The symmetry argument made in the twins paradox is now broken, but what is it about those non-inertial events that prevents us from considering the separate legs of space travel—away and back, as two independent segments of a single inertial worldline? Something must be happening at that transition point. Are the g-forces responsible for the time discrepancy?
This is what I thought after recognizing the asymmetry. But it turns out this is a red herring.
The B worldline reaches the destination and then abruptly stops. At 0.6c, Ben and the spaceship would be flattened in less than a microsecond. Maybe all that time dilation would have been resolved, but Ben didn’t live to tell us about it.
Maybe if there was a more gradual deceleration, the time discrepancy could be explained. Many YouTube videos point to the acceleration as the cause of the twins’ age differences.
I thought this for a while too, and I was ready to sweep it under the general relativity rug.
But after learning special relativity from eigenchris, I no longer believe that gravity and the warped spacetime of general relativity are necessary to explain it.
It is possible to measure time dilation during acceleration and deceleration without general relativity. So I constructed a worldline for Ben that had him gently decelerating to land on a (different) exoplanet. It added time to the overall mission, but kept him alive to complete it.
The deceleration created a force on Ben, but it was limited so that he felt the equivalent of Earth’s gravity- 1g. It took almost 9 months (but he only thought it was 8) to slow down and stop at the exoplanet.
This time, B stops only long enough to collect some samples and blast off again, following a 1g acceleration curve until he reaches the 0.6c cruising speed of his spaceship, at which, like before, he spends 4 years of his proper time.
This time, when reunited, B is 4+1.3+4=9.3. A is 5+1.4+5=11.4 Ben is now 2.1 years younger than Amelia, even more than before! But then, his mission lasted longer so his clocks were in motion longer.
If we look more closely at the deceleration/acceleration segment, we find that the ratio of proper times, B/A is about 0.9 (Ben’s watch ticked 9 times on average to Amelia’s 10). What if the acceleration had been more? Or less? I explored these possibilities and found that they would take less time (or more), but the ratio of proper times remained the same!
Decel/Accel rate
Total Distance, light years
Time, T years
Proper Time t years
Ratio t/T
0.5g
1
2.958
2.727
0.922
1g
0.5
1.479
1.3636
0.922
2g
0.25
0.7395
0.6818
0.922
10g
0.05
0.1479
0.13636
0.922
100g
0.005
0.01479
0.013636
0.922
So no matter what acceleration you applied at the turnaround point, there is no recovering of the time discrepancy between the world paths of A and B.
Returning to the same position cannot be done without incurring an acceleration. It is not possible for the twins to reunite without one or both of them exiting their inertial frame and feeling the forces of acceleration. So in a way, acceleration is necessary for the twins to get back together and compare their elapsed proper times, but it is not the “cause” of the discrepancy. Instead, it is the speed and duration of time spent in motion.
This is what finally convinced me that time dilation was more than just the effects of remote signaling from a fast-moving object, and that there was no explanation to be found in its acceleration. It was truly an effect of objects moving with respect to my local reference frame!
The twins paradox is puzzling because it is not our experience that our personal time is shifting when we move around. It does– we just never notice it. So it seems strange that biological aging can change by taking a high speed trip. But is it really any more strange than the premise that the speed of light looks the same to everyone, no matter how fast they are going? This is the actual source of the paradox. If we can accept that, then we can accept that moving clocks tick more slowly!
Returning to Ben’s experience, I suspect there is more to understand about switching between different inertial frames. That understanding can be found in general relativity theory but I will stop here, having now satisfied my lifelong mystery of the twins paradox.
This has been my attempt to explain it. If it makes sense to you, let me know.
At this point, Ben stops to take his samples and make measurements and survey the exoplanet. He spends a year doing this, but he is no longer moving away from Amelia. Their clocks are each ticking at the same rate. Since they are three light years apart, it will take three years for B’s messages to be received by his twin.
After sending Happy Birthday message #5, Ben takes off for his return trip home. His spaceship is capable of traveling at 0.6c, so it will take 5 years to cover the 3 light year distance. But this is not what Ben experiences. On his calendar it only takes four!
Again, this is because at this speed the units of time for B are longer, so the clock ticks are slower in order for B to observe the speed of light in his proper time as c.
