE OBSERVER OF THE MOTION, NOT SOMEONE IN MOTION WITH THE TIME FRAME BEING OBSERVED. The point of reference becomes the relative plane of observation.
Back in the early 1960’s, the government conducted an experiment using two atomic clocks that were synchronized. One of the clocks remained on the ground and the other was taken for a fast plane ride on a SR72. Upon return of the clock to the location of the clock left on the ground, the airborne atomic timepiece was slower by the exact amount that Einstein’s formula predicted. Why? It all has to do with the absolute speed of light (time light moves a specific distance).
Let’s create an example. Based upon the fixed speed of light, we can accurately calculate the distance that light travels in any given second.
In order to understand just how motion affects the measurement of time given the constant speed of light, let’s construct a light clock mounted on a wheeled carriage. A light clock measures time by reflecting a beam of light off the bottom of the clock body to the top of the clock body and back to the bottom of the clock body. If the clock is stationary, then light travels the path shown in the left sketch. Total elapsed time, 1 second (big clock).
Let's put the same clock in motion.
The clock on the right is in motion relative to you, the observer. If you were able to see the light beam travel during the one second it takes to reach from bottom to top to bottom again, we would see that it is traveling at angles to the plates. So what? Let's review what we know for sure.
1. Light travels at a constant speed in all frames of reference.
2. It took 1 second for the light to travel in the picture on the left.
3. The light traveled further in the picture on the right.
You can tell this by the length of the arrows indicating the path of the light beam. Therefore, if the speed of light does not change and the distance that the light has to travel to complete its cycle becomes greater, then the time to travel the greater distance is longer. THE BASIC FORMULA IS SPEED=DISTANCE DIVIDED BY TIME.
Now, let’s play Einstein for a moment. By using the Lorentz Transform, lets put some real numbers to our little clock picture. We were very industrious and able to get our sun clock to roll along at 90% of the speed of light. You, the observer, are standing still as the clock rolls by. You measure the time that it took the light beam to move its path to be 2.29 seconds. Thrill seeker that I am, happen to choose to be standing on the clock while it is in motion. I will only see 1 second time pass because I am stationary in RELATION to the clock and our clock when stationary to both of us at rest takes one second for the light to travel its path.
So, what do we learn from this? You, while standing by watching the event of the clock and me zip by, aged 2.9 seconds. Because I went riding with the clock, I only aged 1 second during that same interval. (time and space compression). Neat stuff, huh?
Let's look a little closer at what our clock is actually measuring. A clock is designed to record the interval between to events occurring in space. (This is the same definition as a frame of reference.) If the speed of light is held constant universally, time becomes a property required for the defining and existence of the event. ANY OCCURRENCE IS AN EVENT OF TIME AND SPACE (time-space continuum). If you care to improve your knowledge on the subject, there is a neat book out there called SPACE TIME PHYSICS by Taylor and Wheeler.
Theory of Relativity
E=mc^2 forever changed the landscape of physics. Mathematically, since the speed of light is a constant, an increase or decrease in mass at rest is directly proportional to an increase or decrease in system energy. If you combine this relationship with the law of conservation of energy and the law of the conservation of mass, a new equivalence is formed. This then becomes a single law of physics for the conservation of energy and mass. Pretty heavy stuff.
Because of this one formula, we are now able to look at the theories of nuclear fission and fusion. The “c^2” part of that little formula is just an incredibly large number (and, by the way, this is germane to the discussion of the speed of light). Fission is the act of splitting an atom into 2 or more atoms. At the very same instant, a neutron is also released from the bond. Where things get curious is that the 2 atoms and the neutron weigh less than the original atom weighed. So, what happened to the missing mass? This was the heart of the original controversy as to whether or not splitting an atom would start a chain reaction that would destroy the world. The mass converted to kinetic energy, intense heat. The intense part of the heat created by the conversion of mass to energy is that c^2 of the equation.
