3rd Nov '16 12:05:17 PM

**unokkun** Is there an issue? Send a Message

**Changed line(s) 174,175 (click to see context) from:**

So, to recapitulate, Picard and Sisko will forever disagree about whose clock slowed down. The only way to settle the argument is for Picard to turn around and come back so they can compare notes. The only way to do this is for him to accelerate, and that will pick his viewpoint out as the "wrong" one.

**to:**

So, to recapitulate, Picard and Sisko will forever disagree about whose clock slowed down. The only way to settle the argument is for Picard to turn around and come back so they can compare notes. The only way to do this is for him to accelerate, and that will pick his viewpoint out as the "wrong" ~~one.~~

one. [[note]]To be fair, if the universe is [[https://en.wikipedia.org/wiki/Three-torus_model_of_the_universe torus-shaped]] ([[Recap/TheSimpsonsS10E22TheySavedLisasBrain "doughnut-shaped"]], so that after traveling a certain distance, you'll find yourself back where you started), Picard could "catch up" with Sisko without slowing down. That being true wouldn't completely break down relativity (after all, this is a serious proposal for the shape of the universe), but it ''would'' establish an absolute frame of reference, so there ''would'' be a "standing still".[[/note]]

15th Jul '16 9:30:13 PM

**nombretomado** Is there an issue? Send a Message

**Changed line(s) 3,6 (click to see context) from:**

AlbertEinstein's Theory of General Relativity is perhaps the second-most confusing subject in all of modern physics, beaten only by Quantum Mechanics. EEqualsMCHammer applies to this in full force, meaning very few people understand all the [[{{MindScrew}} incredibly weird]] results that Einstein's very, very simple ideas imply. (WritersCannotDoMath means that most of the media isn't helping.)

Relativity was worked out by AlbertEinstein in two pieces, special relativity and general relativity. Bright high schoolers can handle the math of special relativity. The math of general relativity makes graduate students cry, so we'll leave it to [[http://en.wikipedia.org/wiki/Mathematics_of_general_relativity the other wiki]].

Relativity was worked out by AlbertEinstein in two pieces, special relativity and general relativity. Bright high schoolers can handle the math of special relativity. The math of general relativity makes graduate students cry, so we'll leave it to [[http://en.wikipedia.org/wiki/Mathematics_of_general_relativity the other wiki]].

**to:**

Relativity was worked out by

**Changed line(s) 176 (click to see context) from:**

It actually bothered AlbertEinstein quite a bit that non-accelerating viewpoints are better. So much, in fact, that he came up with general relativity to explain the physics of acceleration. But that we'll have to save for later.

**to:**

It actually bothered ~~AlbertEinstein ~~Albert Einstein quite a bit that non-accelerating viewpoints are better. So much, in fact, that he came up with general relativity to explain the physics of acceleration. But that we'll have to save for later.

7th Jun '16 9:34:29 PM

**PaulA** Is there an issue? Send a Message

**Changed line(s) 240,241 (click to see context) from:**

(At low speeds where γ is about 1, this reduces to "momentum = mv", which is SirIsaacNewton's definition for momentum.)

**to:**

(At low speeds where γ is about 1, this reduces to "momentum = mv", which is ~~SirIsaacNewton's ~~Creator/IsaacNewton's definition for momentum.)

26th Jan '16 1:19:23 PM

**FordPrefect** Is there an issue? Send a Message

**Changed line(s) 311,312 (click to see context) from:**

The effect would become more significant near an extremely dense object, such as a neutron star or a [[UsefulNotes/BlackHoles black hole]]. Say we have a 5 solar mass black hole, which would have a Schwarzchild radius of 15 kilometers. (Incidentally, the Schwarzchild radius for a black hole corresponds to where its ''event horizon'' is. Objects that are bigger than their own Schwarzchild radius aren't black holes and don't have event horizons.) Now say you're standing 30 kilometers from the center of the black hole, which is only 15 km above its event horizon. The gravitational time dilation factor here works out to 0.707, which is ''significant'' but not ''huge''. For every 7 seconds that elapse for you, 10 seconds elapse for us folks who are a nice safe distance away from the black hole. You'll have to re-synch your clock when you rejoin us -- assuming the tidal forces from the black hole don't turn you into molecular spaghetti first -- but you're not going to find yourself flung years into the future just by dawdling there for a few hours. If, however, you got closer, such that you were hovering only 1 kilometer above the event horizon, the gravitational time dilation factor would be 0.25, meaning 4 seconds would pass for the outside world for every 1 second you experienced. The effect would get more and more pronounced the closer you got to the event horizon, and when you actually ''reached'' the event horizion the gravitational time dilation factor would be exactly zero. You would be frozen in time right at the event horizon for all eternity, as far as the outside world was concerned.