B continues to transmit the HB signals and A receives them. They are arriving delayed of course– B is still light years away, but they come more frequently now because each year B is closer and the signal transit time is shorter. It is not shorter by 0.6 years; it comes earlier than expected, 0.5 year. This is because Ben is still sending them every year on his birthday, but on his calendar, which is 1-1/4 years apart for Amelia. And in that extended time, he is even closer, so the remaining distance for the signal is shorter and takes less time, 0.5 instead of 0.6 years.
Ben finally arrives home and is now two years younger than Amelia. She has aged 5+1+5=11 years. Ben has aged 4+1+4=9 years. We have just seen how time dilation as a result of B moving close to the speed of light is the cause. His proper time along his world line is less than A’s proper time along hers.
This is not an intuitive result. It feels wrong somehow, but it is how the universe works. It has been confirmed over and over—moving clocks tick more slowly! In 1971 an experiment was run where atomic clocks were flown around the world and then compared to the reference clock on the ground. The differences were tiny but measurable– nanoseconds, and matched the theory.
But when we finally accept that this is how the universe works, we are still left with trying to explain the paradox. Why isn’t the viewpoint from Ben, where A rushes off, the exoplanet rushes in, and he remains stationary in his inertial spaceship frame, an equally valid representation?
And here is the answer—Ben is NOT in a purely inertial frame of reference. “Inertial frame of reference” means NO acceleration, no change in velocity or direction. Ben experienced both. Amelia didn’t.
An inertial frame of reference is a straight line on a spacetime diagram. While it is possible to use the Lorentz transformation to portray Ben as standing still for one of the legs of his journey, it is not possible to transform his full worldline into a single straight path. His path will always have some kink in it that takes him out of the inertial frame.
This is the crux of resolving the paradox. The two worldlines are NOT both inertial frames. There is an asymmetry in the problem. They are not equivalent. B experienced rapid deceleration followed by a change of direction and an acceleration, all of which are non-inertial conditions and not covered by the rules of special relativity.
Paradox resolved: the two perspectives are not physically equivalent, so it is not necessary that the proper times of the two paths match.
I think I can present this topic without math, but I will rely heavily on the visual tool of spacetime diagrams. Spacetime is the hybrid combination of space and time. It might sound abstract and scary, but it is really just recognizing that we travel in space and we travel in time. We can go anywhere in space, but we only go “forward” in time.
We can chart a path of our travels in spacetime on a graph where the x axis represents our relative position in space and the vertical axis is our position in time, labeled t. Yes, space is 3D, but the different directions in space can be folded into a single distance number.
If we stand still, we can plot our path in spacetime as a straight vertical line. Depending on where we are standing, the line will be somewhere on the distance axis. We can call our current position “zero”, and the line will fall on the time axis.
If we start walking to the right, the line will slant to the right as we move to further distances from our starting position. If we walk left, the line will slant left. If we walk faster, the lines will slant more. These plots are called “worldlines”.
When spacetime diagrams are used for plotting worldlines close to the speed of light, we use units of time and distance that match. Time is measured in the usual seconds, or maybe years, and distance is measured in units of how far light travels in a unit of time, so light-seconds (~30 million meters) or light-years (9.5 trillion kilometers). This choice of units results in a spacetime plot for light as falling on the 45-degree line. The speed of light is denoted by the letter c, which on the diagram is one light-year per year.
We can now depict Amelia and Ben’s worldlines. Let’s say that Ben departs at position 0, at time 0, at a velocity 60% the speed of light, 0.6c. Amelia (I will sometimes just call her “A”) follows the green worldline. Ben (“B”) charts a red path to the right with a slope that matches 0.6c—for every vertical unit of time, Ben moves to the right 0.6 units of distance. Five units of time results in covering 3 units of distance. Since the exoplanet is 3 light years away, it will take five years to reach it. Here is a plot with units of years and light-years.
Special relativity is based on a rather unusual premise: that the speed of light is measured to be the same in all inertial frames of reference, no matter their velocity.
A consequence of this premise is that each tick of a moving clock takes longer, at least as seen by a non-moving observer. The moving observer doesn’t notice his clock ticking more slowly, but when he looks back at a non-moving clock, it appears to be the slow one.
The clock in your own inertial frame of reference tells you the “proper time”. Every observer experiences it as the normal passage of time. There is no clue that it appears slower to other reference frames. This effect of the apparent slowing of moving clocks is called “time dilation”.
Units of proper time can be depicted on the spacetime diagram by short vectors that represent one unit of time as experienced by the observer in its frame. Amelia’s units of proper time are shown as head to tail arrows along her worldline on the vertical axis. Ben’s are head to tail arrows along his worldline.
