In Time as the Fourth Dimension?<\/a>, I explained how to calculate the distance (or duration) squared between any two points of spacetime, using a spin-off of the Pythagorean theorem: $$!s^2= (\\Delta x)^2 + (\\Delta y)^2 + (\\Delta z)^2 – (\\Delta t)^2.$$Then I explained the Ten Symmetries of Spacetime<\/a>, i.e. ways to shift or rotate the coordinate system $$(t,\\,x,\\,y,\\,z)$$ that don’t change the formula for $$s^2$$.<\/p>\n Well, it turns out that I lied<\/a>.\u00a0 The formula isn’t actually true, except in the special case that there is nothing in the universe.\u00a0 A significant reservation, I know.\u00a0 Instead, what’s true is that the geometry of space is a field<\/a>, meaning that it varies from place to place, depending on where you are!\u00a0 However, if you zoom in really close at any particular point, it looks similar to the formula I told you.<\/p>\n The field that says what geometry is like at any given place and time is called (brace yourself) the gravitational field<\/em>.\u00a0 In order to describe it, we use something called the metric<\/em>, which indicates what the geometry of spacetime looks like at any given point.\u00a0 The way this works is, suppose we have two points $$p$$ and $$q$$ which are very close to each other.\u00a0 Suppose we want to know the distance between these points.<\/p>\n Since the points are really close to each other, we call the distance between them $$ds$$, where the $$d$$ is just a reminder that we’re using Calculus to study infinitesimal quantities.\u00a0 If you don’t know Calculus, just pretend these are really small numbers.\u00a0 We want to figure out what $$ds$$ is, if we know the infinitesimal coordinate differences $$(dx,\\, dy,\\,dz,\\,dt)$$.\u00a0 The way we do this is by generalizing the heck out of the Pythagorean theorem.\u00a0 I’ll write it down, and then explain what it means: $$!(ds)^2 = g_{xx}\\,(dx)^2 + g_{yy}\\,(dy)^2 + g_{zz}\\,(dz)^2 + g_{tt} \\,(dt)^2 + \\\\ 2[ g_{xy}\\,dx\\,dy + g_{xz}\\,dx\\,dz + g_{xt}\\,dx\\,dt + g_{yz}\\,dy\\,dz + g_{yt}\\,dy\\,dt + g_{zt}\\,dz\\,dt].$$The right-hand side of the equation consists of every possible way of multiplying two of the coordinate distances $$(dx,\\, dy,\\,dz,\\,dt)$$.\u00a0 There are 4 different ways to pick the first $$(dx,\\, dy,\\,dz,\\,dt)$$, and 4 different ways to pick the second, which gives $$4 \\times 4 = 16$$ possible combinations in all.\u00a0 However, multiplication is commutative so e.g. $$dx\\,dy = dy\\,dx$$.\u00a0 So I added terms like that together; that’s where the factor of 2 came from.\u00a0 Taking that into account, there’s 10 terms in all.<\/p>\n The funny $$g$$ things with subscripts are just functions of spacetime, i.e. they are just numbers that depend on where you are, i.e. they are fields.\u00a0 In the special case where we pick these numbers to be $$g_{xx} = g_{yy} = g_{zz} = +1,\\,g_{tt} = -1$$ and the rest zero, we get the geometry I told you about<\/a>, which goes by the aliases “Minkowski space”, “flat spacetime”, and “Special Relativity”.\u00a0 In all other cases we have what is colloquially called “curved spacetime” which is the province of “General Relativity”.<\/p>\n The formula above looks kind of ugly, but we can prettify it by choosing good notation.\u00a0 We collectively refer to all ten of these gravitational fields as the metric<\/em>, denoted $$g_{ab}$$, where subscripts like $$a$$ and $$b$$ can refer to any of the four coordinate labels.\u00a0 (People often call these labels $$(0,\\,1,\\,2,\\,3)$$ instead of $$(x,\\,y,\\,z,\\,t)$$ to avoid confusion, since the metric itself says which of the coordinate directions behave more like space, and which behave more like time, and this can vary from place to place!)\u00a0 Then we write the four coordinate differences $$(dx,\\, dy,\\,dz,\\,dt)$$ collectively as $$dx^a$$, where the superscript says which of the four it is.\u00a0 Finally, we make up a rule called the Einstein summation convention<\/em>, that if we ever see the same letter as both a subscript and as a superscript, we add up all of the four possible ways for them to be the same (i.e. both 0, both 1, both 2, or both 3).\u00a0 These are just changes in how we write things, not substantive changes, but they let us rewrite that long ugly equation like this:$$!ds^2 = g_{ab}\\,dx^a\\,dx^b.$$There, isn’t that much prettier?<\/p>\n Suppose we want to find the distance (or duration) between two points which a<\/em>re NOT infinitesimally close to each other.\u00a0 In that case, we have to choose a path<\/em> between the two points, since the amount of distance (or duration) depends on which path you choose, and in a curved spacetime there’s not necessarily one <\/em>“best” path.\u00a0 This shouldn’t seem that strange, since even in everyday life we know perfectly well that the distance between San Francisco and L.A. depends on which highway you take, and the distance between Tokyo and New York depends on which way around the globe you fly.\u00a0 (It’s totally intuitive for distances, but when the duration depends on the route you take through spacetime, people call it the Twin Paradox<\/a> and act all shocked!)<\/p>\n So this is the first main idea of General Relativity: the geometry of spacetime is a field which varies from place to place.\u00a0 This field affects matter by determining the paths that things take through space and time, but it also is affected by <\/em>matter\u2014we call this gravity.\u00a0 The second main idea is that coordinates are an arbitrary choice; I’ll tell you about this later.\u00a0 The third main idea is the Einstein equation<\/em> which says how matter affects the metric<\/em>.<\/em>\u00a0 I haven’t told you anything about this equation yet, but once I do, you would in principle be able to calculate everything about the gravitational field from that one equation.<\/p>\n There can also be distortions of the spacetime geometry which exist independently of matter.\u00a0 These gravity waves <\/em>are to gravity what light <\/em>is to electromagnetism, ripples in the field which travel through empty space, and can be emitted and absorbed.\u00a0 The propagation of these waves is also determined by the Einstein equation.\u00a0 Since gravity comes from massive objects, gravity waves are emitted when extremely large masses oscillate, for example when two neutron stars orbit each other.\u00a0 We know gravity waves are there, but we haven’t detected them directly.\u00a0 However, we hope to detect them soon with the LIGO<\/a> experiment.<\/p>\n UPDATE: I realized that I never said how you would calculate the distance between two points, once you choose a path.\u00a0 The answer is that you chop the path into lots of tiny little line segments, and find the length of each line segment using the metric.\u00a0 Then you add them all up.\u00a0 If you know Calculus, this can be done using an integral.<\/p>\n","protected":false},"excerpt":{"rendered":" In Time as the Fourth Dimension?, I explained how to calculate the distance (or duration) squared between any two points of spacetime, using a spin-off of the Pythagorean theorem: $$!s^2= (\\Delta x)^2 + (\\Delta y)^2 + (\\Delta z)^2 – (\\Delta … Continue reading