 In this video we want to apply the theory we developed in videos 1 and 2 to one of the most famous predictions of relativity, the so-called twin or clock paradox. Recall that relativity describes events in space and time as seen from two or more inertial reference frames in uniform relative motion. We'll draw a frame quote at rest with blue axes labeled with Latin letters and then a red frame labeled with Greek letters that'll be moving along the x-axis with velocity v. Suppose the flashing lights here represent spacetime events in the blue frame. Special relativity tells us how these events appear in the red frame. We saw the need for a dilation factor beta, which equals one over the square root of 1 minus v squared. Two of the space coordinates, y and z, remain the same but the space coordinate in the direction of motion and time transform in a way that mixes up space and time between the frames. We develop these equations using an elegant graphical representation. Time is plotted horizontally and the single interesting space coordinate is plotted vertically. The rest coordinates are blue and the moving coordinates are red. We can plot advances seen in the rest frame and then read off the corresponding red frame coordinates or vice versa. The 45 degree green lines represent the speed of light in both frames. Now we'll take on the famous twin paradox, also sometimes called the clock paradox. The scenario is that two twins are born on earth. One gets in a spaceship and travels away at high speed for some years, turns around and returns to earth. When reunited the space twin turns out to be younger than the earth twin. This is sometimes called a paradox because if motion is relative why wouldn't the experience of the twins be completely symmetric? We'll find out shortly. To analyze the physics we'll use three reference frames. The blue is the earth or the rest frame. The red is the outgoing frame and the green is the incoming frame. The space twin will ride the red frame away from earth for some time and then jump on to the green frame to return. All we need to do is make a spacetime diagram showing the three frames. Plot the key events as seen by earth twin and then read off those events as perceived by space twin. Because the numbers work out nicely I'll take the velocity to be three-fifths the speed of light. Then the dilation factor beta is five-fourths. We'll see that this tells us that five years for earth twin will correspond to four years for space twin. Here's our experiment as seen by earth twin. Event A, the twins are born and space twin leaves earth. Event B, after five years space twin is three light years away and she jumps from the outgoing red frame to the incoming green frame to turn around. Event C, after five more years space twin returns to earth. Here are the events in earth twin coordinates. A and C occur at earth that is x equals zero and at years zero and ten. Event B, the turnaround occurs midway at year five on earth and since the speed is three-fifths the speed of light at a distance x equals three light years. Between events space twin moves uniformly along the line shown. Let's look at the first half of the trip in detail. Space twin is in the outgoing red frame for five years again as seen on earth. So here we'll plot the constant position and constant timelines for the blue and red frames at spacings of one half year in time and one half light year in space. Since space twin is at rest in the red frame she'll follow a constant red position line. Every time she passes one of the red constant timelines she'll experience one half year of aging. This continues for eight half years, hence four years, at which time space twin reaches the turnaround event. Now she jumps from the red frame to the incoming green frame. Space twin is now at rest in the green frame so she follows a green line of constant position back to earth. Each time she crosses a green constant timeline she experiences another half year of aging. Here's the complete journey in earth twins frame with black dots indicating half year intervals as experienced by space twin. She experiences a trip lasting eight years. Four out and four back. Well to earth twin the trip lasts 10 years. Five out and five back. So he'll be 10 years old while his twin sister is only eight years old. Remember that the dilation factor beta is five over four and we see this is the ratio of time intervals experienced by the twins. Well that's what the theory says and this is all because the speed of light has to be the same in all three frames. This is a truly weird result so let's examine it in more detail. First of all what does it mean to say that someone who is three light years away is four years old? That's what our picture says but how would we verify this? Let's quote good old Albert here. All our space time verifications invariably amount to a determination of space time coincidences. The introduction of a system of reference serves no other purpose than to facilitate the description of the totality of such coincidences. So let's couch the experiment in terms of space time coincidences. Things that could actually be directly experienced by a person. Assume the twins maintain a video connection via radio or light waves. Both record his or her image side by side with the video received from the other twin. What will they see in their videos? In all systems the light or radio waves propagate along a path corresponding to the green diagonal lines on the space time diagram. We can use this to follow light forward or backward in time and either toward earth or toward the spaceship. Let's say it's year four on earth. How old will space twin appear in her received video? We just follow a diagonal line from earth back in time towards space twin and see where it intersects her space time path. In this case it's when she's two years old. So when earth twin is four years old he'll be watching a video of space twin at age two. Likewise when space twin is four we can follow a diagonal line into the past from the spaceship back towards earth. We find that it left earth when earth twin was two. So when space twin is four years old she'll be watching a video of earth twin at age two. We can do this for each twin's birthday. Here the yellow lines represent signals received by earth twin and the orange lines signals received by space twin. Remember that each twin records a video with images of both twins. So when the twins reunite they'll have two videos each showing the two twins. We'll put earth twins video on the left and space twins video on the right. In both cases we'll plot earth twins age in blue and space twins age in red. Okay let's see what we get. Overall they see a very different unfolding of events. In particular space twins video ends after eight years while earth twins video lasts for 10 years. However they both end up in agreement that they saw earth twin age 10 years and space twin age eight years. This is sometimes claimed to be a paradox by the following argument. We treated space twin as the one in motion but relativity says that absolute velocity cannot be measured. Only relative velocity. Therefore we could just as well have taken space twin to be at rest and earth twin to be in motion. Let's think about this. In particular consider the first four years of the respective videos. In four years earth twin sees space twin age two years. Likewise in four years space twin sees earth twin age two years. Their experiences are completely symmetric. It doesn't matter which one is considered to be moving. They both see themselves aging more rapidly. Four years versus two years for the other. Now consider the last two years of the respective videos. This is a bit more complicated because some stuff has already happened to break the symmetry but let's work backwards. In the last two years eight to ten earth twin sees space twin age four years. Four to eight. In her last two years six to eight space twins sees earth twin age four years. Six to ten. Again their experiences are completely symmetric. Running the videos in reverse from their endpoints. Both twins see themselves aging more slowly. Two years versus four years for the other. If we put the first four years together with the last two years we would see a completely symmetric result. Both twins would have aged six years. It would not matter who we consider to be in motion. This symmetry is broken by what happens in between. Here we label the symmetric intervals with green dots the first four years and the last two years in each frame. One thing that jumps out is the symmetric intervals involve communication between only two frames. During the first four years earth twin is receiving signals sent from the outgoing frame while space twin is receiving signals while on the outgoing frame. During the last two years earth twin is receiving signals sent from the incoming frame while space twin is receiving signals while on the incoming frame. However in between there are asymmetric intervals labeled with red dots. Space twin experiences an additional two years receiving signals while on the incoming frame and earth twin experiences an additional four years receiving signals from the outgoing frame. This is the source of the two-year discrepancy in their final ages. The key symmetry breaking event is the frame jumping which would be physically observable as it requires space twin to experience an acceleration. Earth twin does not experience an acceleration. You can add any constant velocity to the entire system and you still end up with the results we obtained. Earth twin always stays in the same frame no acceleration while space twin always jumps frames accelerates. So the experience of the twins is not symmetric. Only one twin experiences acceleration that is jumps frames and that is the one who ages more slowly. There's actually no paradox in the so-called twin or clock paradox. Time dilation has been verified in a massive set of experiments spanning more than half a century. Not only would the GPS system be unable to function without taking time dilation into account but it would be absolutely impossible to design and build modern particle accelerators and to interpret the experimental data they produce.