Ben’s proper time unit being longer than Amelia’s is a result of the premise: the speed of light is the same in all reference frames. As you get closer to the speed of light, in order to measure its velocity (distance per unit time) as the same, either the time units have to get larger, or the distance units have to get smaller, or both (which is what actually happens, but we are focusing on time dilation here). The lengthening of the time step is prescribed by the mathematics of relativity so that the speed of light looks the same to everyone, moving or not.
So now we have time units for each frame. Each tick of the clock moves the person on their worldline in the time direction; Amelia’s is straight up because she is staying put, and Ben’s tick is tilted up and over because he is moving in both time and position. The length of Ben’s tick has been increased to make his speed of light measurement be correct.
We can imagine that Amelia and Ben made arrangements to send happy birthday messages to each other on their common birthday, radio transmissions inserted into the telemetry stream between the spaceship and Earth. We can also imagine that they had a blast when Ben departed on their birthday.
Radio signals travel at the speed of light and so the worldlines of those transmissions fall on 45-degree diagonals. We can plot the first few of them and see what the twins experienced.
Amelia sends her first happy birthday message, HB1, one year after Ben departed. She recognized that Ben was now 0.6 light years away, and since the message had so far to travel, she didn’t expect anything from him until later that year, maybe 7 months or so.
But notice that Ben’s year is a little longer than Amelia’s, so his happy birthday message is sent “late” (even though he thinks he is exactly on time). At the time Ben sends the message, another 3 months has gone by for Amelia. And if he has been traveling for a year and 3 months, he is now more than 0.6 light years away; he is 0.75 light years distant. And from that distance it takes 9 months for the message to make it to Amelia. That additional time, plus the 3 month delay in even sending it, results in a full year before Amelia gets it. As Amelia is sending her second happy birthday message, she finally receives HB1 from Ben!
Ben also doesn’t expect to hear from Amelia each year. He knows he is traveling farther away and the signal needs additional time to reach him. And in fact, it takes two years (on B’s calendar) to receive the first message from A.
I once thought that time dilation was entirely due to the increasing distance and how long it took for the signals to get back. But it is more. B’s clock is running slow, so Ben doesn’t even send his message until three months after Amelia’s birthday. And since it was sent late because of the slowed clock, the message took longer to get back because the ship was farther away when he sent it.
This pattern continues for the rest of the trip to the exoplanet. When he lands, Ben has sent 4 birthday messages, Amelia has sent 5. B has received two of A’s five messages, and A has received 2 of B’s four.
So far we have been showing the trip from Amelia’s perspective. It might be interesting to show how it looks from Ben’s. Relativity theory provides a way to convert back and forth between inertial frames of reference. It is called the Lorentz transform and it accounts for time dilation and length contraction while always keeping the speed of light constant. Here is the spacetime diagram for Ben’s outbound trip where he is considered the “observer at rest”.
You can see that this is NOT a perfect reflection of Amelia’s perspective. She is moving left while Ben is standing still in his reference frame. Ben’s worldline runs for four years on his calendar, but Amelia’s runs for five of hers, which are longer than Ben’s, since to him, she has the moving clock. The messages they send each other arrive at the same interval in each frame, every two years, just as before. At the time Ben reaches the exoplanet, they have each received two messages.
So far there is no paradox, but Amelia and Ben have not been reunited yet, so there is no way to directly compare their relative ages.
When I was studying physics in college, one of the early subjects was Einstein’s special relativity theory. The subject is called “relativity” because it explains the physics of objects moving relative to each other. It is “special” because it only applies to uniform relative motion, not motion induced by gravity, which is covered by “general” relativity, which Einstein described a decade later.
Special relativity replaced Galileo’s and Isaac Newton’s earlier theories, which were superb at explaining falling objects and orbiting planets, but had run into trouble explaining the properties of fast-moving electrons and light.
It is an early subject in the physics curriculum because as students, we were just learning the techniques of calculus and linear algebra; techniques that are helpful, but not required to understand special relativity. Most people are familiar with special relativity, and even if they don’t understand the details, they have heard “E=mc2”, one of the consequences of it. They may also have heard about time dilation, the effect of a moving clock slowing down relative to a stationary one.
