|MadSci Network: Physics|
Dear Ray, This is a great question and it really emphasizes the need to understand two different kinds of acceleration, and the addition of one more relativity postulate called the Equivalence Principle. It is true that the first postulate of Special Relativity states basically that all the laws of physics are the same in all inertial reference frames, and an inertial reference frame is a coordinate system that is not accelerating. (The second postulate being the universality of the speed of light.) This typically means we need to restrict Special Relativity to frames which are moving at a constant velocity and are not near any strong gravitational fields. This avoids any accelerations caused by any forces, gravitational or otherwise. To make the jump to General Relativity we need to include another assumption in addition to the two for Special Relativity. This third postulate is the Equivalence Principle. The Equivalence Principle is sometimes broken down into two parts, the Weak and Strong Equivalence Principles, but the basic idea is that the force of gravity is equivalent to an acceleration of the reference frame. To give an example, imagine two spacecrafts, one resting on the surface of the Earth, where there is a gravitational field causing everything to fall down with an acceleration of 9.8 m/s/s , and the other spacecraft out in space, away from any gravity, but which is accelerating through space with an acceleration of 9.8m/s/s . In both crafts, a “dropped” object would approach the back of the ship with an acceleration of 9.8 m/s/s . In the ship on Earth, the object falls toward the ground due to gravity, but in the ship moving through space, the back of the ship would move forward toward the object. The important point is that for someone in the ship, there is nothing that can be done to distinguish between these two situations. So, the effects of gravity are exactly the same as the effects of the frame accelerating. Why is gravity so important in this example? Why couldn’t some other force be acting on the “stationary” ship and be equivalent to an accelerating frame? Well, it seems that only gravity acts “equally” on everything, that is, with a magnitude proportional to the object’s mass, or measurement of its inertia. All other forces depend on some other property, e.g. the electrostatic force depends on the object’s charge. So, if a space ship was sitting in an electric field, only charged particles would experience a force, and it would be obvious that the whole ship was not accelerating. So, are there any inertial reference frames in a gravitational field, since gravity is like an accelerating frame, and accelerating frames are not supposed to be inertial? As a matter of fact, there are! A frame can actually remove the effects of gravity if it is allowed to free fall. That is, if the frame is allowed to move freely under the influence of gravity. The gravitational field will act on everything in that frame giving it the same acceleration (if we assume the field is uniform within our area of interest). To someone in that frame, it would appear that the frame was out in space, away from any gravity and not accelerating! (At least until the frame hit the ground!) We usually refer to this as “weightlessness”. Therefore, according to General Relativity, we still have inertial frames, but they must be accelerating if they are in a gravitational field. This is a very different idea than in Special Relativity where the frames could not be accelerating. Therefore, from General Relativity, we now see that a freefalling frame is inertial, and therefore all the laws of physics hold in a freefalling frame. One set of the laws of physics being Special Relativity! This must seem like a roundabout way to finally get to your question, but now we are ready to understand it. Is the Earth accelerating through space? Yes, it is in orbit around the Sun. Does Special Relativity hold on the Earth? Yes, it does! Where does it hold if the Earth is accelerating? Since the Earth is in free fall around the Sun … Hold it right there! The Earth is in free fall around the Sun? Well, yes, since the Earth is moving in its orbit under the influence of the gravitational field of the Sun, it is in free fall, actually, that’s what all orbits are. So, for the most part, we don’t notice the gravitational field of the Sun here on the Earth, since we are constantly moving under its influence. (This is not completely true because the Sun’s gravity is not uniform, and so one side of the Earth feels slightly more pull than the other, and along with the Moon’s gravity [mostly the Moon’s gravity] causes our tides.) The acceleration of the Earth around the Sun does not cause the break down of Special Relativity. What about the gravitational field of the Earth? Well, yes it could. If an experiment were done in a freely falling frame on the Earth, then Special Relativity would absolutely hold. In a frame bound to the Earth, General Relativity certainly may cause some changes to a Special Relativistic prediction, however, since on the Earth, gravity acts “vertically”, Special Relativity is still very accurate for “horizontal” experiments. Well, Ray, I hope that answers your question. You are right in that in general all matter is accelerating due to gravitational fields, but that does not preclude using Special Relativity. If we can do an experiment in a frame which is also accelerating due to gravity, then that frame is inertial, and therefore Special Relativity still holds! Thank you for your interest and please let us know if you have any other questions. Sincerely, Jim Guinn Georgia Perimeter College
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