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Stephen Hawking has passed away at the age of 76 on Pi day, 3/14/2018. http://www.bbc.com/news/uk-43396008 https://www.theguardian.com/science/2018/mar/14/stephen-hawking-professor-dies-aged-76?CMP=fb_us

From Light Into Heat. His Radiation Remains. The Journey Complete. ..

A month or so ago, I was asked to provide a logical explanation to the Twin Paradox that showed why symetrical views of time was not a paradox or how was symmetrical views of time completely consistent with the time deviation experienced between the twins. So as promised I fully explain: Why length contraction is the reason why the twins experience time deviation. Why time deviation was not a paradox with the twins symmetrical views of time because length contraction is the asymetrical component that was never considered. Finally, I show how their symetrical views of time really are completely consistent for the entire journey using a trasponder solution with each other. As always, I will be using the example given in Wikipedia so that you can verify the results with those given in that example and I can skip the math for simplicity sake. I will only be providing the logical models that fully explains the paradox. BTW, this is a good example of my previous statements that while logic may be prone to intuitive errors of false premise, such as the Earth is the center of the Universe because everything appears to revolve around the Earth, math is just as prone to intuitive errors of false conclusions. The key points of the problem are: The ship carrying one of the twins goes straight to Alpha Centauri and back. Acceleration is assumed to be an insignificant factor so velocity is a constant 80% of the speed of light in both directions. Alpha Centauri is assumed to be in relativisticly static motion relative to the Earth with a proper distance of 4 light years. Each twin is equiped with a powerful transponder that pings with a source frequency of exactly once per second or 1Hz. The Earth twin sees that Alpha Centauri is a static 4 lightyears away. Therefore, he calculates the trip will take 4/0.8 = 5 years each way or 10 years total. The ships twin has a different perspective of the trip when moving at 0.8c due to length contraction, the distance is only 60% of the proper distance or 2.4 light years away. Therefore from his point of view, the trip will only last 3 light years each way or 6 years total. So when he returns, he experiences 6 years while the Earth Twin experiences 10 years, however that is not the paradox. The paradox is based on the fact that each twin should have symmetrical points of view of their brothers time which is true: When the ships twin is on the outbound leg moving away from Earth at 0.8c then each twin sees their brothers time as moving at 1/3 of normal, or they would each be receiving a transponder ping only once every 3 seconds. When the ships twin is on the return leg, then each twin sees their brothers time as moving at 3x normal or they would each be recieving 3 pings/sec. These time shifts are due to the relativistic redshift which I didnt bother working out the math again, but the formulas are pretty simple and include time dialaion plus normal doppler effect due to lagtime, so that you can verify the results yourself or just refer to the Wikipedia example which uses the exact same problem. The logical resolution to this paradox is the fact that while their views of each others time is symetrical, their views of the distance traveled is asymetrical. The reason why is that the Earth, Alpha Centari, and the space in between the two are all in the Earths inertial reference frame, while the ship plus what is inside the ship is all that is in the ship’s inertial reference frame. The Earth twin sees the ship is length contracted by 60% which has no bearing on the trip The ships twin sees the Earth’s inertial frame as length contracted and as we said, the Earth, AC and the distance inbetween is all included within that inertial reference frame. Therefore, from the ship’s twin’s point of view the distance is length contracted by 60% of 4 light years or 2.4 light years. So with 60% less distance to trave, then the trip takes 60% less time to travel from the ship twins point of view. Another words while their point of view of time is symmetrical, their point of view of distance is asymtrical which accounts for their deviation in time experienced. We can confirm this by correlating their point of view with lagtime. Another words, from the Earth twins point of view, the ships twin would take 5 years plus it would take 4 years for the light (or transponder signal) to get back to Earth from Alpha Centauri: 5 years + 4 years = 9 years That means the Earth twin would expect to witness the ship actually execute the turnaround 9 years after the ships departure. Or when the transponder signal recieved back on Earth would change from 1 ping every 3 seconds to 3 pings/second then the ship would have executed the turn which would happen 9 years after the origional launch. If you do the math and count the pings received during those 9 years at 1 ping/3 seconds adds up to: 9 years x 1/3 = 3 years which is actually what the ships clock would read at the turn around by both twins. Events that include both a time and a place must always be consistent to any inertial frame. On the trip back to Earth, the journey would only seem to take 1 year as perceived by the Earth twin so: 9 years + 1 year = 10 years total time as expected by the Earth twin. However, during that 1 year, the transponder is pinging 3 times per second so it adds up to 3 years 1 year x 3 = 3 years 3 years + 3 years = 6 years total. The Ships twin experiences something different. He hears the transponder received from Earth ping once every three seconds, and since the outbound leng only takes 3 years, he sees the Earth clock as counting only 1 year. 3/3 = 1 year On the return leg, the ship’s twin experiences three years worth of pings that are pinging at 3 pings per seconds, therefore the Earth clock advances 9 years during his 3 year return leg. 3 years x 3 = 9 years 1 year + 9 years = 10 years which is the time elapsed back on Earth. So not only is the math consistent, it sould not even be a surprise to either twin that their brother has aged differently.

Tip for anyone having the same issue: Save the picture to your hard drive and drag&drop it into a google graphics search. It works wonders

It is a sad day, the world lost a great mind. His sense of humor was a great addition to his work - his singing of the Monty Python "Galaxy Song", the poker scene on Star Trek The Next Generation, his appearance on episodes of "Big Bang Theory" and the Simpsons...countless interviews and comedy shows he appeared in. He had a great life considering that he made it to 76 when he was getting 2 years max at 22 years of age. On top of his work in physics he left 3 children, a major hollywood film was made as his biography - who could ask for a more fulfilling life.