Sometimes, when I try to describe to others what I do as a color scientist, I am asked if I can fix their photos. Usually it is to make their printer look more like their monitor, but a few years ago it was a friend asking about how to correct his underwater pictures while scuba diving. It turns out that this is an unsolved problem in color science and it was intriguing enough that I spent some time studying it, and worked out a solution for a simple geometry using my friend’s images as test cases.
Five years later I decided to submit a summary of this work to the international Color Imaging Conference, this year held in Lillehammer Norway. Remarkably, it was accepted, and doubly remarkably, it was runner up for the coveted “Cactus Award” for interactive poster papers.
I was lucky to have ended up at this observing location with such excellent weather. When planning to view total eclipses, I am advised to arrange for other activities as well; the eclipse itself is subject to fickle viewing conditions (my one prior total solar eclipse effort was thwarted, but the travel experience was rewarding nevertheless).
While I was mesmerized by the experience and NOT taking pictures, my script-driven camera captured more than just background stars for measuring gravitational deflections. Here are my three favorites.
My HDR composite spanning 14 stops of exposure and showing some of the structure in the corona.
A shot at third contact showing some prominences and the emergence of two “beads”
The end of totality marked by the “diamond ring” left a lasting impression; a signature of the unique event we had just witnessed.
It is a bit disappointing to be unable to show a clear gravitational signal, even with all of the successful exposures that were taken, but I recognized the difficulty of this measurement early on. In addition to the variables I anticipated, there are some additional uncertainties that I now recognize.
Here is my updated list of confounding variables:
Lens distortion. This seems to be the largest one, measuring many arcseconds by the time one reaches the edge of the frame. There are two ways to combat it: take the before and during images with the exact same center and orientation on the sky (which I found impractical with my equipment), or calibrate the lens. I did the latter, but found that this too was sensitive to overall gain/magnification assumptions.
The variation in positions due to the stars twinkling is about 2 arcseconds. I attempted to mitigate this by multiple exposures, but more were needed than I obtained.
Centering and orientation. I did the best I could to position and orient the reference stars to the coordinate of the sun’s center during mid-eclipse, but the rigid transform technique requires an adequate collection of uniformly distributed stars. With only five or six, there may have been biases in this calibration of the order of arcseconds.
Temperature variation. My reference images were taken during Minnesota spring and summer evenings, usually light-jacket weather. The eclipse pictures were taken at midday in Idaho. Although we noticed a cool-down during the eclipse, only a few of us felt the need to put on our fleece. Silicon has a thermal sensitivity of 2.5 ppm per degree K, which could account for some of the error. Overall however, this is small compared to the uncertainties displayed, since even a 10-degree K difference would show an error of 0.02 arcseconds per thousand radial distance. There may be other artifacts of temperature change however, see the following regarding focus.
Focus variations. As one focuses the image on the detector, there is a geometric gain involved. I measured the travel on my telescope focuser for one turn of the fine-focus knob. I also noted that best focus could be determined to within 1/8 turn of that knob. This worked out to be 0.3mm, which, at the focal plane of a 480mm lens is about 0.6 arcseconds per thousand, a significant amount!
Algorithm sensitivities. The before images were taken at night, sometimes with a partial moon providing background illumination of the sky. The during images were taken in the presence of the corona, a strong offset of the background level, and one which has a directional gradient as well. It is possible that the first stage of processing, starPos.m, could have been influenced by this difference. I do not have any estimates of its sensitivity.
Published star positions. I used reference star locations from Stellarium, which uses the latest publicly available database of star locations. I used J2000 epoch numbers out of habit, but perhaps I should have used current date coordinates. This would only affect the errors comparing my observed positions with the published ones, not the errors between observed “before” and “during” star positions.
While I am not surprised at the failure to find the gravitational deflection signal, I am disappointed I did not get a bit closer. Regardless, it has been a wonderful project to undertake. I learned much and re-learned more. I hope the descriptions of the process have been enlightening. If you have read this far, perhaps you have found the narration worthwhile or even enjoyable. Best wishes and clear skies to all future solar eclipse observers!
I was able to obtain 35 photos during totality that were candidates to locate stars in the field. The exposures ranged from 1/60 to 2 seconds, but it became clear after applying the detection procedure starPos.m, that only the longest exposures, 1 and 2 seconds, would yield detected stars. The inner regions of the corona were just too bright and irregular for the algorithm to find them.