**to:**

The effect would become more significant near an extremely dense object, such as a neutron star or a [[UsefulNotes/BlackHoles black hole]]. Say we have a 5 solar mass black hole, which would have a Schwarzchild radius of 15 kilometers. (Incidentally, the Schwarzchild radius for a black hole corresponds to where its ''event horizon'' is. Objects that are bigger than their own Schwarzchild radius aren't black holes and don't have event horizons.) Now say you're standing 30 kilometers from the center of the black hole, which is only 15 km above its event horizon. The gravitational time dilation factor here works out to 0.707, which is ''significant'' but not ''huge''. For every 7 seconds that elapse for you, 10 seconds elapse for us folks who are a nice safe distance away from the black hole. You'll have to re-synch your clock when you rejoin us -- assuming the tidal forces from the black hole don't turn you into molecular spaghetti first -- but you're not going to find yourself flung years into the future just by dawdling there for a few hours. If, however, you got closer, such that you were hovering only 1 kilometer above the event horizon, the gravitational time dilation factor would be 0.25, meaning 4 seconds would pass for the outside world for every 1 second you experienced. The effect would get more and more pronounced the closer you got to the event horizon, and when you actually ''reached'' the event ~~horizion ~~horizon the gravitational time dilation factor would be exactly zero. You would be frozen in time right at the event horizon for all eternity, as far as the outside world was concerned.

26th Jan '16 1:16:24 PM

**FordPrefect** Is there an issue? Send a Message

**Changed line(s) 289 (click to see context) from:**

Einstein's second brainwave it just as important as the first, and arguably more intuitive: ''Velocity is relative''. There is an UrbanLegend that he came up with this while sitting on a stationary train, looking out of the window. The train beside his began to pull away, and he realized that his train pulling away would appear exactly the same from the other train, as the other train pulling away appeared to him. (Apart from the acceleration, which will be discussed later.)

**to:**

Einstein's second brainwave ~~it ~~is just as important as the first, and arguably more intuitive: ''Velocity is relative''. There is an UrbanLegend that he came up with this while sitting on a stationary train, looking out of the window. The train beside his began to pull away, and he realized that his train pulling away would appear exactly the same from the other ~~train, ~~train as the other train pulling away appeared to him. (Apart from the acceleration, which will be discussed later.)

16th Oct '15 11:46:35 PM

**nombretomado** Is there an issue? Send a Message

**Changed line(s) 124,125 (click to see context) from:**

seconds pass. The 14-year travel time mentioned for ''PlanetOfTheApes'' above was obtained by plugging in ''t''=1000 years (to speed up) and ''g''=10 meters/second/second, and doubling (to slow back down so you don't crash into your destination).

**to:**

seconds pass. The 14-year travel time mentioned for ~~''PlanetOfTheApes'' ~~''Film/{{Planet of the Apes|1968}}'' above was obtained by plugging in ''t''=1000 years (to speed up) and ''g''=10 meters/second/second, and doubling (to slow back down so you don't crash into your destination).

16th Oct '15 11:46:19 PM

**nombretomado** Is there an issue? Send a Message

**Changed line(s) 70,71 (click to see context) from:**

In ''PlanetOfTheApes'', George Taylor travels away from Earth for 2,000 years. Presumably he spent half his time speeding up and half his time slowing down. If you accelerate at 10 meters/second/second for 1,000 years, you get up to 99.999955% of the speed of light. Time dilation does take effect, so George Taylor should have aged rather less than 2,000 years. However, it takes a while to kick in--the astronauts should have experienced a trip of about 14 years, not 18 months.

**to:**

In ~~''PlanetOfTheApes'', ~~''Film/{{Planet of the Apes|1968}}'', George Taylor travels away from Earth for 2,000 years. Presumably he spent half his time speeding up and half his time slowing down. If you accelerate at 10 meters/second/second for 1,000 years, you get up to 99.999955% of the speed of light. Time dilation does take effect, so George Taylor should have aged rather less than 2,000 years. However, it takes a while to kick in--the astronauts should have experienced a trip of about 14 years, not 18 months.

11th Sep '15 5:01:38 PM

**FordPrefect** Is there an issue? Send a Message

**Changed line(s) 29,30 (click to see context) from:**

Second, Mars will get brighter and Earth will get dimmer. Here's one way to think about this. Suppose Boothby stays behind on Earth and flashes one pulse of light at the ''Enterprise'' every second. Since the ''Enterprise'' is moving away, a second after O'Brien sees the first pulse, the ship will have moved away from Earth, so the second pulse won't have caught up--it will take more than a second for him to see the second pulse. If you picture now a strobe light flashing bursts every millisecond, then from O'Brien's viewpoint, this delay will "smear out" until it looks like a steady, dimmer light.[[note]]Relativistic effects (length contraction and time dilation) will change the picture, but not much; O'Brien will still see the the light pulses spaced out, just not as much as Sir Isaac Newton would expect.[[/note]]

**to:**

Second, Mars will get brighter and Earth will get dimmer. Here's one way to think about this. Suppose Boothby stays behind on Earth and flashes one pulse of light at the ''Enterprise'' every second. Since the ''Enterprise'' is moving away, a second after O'Brien sees the first pulse, the ship will have moved away from Earth, so the second pulse won't have caught up--it will take more than a second for him to see the second pulse. If you picture now a strobe light flashing bursts every millisecond, then from O'Brien's viewpoint, this delay will "smear out" until it looks like a steady, dimmer light.[[note]]Relativistic effects (length contraction and time dilation) will change the picture, but not much; O'Brien will still see the ~~the ~~light pulses spaced out, just not as much as Sir Isaac Newton would expect.[[/note]]

1st Jul '15 8:27:11 PM

**nombretomado** Is there an issue? Send a Message

**Changed line(s) 94,95 (click to see context) from:**

Suppose that [[Series/StarTrekTheNextGeneration Captain Picard]] and [[Series/BattlestarGalacticaReimagined William Adama]] fly past each other. They are in deep space, far from any planets, so they have no notion of who is "really" standing still and who is "really" moving. The ''Enterprise'' passes the ''Galactica'', so all they know is that the ''Enterprise'' is going some amount ''v'' faster than the ''Galactica''.

**to:**

Suppose that [[Series/StarTrekTheNextGeneration Captain Picard]] and ~~[[Series/BattlestarGalacticaReimagined ~~[[Series/BattlestarGalactica2003 William Adama]] fly past each other. They are in deep space, far from any planets, so they have no notion of who is "really" standing still and who is "really" moving. The ''Enterprise'' passes the ''Galactica'', so all they know is that the ''Enterprise'' is going some amount ''v'' faster than the ''Galactica''.

19th Aug '14 10:01:43 PM

**Uchuujinsan** Is there an issue? Send a Message

**Changed line(s) 60,62 (click to see context) from:**

** This above equation only applies if velocity can be assumed to be zero. For a mass at nonzero velocity the appropriate equation becomes E[[superscript:2]]=m[[superscript:2]]c[[superscript:4]]+p[[superscript:2]]c[[superscript:2]]

*** And then the reason that it simplifies down to E=mc[[superscript:2]] is that the full equation has E as the hypotenuse of a right-triangle, and applies the Pythagorean theorem. E is the hypotenuse, and the legs are mc[[superscript:2]] and pc, respectively.

*** And then the reason that it simplifies down to E=mc[[superscript:2]] is that the full equation has E as the hypotenuse of a right-triangle, and applies the Pythagorean theorem. E is the hypotenuse, and the legs are mc[[superscript:2]] and pc, respectively.

**to:**

** This ~~above equation only applies if velocity can be assumed ~~is often corrected to ~~be zero. For a mass at nonzero velocity ~~E[[superscript:2]]=m[[superscript:2]]c[[superscript:4]]+p[[superscript:2]]c[[superscript:2]]. In fact, both formulas are correct and apply to moving objects, the ~~appropriate equation becomes E[[superscript:2]]=m[[superscript:2]]c[[superscript:4]]+p[[superscript:2]]c[[superscript:2]]~~

*** And then the reason that it simplifies down to E=mc[[superscript:2]]difference is that in the ~~full equation has E as the hypotenuse of a right-triangle, and applies the Pythagorean theorem. E ~~original formula E=mc[[superscript:2]] m is the ~~hypotenuse, ~~relativistic mass while in the other formula it's the rest mass, often just called mass. Basically both are the same formula, just written differently, and ~~the legs are mc[[superscript:2]] and pc, respectively.~~

can be changed into each other with a little math.

*** And then the reason that it simplifies down to E=mc[[superscript:2]]

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