<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Isn't saying that the coordinate system is expanding the same thing as saying that space is expanding? In either choice of wording, the implication is that the objects are staying in the same 'place' but the distance between them is growing. In fact we must say that the distant galaxy is staying in the same coordinates otherwise those galaxies would be travelling faster than light relative to us and that would violate our current understanding. Seems like a bit of a paradox. Another paradox, to me anyway, is when in some models it is stated that there is a critical mass density of the universe which when reached will start to reverse the expansion. Wasn't the mass density in the early universe far greater than it is now and it didn't prevent the expansion then what make us think that it would in the future. The forces at work (expansion, gravity) seems to be separate things. I don't think there is any known way to either expand or contract the manifold. <br />Posted by UncertainH</DIV></p><p> You can think of expansion in either of two ways. You can think of the manifold itself expanding locally (the expansion is not uniform and that is one reason that it is said that the space within a galaxy or a local group is not expanding). You can also think of the expansion as being a change in the metric, so that distance increases because the way in which it is measured is changing. These are really the same thing since you need both the metric and the underlying manifold to speak sensibly of distance.</p><p>You need to think about this rather abstractly. You can't really say that the galaxies are "in the same place" and the distance between them is growing, although that is a valid perspective. The reason that you need to be careful with such a statement is that there is no good definition of "absolute space" or "absolute rest". If you arbitrarily fix your coordinate system on a particular galaxy then you can say that it is "fixed" but that is simply an artifact of your selection of a reference frame. It is OK to do that, but you need to recognize that you have indeed selected a specific reference frame for your discussion.</p><p>But basically saying that the coordinate system is expancing, saying that the metric is changing, and saying that space is expanding are all different ways of verbalizing the same thing. And there are ways to expand or contract the manifold in these senses -- that is what a cosmological constant does.</p><p>One also ought to be a little careful with the terminology of "metric". The term "metric" is used in two ways in mathematics, and sometimes the useage is confused in discussions of physics. A metric space is a topological space in which the topology arises from a distance function. A distance function is a function that assigned to each pair of points (x,y) a non-negative real number d(x,y) that satisfies the intuitive properties of distance: d(x,y)=0 if and only if x=y, and d(x,z) is less than or equal to d(x,y) + d(y,z). The functin d is called the metric of a metric space. In the case of manifolds, and in special relativity, the word metric is meant to provide an inner product that is defined on tangent vectors, From that inner product one can define a notion of distance along a path, via a line integral, and from that obtain a notion of distance. That definition of distance satisfies the requirements of the "d" funtion of a metric space if the inner product is positive-definite. In the case of relativity the inner product is not positive definite but rather is Lorentzian, and that is the source of the classification of points into "timelike" and "spacelike" separation from a fixed point. One still gets a generalized notion of distance but it does not fit all of your intuitive ideas regarding Euclidean distance, although in a coordinate patch, one can separate time and space coordinates and the metric is positive definite with respect to the space coordinates, and the geometry is locally Euclidean there.</p><p>You are correct in the assumption that the Big Bang model predicts a very high density of matter and energy in the early universe. And yes that ought to cause a contraction under ordinary circumstances and with appropriate initial conditions. But the initial conditions of the Big Bang were apparently not ordinary, and we are not sure what physical laws dominated that early universe either. The scalar field of inflatin theory would have driven rapid expansion for the first fraction of a second, and imposition of appropriate boundary conditions would cause expansion to continue in any case until something (like gravity) slowed or even reversed it. We simply do not know enough to explain the mechanism, so we must rely on the observations of the current behavior of the universe and then use general relativity in reverse to determiine what happened earlier.<br /></p> <div class="Discussion_UserSignature"> </div>