The other nuclear event corresponding to Einstein’s theory of relativity is that of fusion. If you were to take a hydrogen atom and fuse it with a helium atom by applying some extremely high temperatures, the result is a heavier atom. However, there is again a discrepancy between the weight of the two initial atoms and the newly formed atom. The missing mass in this case is again converted to raw energy.
The energy mass unification of fusion is frequently misunderstood. Many people believe that the mass increases as the system approaches the speed of light. (This misconception stems from the concept of time and mass compression as briefly discussed earlier.) Obviously, if this is a misconception, then that belief is in error.
Let's get in a rocket and attempt to travel to the speed of light. Two things occur with acceleration. 1, you must increase speed and 2, you must overcome the friction created by the motion of the mass of the rocket. Because the system’s resistance to acceleration is a measurement of the system's energy and momentum, there is no reference to mass.
- Energy must be added to the system to increase the ship's speed.
- More of the added energy goes towards increasing the system's resistance to acceleration.
- Less of the added energy goes into increasing the system's speed.
- Eventually, the amount of added energy required to reach the speed of light would become infinite.
If you have hung with my less-than-perfect analogies thus far in attempts to describe the physics involved, then this next point should be rather easy to handle. There is no simultaneity between 2 events when viewed in different frames of reference.
You and I are witness to an event. Two identical cannons choose to fire at each other at precisely the same time. The cannon projectiles meet exactly half way. Now, you start to ride a bike super-fast along the path that the canons fire in. As they shoot again, you are at the exact midpoint between them. Unfortunately, you forget to duck. Which projectile will hit you first? Even though you are in the exact midpoint, the canon projectile you are riding toward will collide with you first because you are in motion toward it. As a point of curiosity, from my perspective, you were struck simultaneously by both projectiles. Slim comfort. Two different observation points of the same event yield 2 different results.
The ultimate conclusion is that because of Special Relativity, two different observers have equal right to view the same event with respect to their individual frames of reference.
Paradoxes
Now that we have looked at the basics of the theory, lets look at a couple of examples of paradoxes that have plagued the Theory of Relativity since inception.
The most famous of these is the Twin Paradox. Suppose we are twins. You and I share the same reference frame with each other on the earth. You are sitting in a spaceship and I am standing on the ground. We each have identical watches that we now synchronize. After synchronizing, you blast off and speed away at 60% the speed of light. As you travel away, we both have the right to view the other as experiencing the relativistic effects (length contraction and time dilation). For the sake of simplicity, we will assume we have an accurate method with which to measure these effects. If you never return, there will never be an answer to the question of who actually experienced the effects. But what happens if you turn around and return to the earth? Both would agree that you aged more slowly than I did, thus time for you was slower than it was for me. To prove this, all you have to do is look at our watches. Your watch will show that it took less time for you to go and return than my watch shows. As I stood there waiting, time passed faster for me than it did for you. Why is this the case if both of us were traveling at 60% the speed of light with respect to one another?
The first point to understand is that acceleration in Special Relativity is a little tricky. In Special Relativity, you can describe the acceleration in terms of locally "co-moving" inertial frames. This allows Special Relativity to view all motion to be uniform, meaning constant velocity (non-accelerating). While both did view the other as shrinking and slowing down, the person that actually underwent the acceleration to reach the high speed is the one that aged less. If you dig deeper into the world of SR, you will realize that it's not really the acceleration that is important; it's the change of frame. Until you and I returned to a frame of reference where our relative motion was zero (where we are standing beside each other) we would always disagree with what the other said he saw. As strange as this seems, there really isn't a conflict - both did observe that the other was experiencing the relativistic effects. One technique that is used to show the dynamics of the Twin Paradox is a concept is called the Relativistic Doppler Effect.
