Alfred Centauri

"If it is infinite, how can it be expanding?" I think the above is a bit confused. The first "it" appears to be referring to the universe while the second "it" appears to be referring to spacetime since, in GTR, it is spacetime that is expanding. But, according to general relativity, spacetime is a dynamic entity, a *part* of the universe that acts on matter, "telling matter how to move" while matter acts on spacetime, "telling spacetime how to curve".

Infinity is not a number and so expressions like "an infinite distance" are not meaningful. Consider the case of the two javelins: for all time, the distance between the javelins is finite. There isn't even a potential to "go an infinite distance". One could properly say (under certain assumptions) that the distance increases without bound, i.e., there is no limit to how large the distance between the javelins may become. However, that distance, no matter how large, will *always* be finite; the distance can never *be* infinite. Is it proper to speak of a "size" of the universe? I'm not certain that it is. I think that it is instructive to think about the question "how distant can two physical objects be from each other?" Thought about this way, it's clear that the distance in question, no matter how large, is finite. It's also clear (under certain assumptions) that no matter how large the distance is between the objects, that distance could be larger still. IOW, to say that the universe is finite is, I think, to say that there are no actual infinities. There are no "infinite distances" between physical objects, no "infinite areas" of physical surfaces, no "infinite volumes" of physical containers, etc. That is not the same as saying that these quantities cannot be arbitrarily large, just that they are always finite, i.e., they are always expressible as a number.

@ Capitalism Forever, you ask how could accepting a "sacrificial offering" from an altruist hurt one's self-esteem. From Galt's speech: "There is no conflict of interests among men who do not desire the unearned, who do not make sacrifices nor accept them, who deal with one another as traders, giving value for value. The principle of trade is the only rational ethical principle for all human relationships, personal and social, private and public, spiritual and material. It is the principle of justice. A trader is a man who earns what he gets and does not give or take the undeserved. So, I contend that accepting the gift would be harmful to one's self-esteem if, in one's own judgment, one believes the gift is a (true) sacrifice and/or one is undeserving. However, I think it's true that, in reality, many/most so-called altruists do not understand self-sacrifice in the way Rand did. That is to say, to many, self-sacrifice is denying ("sacrificing") one's desire for the material for some greater "spiritual" value. In other words, it is more likely that the giver in this scenario believes, in their own warped judgment, that giving away their possessions to you benefits them in some non-material way. As Rand wrote, true self-sacrifice involves denying that too: "“Sacrifice” is the surrender of that which you value in favor of that which you don’t." If the giver truly doesn't value his possessions, where is his sacrifice?

On one hand, it is reasonable to argue that accepting the gift will not be in your long term self-interest for a number of reasons, e.g., it might be harmful to your self-esteem or there may be altruist "strings" attached to the gift. On the other hand and barring such considerations, would an appeal to the trader principle resolve this? The altruist values self-sacrifice so, by accepting his presumably valuable (to you) gift, you trade value for value. Here's a twist. Let's say you needed a kidney and, like in the news recently, an altruistic stranger offers you one of theirs. Is it immoral to take it?

I suppose that what you are asking is along the lines of 'are negative numbers physical' in the sense of 'here is one apple, now show me negative one apple'. That might be tough but negative numbers can be given a more physical interpretation by considering antimatter. For example, under certain conditions a particle and its antiparticle can be created. Where there was no matter, now there is matter and antimatter. Think of it like this: where there was 0, now there is a +1 and a -1. (I bet I'll get some wisecracks for that one...)

Try it this way instead... Let sqrt(x) denote the positive root of x -1 = -1 +sqrt(-1)=+sqrt(-1) +sqrt(-1/1)=+sqrt(1/-1) +sqrt(-1)/+sqrt(1)=+sqrt(1)/+sqrt(-1) +[sqrt(-1)/sqrt(1)]=+(-1)[sqrt(1)/sqrt(-1)] (Why? Recall that 1 / i = - i) And so we cross-multiply and get: +[sqrt(-1)sqrt(-1)]=+(-1)[sqrt(1)sqrt(1)] +(-1) = +(-1)

If a = b, then a - b = 0 = 0.000... Let a = 1 = 1.000... Let b = 0.999... a - b = 0.000... ==> 1.000... = 0.999... = 1 ;>)

I disagree as I'm well aware that the product rule must be used. Starting with: p = mv where m = (gamma) m_0 and m_0 is the mass measured in the rest frame. We have... F = dp/dt = m dv/dt + v dm/dt = m a + v dm/dv dv/dt = (gamma) m_0 a + v m_0 d(gamma)/dv dv/dt Thus, the force vector and acceleration vector are not parallel as in Newtonian mechanics. In the special case where the force and velocity vectors are parallel, the above simplifies too: F = (gamma)^3 m_0 a (I've used scalar quantities since we're in 1-D) Now, taking as given that inertia is the resistance to change in motion for an applied force (F / a), we have that the inertia is: F / a = (gamma)^3 m_0 Which clearly increases without bound as the velocity tends to c. Of course, m_0 is a constant. Regards, Alfred

More precisely, inertia is defined as that property of an object that resists _changes_ in motion. In Newtonian mechanics, a _change_ in the motion of an object is due to a force acting on that object. The _magnitude_ of that change in motion is proportional to the applied force and is inversly proportional to the inertial mass of the object. This inertial mass is a fundamental property of the object and is constant. In relativistic mechanics, the inertia of an object is _not_ constant but is instead a function of velocity. At zero velocity, the intertia is given by the 'rest mass' of the object. As the velocity of the object increases, the inertia increases without bound as the velocity approaches the speed of light.

Allow me to point out that the term 'straight line' can be misleading when discussing GR. In Euclidian geometry, the shortest path between two points is along the 'straight line' connecting the two points. Within a curved geometry such as the surface of sphere, a straight line is somewhat ill-defined. In this case, we look to the property of 'shortest path' to define our generalized 'straight line' or geodesic. In the case of the surface of a sphere (such as the Earth), the geodesics are arcs of great circles. GR is based on Special Relativity (SR) which joins space and time into a 4-dimensional structure called 'spacetime'. According to GR, the geometry of this spacetime is 'curved' and free (unaccelerated) objects trace out paths (world lines) that are geodesics, the generalized 'straight lines', of this curved geometry. These 4D paths in spacetime appear as accelerated motion when projected onto the 3D space we perceive.

Actually, it has been observed (not proven) that, in a vacuum, two 'nearby' objects fall at the same _rate_ in a gravitational field _regardless_ of the mass of the individual objects. Within the context of Newtonian Gravity, this observation is 'explained' by setting the 'inertial mass' in F=ma equal to the 'gravitational mass' in F=GMm/r^2. It is the use of the symbol 'm' for both inertial mass and gravitational mass as well as the unqualified use of the word 'mass' that leads to questions like yours.