<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Yes the metric can change locally, in fact it is defined locally.There are two versions of the word "metric" that you might find in mathematics.One is in the definition of a "metric space" where a metric, d is a function that satisfies the intuitive notions of distance, d assigning to any two points x and y a non-negative real number, " the distance from x to y" that satisfies d(x,x)=0, d(x,y)=d(y,x)>0 if x and y are distinct and d(x,y) >/= d(x,z) + d(z,y). A metric space has a topology generated by the metric, and you can take limits in metric spaces in the same way that you do in calculus classes. However, that definition of a metric is NOT what is meant by a metric in relativity.General relativity takes place on what is called a 4-manifold, which is a topological space that locally looks like ordinary Euclidean 4-space. It comes equipped with a notion of differentiability and with a notion of tangent vectors (it takes some work to really define and understand these notions but think about them intuitively for the purpose at hand and you will be OK). If you have a curve on this manifold you can measure distance along that curve by approximating it with tangent vectors and adding up the lengths of the tangent vectors -- if you have some way of determining the length of tangent vectors.So, how do you determine the length of a vector. In ordinary vector analysis, you take the dot product of the vector with itself and then take the square root. So what you need is a notion of a dot product, or what is also called an inner product. What is an inner product? It is just a quadratic form, usually required to be positive definite -- a matrix A that is positive definite. Then if X and Y are column vectors the inner product of X and Y is just XT A Y where XT is the transpose of X. If you have such a matrix defined at each point of a manifold, it is called a metric, and cal measure the length of vectors based at the point corresponding to the metric. So you see that the metric is defined locally, and since it determines the local geometry, that geometry is also local in nature and can vary from region to region over the manifold. There are some other technicalities that basically require the quantities in question to vary smoothly, so that you can make sense of derivative of themGeneral relativity has one more wrinkle. The metric is not positive definite, but is Lorentzian. That complicates things a bit, but the notion that the geometry is locally defined remains valid.To do this rigorously takes quite a bit more machinery and work, but I hope the paragraphs above communicate the basic ideas. <br />Posted by DrRocket</DIV></p><p> </p><p><font size="2">Thanks. Took a few reads but I think I got it. It's the "potato" analogy explained. Just got into this recently but that was a hurdle. The whole was plenty without the sum of the parts. The "smoothing" from one metric to another is explained by the differential nature.</font><br /></p>