The Doppler effect basically says that there is an observed frequency shift in electromagnetic waves due to motion. The direction of the shift is dependent on whether the relative motion is traveling towards you or away from you (or vice versa). Also, the amplitude of the shift is dependent on the speed of the source (or the speed of the receiver). A good place to start in understanding the Doppler effect would be to first look at sound waves. There is a Doppler Shift associated with sound waves that you should recognize easily. When a sound source approaches you, the frequency of the sound increases and likewise, when the sound source moves away from you, the frequency of the sound decreases. Think about train blowing its whistle. If you are stationary in the same frame of reference as the train and it is stationary, the whistle will have a uniform pitch. Now, lets put the train in motion. As the train approaches, you hear the whistle tone as a high note, perhaps as much as 1 or 2 notes higher in sound than it was while you and the train were at rest. When the train passes you, you can hear the whistle tone change to a lower note. Why?
As you put sound in motion, it tends to compress the side facing the motion. As you compress the sound, the pitch or tone of the sound increases. After the sound has passed, the rear portion of the sound waves are literally being pulled along and the trail out, opening up in a wave pattern like the wake of a boat moving across a lake. Where this opening up occurs, the sound is decompressed and the pitch or tone lowers.
This Doppler shift also affects light (electromagnetic radiation) in the same manner with one critical exception; the shift will not allow you to determine if the light source is approaching you or if you are approaching the source and vice versa for moving away.
Now let's apply this information to the Twin Paradox. Recall that you sped away from me at 60% the speed of light. I picked this arbitrary speed, because the corresponding relativistic Doppler shift ratio is "2 times" for an approaching source and "1/2" for a source that is moving away. This means that if the source is approaching you, the frequency will appear doubled (time is then halved) and if the source is moving away from you, the frequency will appear halved (time is then doubled). (similarly I could have used any speed for the paradox; for example, 80% the speed of light would have led to a Doppler shift of "3" and "1/3" for approaching and moving away respectively). Remember, the direction of the shift is dependent on the direction of the source, while the amplitude of the shift increases with the speed of the source.
Let's take another trip, but this time you will travel 12 hours away and 12 hours back, as measured by your clock. Every hour you will send a radio signal to me telling me the hour. A radio signal is just another form of electromagnetic radiation; therefore, it also travels at the speed of light. What do we get as you travel away from me? When your clock reads "1 hour" you send the first signal. Because you are moving away from me at 60% of the speed of light, the relativistic Doppler Effect causes me to observe your transmission to be ½ the source value. From our discussion above, ½ the frequency means the time it takes is twice as long, therefore, I receive the your "1 hour" signal when your clock reads "2 hours". When you send your "2 hour" signal, I receive it at hour 4 for me. So you can see the relationship developing. For every 1-hour signal by your watch, the elapsed time for me is 2 hours. When your clock reads "12 hours" you have sent 12 signals. I, on the other hand, have received 12 signals, but they were all 2 hours apart…thus 24 hours have passed for me. Now You turn around and come back sending signals every hour in the same manner as before. Since you are approaching me, the Doppler shift now causes me to observe the frequency to be twice the source value. Twice the frequency is the same as ½ the time, so I receive your "1 hour" signals at 30min intervals. When the 12-hour return trip is over, you have sent 12 signals. I have received 12 signals, but they were separated by 30 minutes, thus 6 hours have passed for me. If we now total up the elapsed time for both of us, we see that 24 hours (12 + 12) have elapsed for you, but 30 hours (24 + 6) have elapsed for me. Thus, I am now older (and possibly wiser{?}) than you, my identical twin. If you had traveled farther and faster, the time dilation would have been even greater.
This time you travel 84 hours out and 84 hours back (by your clock) at 80% the speed of light. The total trip for you will be 168 hours, and the total time elapsed for me will be 280 hours; you were gone for 1 week by your clock, but I waited for 1 week 4 days and 16 hours by my clock. Remember that I will receive your outgoing signals at half the frequency which means twice the time. Therefore, I receive your 84 hourly signals every 3 hours for a total of 252 hours (3 is the Relativistic Doppler shift for 80% the speed of light). Likewise, I receive your return trip 84 hourly signals every 20 minutes for a total of 28 hours (20 minutes is the 1/3 Relativistic Doppler shift for the return). Now you know the total round trip from my perspective, 252 + 28 = 280 hours or 1 week 4 days and 16 hours. You, on the other hand, traveled 84 hours out and 84 hours back for a total of 168 hours or 1 week.