This left 15 images to work with. The camera orientation was good in that there were many candidate reference stars in the frame. Here is the mapping of a mid-eclipse exposure (3496):
Here is a map of the located stars. The color codes indicate the channels (red, green or blue) in which the star was found. White indicates being found in all channels. The number of located stars for this image are: 6 red, 9 green, 5 blue.
The detected stars are then mapped to our virtual camera view and correlated against our list of reference stars (imagePos.m). They are then more precisely aligned to the center of the sun. The lens distortion is removed at this stage (radialAlign.m).
The uncorrected lens positions look like this:
Lens corrected:
The errors are still rather large, but by collecting the statistics from all of the “during” frames, we can see how they land with respect to their published reference positions. The standard deviations are consistent with atmospheric seeing, but the differences in average positions indicates other sources of error.
If we average all of the “before” images I took, and see how they compare to the published star locations, we get a similar wide ranging plot, the variances are again consistent with the seeing (but the average errors are not):
If we take the difference between these two data sets, we should see the gravitational deflection signal we are looking for. Unfortunately, it is lost in the noise.
I can make a plot similar to Eddington’s that shows the average measured deflection of these reference stars, but I will not claim that it demonstrates gravitational deflection.
This was written prior to eclipse day as I was contemplating how to compare the two image sets. I include it here to keep the thought sequence intact.
When we apply steps EE-1, 2, and 3 to both the before images and the during images, we will have a set of radial distances to compare. In the best of conditions, the distances will be very close to each other. There will be measurement noise and it is unlikely that the subpixel difference we are looking for will be immediately obvious from the measurement of any single reference star.
There are some tests we can make ahead of time to see what to expect. For example, we can compare an image taken on different (before) dates to see if they show consistent positions of the reference stars. We can compare images taken in a single session to measure the effects of atmospheric seeing and other factors.
To this end, I have made various images of this field of the sky over the months preceding the eclipse. They are among the least interesting of the astrophotos I have ever taken, since they show a single bright star (Regulus), and not much more. There are no deep sky objects of interest in this particular patch.; no galaxies, no nebulas, no star clusters, no Milky Way field. However, there are stars that can be detected, even in metropolitan light pollution, that match up with the Stellarium reference star database.
I took exposures for two consistency tests. One is multiple exposures of the same exact scene, with no changes in any settings. Ideally, the images would be identical, but if not, would be a measure of the dynamic atmospheric distortions, or perhaps the mechanical vibrations of my camera-telescope-tripod setup.
The second test changes the camera angle. The center of view remains approximately the same, but the camera rotates to 45 and 90 degrees. Each change requires a re-focus. If my frame centering and lens radial corrections were accurate, the stars should remain in the same locations.
Even if my lens corrections were imperfect, the first test: multiple exposures with no changes, should pass. Perhaps the lens corrections did not place the detected stars at their exact reference locations, but at least they should all fall at the same place. I could compare them against their reference, which might show an error, but if I compare them against each other, the differences should vanish.
The second test, comparing images of the same field but at different angles (portrait vs landscape etc), is a simulation of pictures taken at different times. It is a “best case” test: everything is the same except the camera was rotated and refocused, compared to the real-world case for the “after” image where not only is the angle different, the telescope alignment will have a slightly different center, the angle in the sky (and its atmospheric refraction) will be different, the temperature, elevation, and air pressure will be different, and many other uncontrolled (and unknown) variables will differ from those of the before image. This test tells us the best we can expect from comparing before and during images.
Here are the results of the first test, where a second image is compared to the first, all else being equal. The stars should be detected at the same exact positions, and yet they aren’t.
The differences between the positions of the same stars in two successive exposures. There are approximately eight stars detected (in green, fewer in the other channels). This is an indication of the variations introduced by the atmospheric seeing.
The standard deviation of this comparison was 2.3 arcseconds. The comparisons of other successive frames yielded standard deviations of 2.1 and 1.6. It appears that the consistency of star positions as detected by my equipment is about 2 arcseconds. This is consistent with reports of the atmospheric seeing in Minnesota.
I was curious about how to characterize “seeing” and found these interesting links (there is always too much to explore and investigate to the depth I would like):
There is another domain of relevant knowledge: how to determine if the distribution of observations is the same, or different, from another. I will be comparing position measurements from the “before” condition to the “during” condition when the sun’s gravity will have a possible influence. How can we tell if the measurements are from truly different conditions, rather than the normal variations caused by noise? Here are some links I explored to try to answer this question:
