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Janus last won the day on March 14

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  1. Moving at the speed of light

    The simple answer is that the universe works in such a way as to prevent your ever exceeding c relative to your starting point. You are essentially asking what prevents you from constantly accelerating at 1 m/s2 until you exceed the speed of light. If the Universe operated under the Rules of Newtonian physics, nothing. After 7 years, 21 days, 20 hrs, 47 min and 38 sec, you would be moving faster than light. However, we don't live in a universe that follows Newtonian rules, but follows Relativistic ones instead. And one of those differences in rules is in how velocities add together. Under Newton if you want to get the sum of two velocities, you simply add them together like this, w=u+v. Thus if you were moving at 1 m/s relative to some reference and then added 1m/s to your current speed, you would be moving at 1+1=2m/s relative to the initial reference. However, it turns out that this isn't correct. the right way to add the velocities is by w= (u+v)/(1+uv/c2) where c = the speed of light in a vacuum or 299,792,458 m/s Now when you add 1 m/s to 1m/s you get a resultant velocity of 1.9999999999999999777469988789276 m/s almost, but not quite 2 m/s At low speeds, this doesn't amount to much, but as the speed increase, the difference starts to mount up. If you were moving at 0.1c and increased your velocity by 0.1c, you would be moving at .198019802 c relative to the point you were initial moving at 0.1c relative to. If you boost you speed by another 0.1c, you will now be moving at .2922330097c relative to the initial frame. Do this 7 more times, and instead of moving at c relative to the initial frame like you would under Newton, you would be moving at 0.7629989373 c. Each time you add change your speed by 0.1 c relative to your current speed as measured by you, you add less than 0.1c total change in your velocity. And the closer you get to a total of c, the less change in total velocity you'll end up with, and no matter how you try and add up the velocities, the resultant velocity will always end up being less than c.
  2. The use of acceleration due to gravity here has nothing to do with relative motion, it is a more universal expression for comparing the force of gravity, since it does not rely on the mass of the object undergoing the acceleration. As far as your next post goes: The first part holds in that a gravitational potential does imply a gravitational force, however, this does not in turn imply that objects at different heights must experience different accelerations for there to be a difference in potential. The difference in potential is the integral of the force over a distance and does not require that the force differs over the distance. In this case, the space traveler would have to combine two separate calculations to determine how fast the Earth clock ticks relative to his own at any given instant. One that factors in his position relative to the Earth's gravitational field, and another to determine its relative position in the "acceleration field". The main difference being that the Earth's gravitational field strength decreases with distance from the Earth, and the strength of the "acceleration field does not. This would be in addition to any time dilation due to relative velocity differences. (if you choose the instant in which the ship has just stopped its velocity away from the Earth and is just going to start back to Earth, At that moment it will be at rest with to the Earth, and you will only need to use the first two calculations.)
  3. 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.)
  4. 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. )
  5. Which way is up?

    Neutrinos were hypothesized before they were ever detected, but we had detected other particles such as electrons, protons and neutrons. This did not mean that neutrinos were likely to turn out to be one of these particles. The very reason they were hypothesized was the fact that none of these particle had the properties that the neutrino needed to have. In the same way, quantum fluctuations do not behave in a manner the is consistent with them fulfilling the role of the hypothesized gravtion.
  6. Which way is up?

    It is "specific energy" or energy per unit mass. In this case it would be joules/kg
  7. Which way is up?

    The fact that there is zero gravitational force at the center of the Earth does not mean that an object at the center of the Earth isn't at a lower gravitational potential than an object on the surface. A difference in gravitational potential is tied to the energy needed to make the entire trip between two points and not what happens at the start or finish of the trip. Using the standard reference which puts zero gravitational potential at an point an infinite distance from the gravity source, the gravitational potential from the surface of a sphere outward is found by -GM/r, where r is the distance from the center of the sphere. As r gets smaller the potential goes more negative and thus becomes lower. For a point inside a uniformly dense sphere, the potential is -GM [ (3R2-r2)/2R3] which reduces to -3GM/2R at the center. -3gm/2R is more negative than -GM/r, so the gravitational potential at the center is lower than at the surface. Here's a graph for the gravitational potential for the Earth (assuming a uniform density) starting at the center and moving out to some distance above the surface. the green line marks off the radial distance of the Earth's surface. The exact shape of the curve on the left of the green line will be different for the real Earth due to the variance of density with depth, but will still be lower at the center than any other point. Time dilation is related to the difference in gravitational potential and thus a clock at the center of the Earth, being at the lowest potential will run the slowest. Going back to what I said about the potential difference applying to the entire trip, If you take a clock and move it from the center of the Earth to that point an infinite distance from the Earth, it will start at zero g and end at zero g, but it will have raised its gravitational potential by a fair amount and will run faster at the far removed point than it did at the center of the Earth.
  8. As far as planets orbiting stars goes, the equation for an orbiting clock would give the time dilation for any given planet as measured by a distant observer. (As long as you are dealing with circular or nearly circular orbits, adjustments would have to be made to deal with highly elliptical ones. Of course for most normal stellar systems this going to be a very small effect (for the Earth it works out to be about a difference of 1/2 sec per year.) And since our detection of extra-solar planet are by indirect means (measuring it parent star's wobble or it's transit in front of the star) and not direct observation of light from the planet, this is not something we presently would measure.
  9. But now you have to invoke GR and gravitational time dilation. And this has nothing to do with what the clocks "experience" but with their relative positions in the gravitational fields. For example consider two satellites orbiting the same body at different altitudes. Both are in free fall and experience 0 g. If this is what determined the gravitational time dilation difference between them, then it should be zero, and the only time dilation would be from their orbital velocity difference, and if R is the radius of a clock's orbit then its time dilation factor should be found by. T = t0 sqrt{1-GM/Rc2} since orbital velocity is v = sqrt{GM/R} However the actual time dilation for an orbiting clock is T = t0 sqrt{1-3GM/Rc2} The difference being due to the fact that one of the orbiting clocks is higher in the gravitational field than the other. They are accelerating in the sense that they are changing velocity with respect to some observer at rest with respect to the gravitational field. They are not accelerating in the sense of being in an accelerated frame, as they would not measure clocks in the direction they are falling as running fast, but would instead measure their respective tick rate depending on their relative position in the gravitational field. Put another way, if you had a rocket gaining velocity towards a clock due to its firing of its own engines and far removed from any gravity field, it would measure that clock as running fast due to the rocket's acceleration as an additional factor along with the time dilation due to the difference in velocities. With a rocket falling towards the surface of a planet in free fall, the rocket would measure a clock on the ground as running slow due it's lower position in the gravity field in addition to the time dilation caused by the difference in velocities.
  10. Turning does break the reciprocal effect. This is because measurements made in an accelerated frame differ from those made in an non-accelerated (inertial) frame. If you are in an inertial frame all you need to know to determine the time dilation of another clock is its relative velocity to you. If you are in an accelerated frame, you additionally have to take into account three new factors: The magnitude of the acceleration, the direction of the other clock relative to the acceleration, and the distance to the clock along the line of acceleration. Both the acceleration magnitude and distance effect the degree of difference in tick rate you will measure. The direction determines whether it is a faster tick rate or a slower one. If the clock is in the direction of acceleration, you will measure it as ticking fast, if it is in the opposite direction, you will measure it as ticking slow. This measurement holds for clocks whether they share your accelerated frame or not. (This means that if you are in an accelerating rocket, a clock in the nose will run faster than one in the tail, even if they don't move relative to each other in the rocket frame.) With the twin paradox this means that when the traveling twin turns around, he is both a great distance from the Earth twin and accelerating towards him. This combination results in the Earth twin clock advancing very quickly during the turnaround period according to the traveling twin, which more than compensates for the fact that it ticks more slowly during the coasting periods of the both the outbound and return legs. The Earth twin, which remains in an inertial frame throughout only measures the traveling twin's clock run slow due to relative velocity.
  11. In the past you might have said that, but the term "relativistic mass" is no longer used. The energy increases asymptotically and energy has properties that used to be only associated with mass.
  12. I'm at a loss to how your first statement above relates to the section of my post you quoted. I was referring to relativistic effects only. In the early days of Relativity, they used the term "relativistic mass" to refer to the apparent mass increase of a moving object to distinguish it from the "rest mass" of the object or its mass as measured when at rest with respect to the observer. That term has fallen out of favor. Today, "mass" is used to mean only the "rest"mass. "Relativistic mass" is now just energy which has some "mass equivalent" properties. Its a matter of convention in terminology to avoid confusion. Thus Is not correct by modern usage of the term "mass" which refers to the "rest mass" which is invariant across reference frames. The length contraction part is correct because it is reciprocal.
  13. Actually, no. While it is common to see it said that mass increases with velocity in many popularizations of SR, modern physics tends to treat mass as an invariant property (this avoids the confusion caused by having to distinguish between rest mass and relativistic mass. What used to be called relativistic mass is now just included under the label of energy. (with the understanding that some of the properties expressed by mass alone under Newtonian physics are now also expressed by energy.) So as an object's velocity increases with respect to you its kinetic energy relative to you increases, but if you change your own velocity, your kinetic energy relative to yourself remains 0. This is no different than for Newtonian physics. If you collide with a bullet with a relative velocity of several 100's of meters/sec, it doesn't matter if you are at rest with respect to the ground and the bullet was traveling, or the bullet was suspended at rest with respect to the ground and you were traveling with respect to it. The impact between bullet and yourself will be the same, and you will suffer the same consequences. The difference Between Newton and Relativity is that for Newton the KE increases by the square of the relative velocity and thus approaches infinity only is the velocity approaches infinity, while in Relativity, the KE increases by an asymptotic function and approaches infinity as the relative velocity approaches c. Length contraction (and time dilation) result from the fact that observers in relative motion with respect to each other measure time and space along different axis in space-time.( an analogy is to map time as being in the front-back direction and space in the left-right direction. people facing in different directions measure left-right and front-back relative to themselves and differently from each other. In Relativity, we are dealing with 3 spacial and one time direction and it is relative motion that creates the difference in the space-time axis of the observers.
  14. Water has calories

    Reminds me of the "Scotch on the Rocks" diet. It was based on the idea that there were a smaller number of Calories in a scotch on the rocks than the calories your body expends bringing the consumed drink up to body temp. The trick was in that fact while perfectly true, a Calorie is actually a kilocalorie and equal to 1000 calories.
  15. How can you measure distances across space?