robertj@garfield.UUCP (Robert Janes) (07/17/85)
Just a few comments on an article posted to net.sf-lovers which may be of interest to physics people on the net. The comments reflect my own probably inadequate understanding of the subject but here goes anyways. >Subject: RE: FLT >From: jagardner@watmath.UUCP (Jim Gardner) >Reply-To: jagardner@watmath.UUCP (Jim Gardner) >Organization: U of Waterloo, Ontario . . >A little relativity theory: we begin with the basic law of >physics F=ma. This equation is a >special case< of the the basic law law of physics called Newton's Second Law. Newton original framed somewhat in this fashion: " force is porportional to the change in the quantity of movement". Quantity of movement is what we now call momentum and thus the basic law is reformulated as: F = (dp)/(dt) where p = momentum and t = time ( you run into certain problems when you try to establish what you mean by time in high velocity situations ) In classical mechanics and in relativistic mechanics if we are careful about what we mean by m we have: p = mv apply ing the product rule we get F = m*dv/dt + v*dm/dt where if m is constant and we denote dv/dt by a we have F = ma. Newton however made the assumption that m was independent of v which is not in fact the case as was shown by Einstein. In fact mass ( as measured by an observer not moving with the object ) does depend upon v as follows m = m0/sqrt( 1 - (v/c)**2) Using this we see that m increases assymptotically as v->c. Thus the basic law is still quite sound if properly applied! Einstein did though as a consequence of his work on time, velocity and electromagnetic bodies show that we had to refine our understanding of the relationship between mass and velocity. >Tachyons get around the problem by _starting_out_ going faster >than the speed of light. Since they're already past the boundary, >you don't run into the infinite force problem, so they can happily >do whatever they want. >Jim Gardner >University of Waterloo But then we have the following problem: if v>>c then 1-(v/c)**2 < 0 hence sqrt( 1 -(v/c)**2) is imaginary !! What is imaginary mass and why is it necessarily more tangible and acceptable then infinite mass ? There is no scientific evidence for the existence of tachyons nor do any of the current models of physics require their existence so while they may exist there is no reason to believe they do save for ( at present ) wishfull thinking. I don't read net.physics so please send any critiques or corrections by mail. I would be interested in followups relativity theory is a favorite area. cheers Robert Janes Memorial University of Newfoundland
jagardner@watmath.UUCP (Jim Gardner) (07/18/85)
[...] A little while ago I posted an article talking about two interpretations of a basic principle of special relativity that says the faster an object is going, the more force you need to apply to make it accelerate. I don't want to get too far into physics in net.sf-lovers, but there is an important law of philosophy of science here, so one more article to clear things up. In article <3295@garfield.UUCP> robertj@garfield.UUCP (Robert Janes) writes: >Newton however made the assumption that m was independent of v which is >not in fact the case as was shown by Einstein. In fact mass ( as measured >by an observer not moving with the object ) does depend upon v as follows > > m = m0/sqrt( 1 - (v/c)**2) > >Using this we see that m increases assymptotically as v->c. >Thus the basic law is still quite sound if properly applied! Einstein gave many things to the physics world and one that is still not adequately appreciated today is the concept of "operational definitions". Science was very close to relativity at the turn of the century; the experimental results (like the Michaelson-Morley experiment) were there; the mathematics was there; the recognition of the problem was there. Einstein gets the credit because he said, "Let's look at the way people _measure_ time." Once you look at the _operation_ of taking various kinds of measurements, you see how such operations lead to measuring time dilation, length contraction, and so on. Operationally, how does one measure mass? One subjects an object to a known force and sees how it accelerates, then one applies F=ma. If the object is more or less stationary in your own frame of reference, you can for example put the object on a weigh scale. The force of gravity acts on the object, accelerating it downward a small distance before the known force of the springs in the scale decelerate the object. The distance that the object has moved (entirely a function of the acceleration given by the two forces) is used to determine the object's mass. When an object is moving relative to your frame of reference, physicists measure its mass the same way. They subject the object to a known force and see how much it accelerates. For example, a particle in a particle accelerator is subjected to a known electrical force and various techniques are used to see how fast it ends up going. Observations and special relativity both show that the particle behaves _as_if_ its mass were greater than its rest mass (the formula given above is correct). However, the reason physicists get this result is that they believe in F=ma (or its generalization F=dp/dt)! They work out the mass from the observed acceleration from the known force (or sometimes the force observed to be necessary to obtain a given acceleration). I'm certainly not proposing that we do away with F=ma or that it is even incorrect. All I'm pointing out is that the definition of mass and the formula for calculating mass has F=ma built right into it. Physicists decided that F=ma (a mathematical equation) was more important to preserve than the concept that the mass of an object is fixed. They changed the definition of mass rather than changing the equation. Mr.Janes goes on to talk about tachyons and applies the mass equation to them. > But then we have the following problem: > > if v>>c then 1-(v/c)**2 < 0 > hence sqrt( 1 -(v/c)**2) is imaginary !! > > What is imaginary mass and why is it necessarily more tangible and > acceptable then infinite mass? I don't claim that tachyons exist. But if they did exist, there is no reason why the given formula for mass would apply. It does not apply to particles going at the speed of light, since the formula would involve division by zero. Nevertheless, there are many many "things" that move at the speed of light (light being a prime example). We can say that the mass formula is not correct for "things" whose speed is >= the speed of light; or we can say that mass is not a meaningful concept for such things. Either way, we cannot say that the given formula argues against the existence of tachyons, since it argues against the existence of light too, and we know that light exists. The formula does provide substantial argument that you won't reach light speeds through acceleration from slower than light speeds, but it says nothing about things that are already at light speeds or faster. Jim Gardner, University of Waterloo