Here, we will view your travel from a different perspective. This trip will be 15 hours out and 15 hours back. This time, I will send you a signal every hour on the hour based upon my clock. What will you see? By the time the first leg of your trip is completed, my clock says you have been gone for 15 hours and I have sent 15 signals. You, on the other hand, will say that you received 6 signals separated by 2 hours (relativistic Doppler shift) for a total of 12 hours. Where did the other 9 signals go? Well, they are still on their way to you. As you turn around and head back you will run into the 9 missing signals plus as you continue onward, you will ultimately receive the next 15 signals I send out. You will receive 24 signals that are 30 minutes apart. I sent 30 signals in 30 hours and you received a total of 30 signals in 24 hours. The result is the same as before, but we do not agree on when the first leg of the trip ended and the last leg began. So from this we can conclude that the change of frame for you is what distinguishes you from me. For me, nothing changes at all (reference frames remain the same). You, on the other hand, change frames. First you move away and then you move back. This change you make is the change in symmetry that removes the paradox.
Before going on to the next concept, we need to make a couple of things very clear about special relativity. As we understand the physics surrounding the speed of light, anything, and here we are talking about mass, approaching that barrier from a slower speed is doomed because of length contraction and time dilation. This terminus of mass however, does NOT restrict objects from moving faster than the speed of light on the other side of the barrier. Think of the speed of light in the same frame again as sound, a barrier. When jets go beyond the speed of sound, they literally break the sound barrier. They are traveling completely in front of the sound they generate. The sound in our earlier diagram would in reality look just like a vertical flat line at the time you pierce the barrier. The problem is that at the speed of light barrier, that vertical flat line leaves no room for the third dimension required to define mass and so mass ceases to exist. Oops!
There has been no specific confirmation as to the existence of things beyond the speed of light, but there are theories about the particles called tachyons that have the ability to exist there. Someday, someone will discover their existence.
Secondly, velocities from a different frame of reference can not be summed. For example, if I run 5 miles/hour and at the same time, throw a rock 5 miles/hour, the only reason you (standing still) can say the rock is traveling 10 miles/hour is because the speed is so small with respect to the speed of light. We use the Lorentz Transformations to transform from one frame to another using the relative velocity of the frames. These transformations tell us mathematically that while at slow speeds the error in straight addition is much too small for us to detect, at very fast speeds, the error would become quite large. So classical mechanics, which teaches us to sum these velocities, is actually incorrect. We can do it, but it's a case of getting the right answer for the wrong reason.
Simultaneity (or lack thereof) is a terrific tool for understanding many of the paradoxes associated with and discussion of special relativity. And, if I am to be thorough, simultaneity must be considered for all events between separate frames of reference. Let's re-visit the twin paradox. (One more time, you travel out 12 hours at 60% the speed of light and return at the same speed). Basically, there are three frames of reference to consider. First, we are on the earth with no relative velocity between us. Second, you embark on the outgoing leg of your trip. Thirdly, you (after instantaneously turning around) embark on your return leg of your trip. I am using the same example as before, except I am using numbers from the Lorentz Transforms as opposed to the Relativistic Doppler Shift to explain the observed phenomena.
1st frame:
You and I each agree on everything we observe. This should be easy to understand since there is no relative velocity between us. We are in motion together.
2nd frame:
You travel out 12 hours by your clock. With the two postulates in mind, we realize that you observe time dilation for your outgoing trip. Thus, if you record 12 hours, I will record 15 hours. Remember that at 60% the speed of light, the time dilation will be 80%. Therefore, if you record your time to be 12 hours, this is 80% of what I record - 15 hours. But what do you observe for my time? You see the time dilation as effecting me; therefore, you measure your trip to be 12 hours, but you observe 9.6 hours (80% of your clock's time) for my time.
2nd frame totals:
I measure my time to be 15 hours, but your time to be 12 hours. You measure your time to be 12 hours, but my time to be 9.6 hours.