No, it isn't. To put numbers to my earlier example: Acceleration due to gravity is GM/r2 Gravitational potential is -GM/r For the Earth, the difference in acceleration at the surface and 1 earth radius above the surface is 7.35 m/s2 The difference in potential is 31255879.6 joules/kg For our 4x Earth mass planet, the difference in acceleration between surface and the same altitude above the surface is 5.45 m/s2 The difference is potential is 41674506.2 joules/kg This is 33% larger than that for the same altitude difference for the Earth, while the difference in acceleration is only 74% as much. You can even have a difference in potential without any difference in acceleration. In a uniform gravity field, the difference in gravitational potential would be found by gh, where g is the acceleration due to gravity throughout the whole field and h is the height difference between them. In this case, there is no difference in acceleration over the region being measured (gravitational potential over small height differences approach this ideal, as g changes insignificantly over the region considered.)

To build on what swansont has already alluded to. The fact that our Earth observer and space traveler experience the same local acceleration is not the determining factor in terms of the time dilation each would measure in other clocks. As swansont said, gravitational time dilation is due to the difference in gravitational potential, or in other words, the total effect of the gravitational field between the position of the two clocks. One way to visualize this is to consider how fast would something dropped from the higher altitude be moving when it reached the lower altitude. That would be a measure of the difference in gravitational potential. So if we take an object and drop it from a altitude 1 earth radius above the surface, when it starts its fall it will experience 1/4 the acceleration it does at the surface and it will hit the ground moving at some speed. A clock placed at this altitude will run faster than one on the surface. Now if we put a clock on the surface of the world with 4 times the mass of the Earth and twice the radius, it will experience 1g just like a clock on the surface of the Earth. If we put an object 1 earth radius above the surface, it will experience 4/9 the acceleration as the surface. If we drop an object from this height, it will hit the surface moving faster than one dropped from an equal height on the Earth. In addition, a clock placed here will run faster than one on the surface by a greater factor than the difference in rate between the two clocks separated by the same altitude in the Earth scenario, even though the difference in acceleration experienced by the Earth clocks is larger than the difference for the second set of clocks. When applying this to our accelerating space traveler, to use the equivalence principle, we have to consider what the equivalent gravitational field to his acceleration would be like. In this case, it would be a uniform gravity field that extends to infinity along the line of acceleration that does not diminish in strength with distance. Clocks that are in the direction he is accelerating will run fast, and clocks in the opposite direction will run slow. The greater the distance between them and these clocks, the larger the difference in their tick rates. Thus for our space traveling observer, As he accelerates at 1 g away from the Earth, not only does his speed relative to the Earth increase causing him to measure a time dilation in the Earth clock, but the Earth is getting increasingly further away in the direction opposite to his acceleration vector. As a result, he would measure an additional increase in the slowing of the Earth clock tick rate. When he changes his acceleration in order to decrease his relative velocity and then accelerate back towards the Earth, the magnitude of the acceleration remains unchanged. However, the Earth's position relative to the acceleration vector does change. Now it is in the direction of the acceleration, and thus according to our traveler, the Earth clock runs fast. This is how the equivalence principle would be applied in this case. One thing to note is that this "equivalent gravity" due to acceleration is only measured by our accelerating observer. Our Earth observer would only measure time dilation due to the difference in relative velocities. (Both observers would also measure any difference due to relative positions in Earth's gravity field, but depending on the exact scenario, this can be insignificant. If you are using scenarios of high fractions of c over light year distances, this additional factor will likely be smaller than the rounding errors in your calculations. )

A giant amongst men: Given 2 years to live at the age of 21 years, he instead reached out into the Universe, with his knowledge, humour and example. I'm drinking a can of Fosters to his memory.

In looking into the background of this report I came across an excellent paper, "Late Pleistocene human population bottlenecks, volcanic winter, and differentiation of modern humans". Although this paper is two decades old it gives a clear picture of the landscape of competing ideas into which the Toba eruption global winter and its impact on humanity, was inserted. While doing this speedy literature search I ran across another paper, from about five years ago, that also cast serious doubt on the Toba Eruption/Bottleneck connection. Life being what it is I can no longer locate it. If I find it during a further search I'll post a link here.

Today I learned that plants have a way to use the entire visible spectrum. The literature and our present examinations indicate that the intra-leaf light absorption profile is in most cases steeper than the photosynthetic capacity profile. In strong white light, therefore, the quantum yield of photosynthesis would be lower in the upper chloroplasts, located near the illuminated surface, than that in the lower chloroplasts. Because green light can penetrate further into the leaf than red or blue light, in strong white light, any additional green light absorbed by the lower chloroplasts would increase leaf photosynthesis to a greater extent than would additional red or blue light. Based on the assessment of effects of the additional monochromatic light on leaf photosynthesis, we developed the differential quantum yield method that quantifies efficiency of any monochromatic light in white light. Application of this method to sunflower leaves clearly showed that, in moderate to strong white light, green light drove photosynthesis more effectively than red light.https://academic.oup.com/pcp/article/50/4/684/1908367

That's us, silly and hilarious. D Trump doesn't represent me, I'm Canadian. And as an objective observer, I know who is the bigger whack job of the two. Lets try to keep that in mind.