Obviously, the event, which is the end of the outgoing trip, is not simultaneous. You think my time is 9.6 hours but I think your time is 15 hours. On top of that, we both think that your time is 12 hours, which doesn't agree with either of the first two times. Lost?
3rd frame:
From my perspective, nothing new has happened. I remained in my initial frame of reference and you returned at the same velocity you left with. Therefore, I measured the return trip to take 15 hours for your frame (same as the outgoing trip) and observes the trip to take 12 hours for you. From your perspective, you encountered a major change. You actually changed frames from one of traveling out to one of traveling back. Now, at the start of the return trip, when you look at both of our clocks, you observe your clock to read 12 hours and my clock to read 20.4 hours. Think about this. You now show that my clock has jumped ahead from 9.6 hours to 20.4 hours. How can this be???? When you changed from the 2nd frame to the 3rd frame, the established symmetry between you and me was broken. Thus, we each view our own time as having no change. And since you were the one that actually changed frames, you showed more elapsed time for me. From here on out, it is business as usual. The return trip is clocked at 12 hours by you, but you observe 9.6 hours for me. Again, let's clean this up…
3rd frame totals:
I measure my time to be 15 hours, but I measure your time to be 12 hours. You measure your time to be 12 hours, but you measure my time to be 9.6 hours. Remember, this 9.6 is only for the return trip after the frame change.
Trip totals:
I measured your time to be 15 hours for the outgoing trip + 15 hours for the return trip…30 hours.
I observed your time to be 12 hours outgoing + 12 hours return …24 hours.
You measured your time to be 12 hours outgoing + 12 hours return…24 hours.
You observed my time to be 20.4 hours (after outgoing trip and frame change) + 9.6 hours for the return trip…20.4 + 9.6 = 30 hours.
Can you find any events in which both of us agree on the time for both ourselves and the other?--- No, you can't. ---The lack of simultaneity is the key to the paradox. Both of us are measuring and observing. Unfortunately, we are not measuring and observing the same events. It is impossible for us to consider something like the end of the first leg as simultaneous when we each view it occurring at different times for me. It's interesting to note that the results are the same as the Relativistic Doppler shift results. Is there a pattern here? Relativity allows for various methods to be employed to resolve the problems. For this case, use of space-time diagrams (there's those words again) would clearly show every point that we have talked about. I have merely used the Lorentz transforms in combination with the Relativistic Doppler effect.
Many people have trouble with the twin paradox because of the way in which the frame change is handled. In this case, the jump on your clock for me after the frame change (9.6 to 20.4 hours) is the problem. There really is no problem here. If you want to integrate the acceleration to use various inertial frames during the turn around, it can be done (with the same results). Another common approach is to imagine someone else in space that passes you just when you reach the point of your turnaround. This person is heading towards me at the same speed that you were traveling, so there is no need to consider you any further. The key fact is that if we then went back in the substitute's frame and looked at his clock for me, it would show that some amount of time had already been recorded when the substitute began his trip towards me. How far back should we go? Since you traveled out 12 hours on the outgoing trip, we should go back 12 hours in the substitute's frame. At this starting point for the substitute, his clock for me would read 10.8 hours. This is extremely important. It clearly shows that both of us or you and the substitute observe the other as having slower times. The big shift occurs when the frame of reference is changed. This means that both observe the other to have a slower time during the actual outgoing and return trips, but there is a shift during the frame change that more than makes up for your account of my slowly running clock. After the frame change, the damage has been done. You will still observe my clock to run slow, but it will never slow down enough to compensate for the 10.8 hours that were perceived during the frame change. Is this time jump a physical occurrence? No. The time jump occurs because when you change frames, you are no longer using the same event as a reference. When you made your turnaround, the event in my frame that you thought was simultaneous with your turnaround changed. Your frame change caused this confusion because your new frame uses a different time for the event in my frame. More clearly, the turnaround event in my frame has a different time value for the outgoing leg and the return leg, as perceived by you. Keep in mind that in the above references to my frame, I'm really talking about what you think my frame time would be. This time difference is only apparent to you because it is YOUR frame change that causes the discrepancy. In my frame, nothing changes for me when you change frames. Here again, by realizing that the two events are not simultaneous, the paradox is resolved. The point I am trying to emphasize is that there are a variety of ways to handle the paradox. All of the methods yield the same result, but if you actually consider the simultaneity of the situation, then the how's and why's become more clear.
Now that you have been introduced to the concepts of the theory, let's take a quick look at the relation between time travel and Special Relativity. If you remember the result from the twin paradox, you should agree that traveling into the future is possible, even at the speeds that our astronauts travel. Granted they would probably only be gaining a few nanoseconds, but when they return, the time on earth is ahead of their system time. Thus, they have returned to the future. As far as traveling back in time, Special Relativity is not as gracious as it is with moving forward. Let's take a look at this approach…
Many creative minds have wondered that since time slows down as you approach the speed of light, if you could find a way to travel faster than the speed of light, could you travel back in time? If I am to believe that special relativity is correct, then I am also to believe that the following events would occur. In order to travel faster than the speed of light, I assume that you would at some point have to travel at exactly the speed of light. For example, you can not travel 51 miles/hour without having traveled 50 miles/hour at some point, of course, this is providing that you were traveling 50 miles/hour or less to begin with. Now Special Relativity tells us that at the speed of light, time stops, your length contracts to nothing, and your resistance to acceleration becomes infinite requiring infinite energy (as observed by a frame of reference that is not in motion with the system). These conditions do not sound very conducive to life. Thus, I conclude that time travel into the past, using the concepts of SR, has some severe issues to overcome.
What do we conclude from all this nearly circular logic? Special Relativity deals with contractions and dilations that are not in agreement with our commonsense views of the universe. In fact, they almost appear ludicrous. Yet, there have been several observations that agree with the predictions of Special Relativity. So, until the theory is proved wrong or a simpler theory produces the same results, Special Relativity will maintain its position as the best theory out there.
Here are five concepts you have discovered:
- There is no such thing as an absolute (completely stationary) frame of reference.
- The laws of physics apply equally to all frames of reference.
- The speed of light is constant in all frames of reference.
- There is no simultaneity of events between separate frames of reference.
- You are never too old to learn.
If you pursue a better understanding of Relativity than I can render, do not fall prey to these errant statements:
- Time slows as speed increases. (Only when viewed by another frame of reference)
- Objects shorten as speed increases. (Same as above)
- Relativity can't handle acceleration. (Biggest misconception about Relativity)
- Mass increases with speed. (Energy required to increase speed increases, not the rest mass)
- Nothing can travel faster than the speed of light. Crossing the speed of light barrier from either a faster or a slower speed is the problem, not whether or not things can travel faster.
The beauty in the theory of special relativity is that it gives us laws from
which we can unite space and time and also energy and mass. Special relativity
is definitely a thinking person's (politically correct term here) playground.
Conclusion
Now, where do we go from here? Obviously there are some basic barriers to zipping into two things being able to occupy the same space at the same time. Yet, that is exactly what happens (in theory) with objects traveling faster than the speed of light. Time also changes in that its reaction will be the same as light. (Remember the earlier examples) If mass compresses, so does time. Here we have the same type of relationship occurring. If mass can re-distribute itself so that numerous objects can occupy the same space at the same time, then all time happens at the same instant (if viewed from the outside in a separate time frame) Time, from this perspective has big open spots in it just like the spaces in-between atomic particles. Just like our travels toward the speed of light, with different points of reference, different things are happening. We see time as linear, yet time can have more than one frame of reference. Now you can understand how God is able to view all things everywhere at one time. I personally found this to be one of the most fascinating concepts I was ever able to comprehend.
We are limited to our world of finite time and space restrictions. I have on numerous occasions revisited this problem and have, as of yet, not been able to come up with anything that gives me a better explanation.
Food for thought.
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