![]() ![]() Stretch the rope taut, and rotate around the center point, tracing out the position of the free end of the rope. Choose any point, and tie one end of a length of rope to that point. To use this formula, we make a circle in a particular way. $$Circumference = 2 \times \pi \times Radius$$ We remember the number π - pi, roughly 3.1416 - showing up in the formula One familiar fact from this exploration is the formula for the circumference of a circle in terms of its radius. ![]() In high-school geometry class, we typically explore two-dimensional space, which is almost always assumed to be flat. Leave the page down, and you see a flat surface. Turn up half the page in a book, and you will see that it is curved. We also recognize a curved surface when we see it. Movie screens, pages in a book, and the surface of an apple are all examples. We are all familiar with two-dimensional surfaces. To understand this warping, it helps to think about warping in two-dimensional space. One-dimensional straight lines became one-dimensional curvy lines two-dimensional flat sheets became two-dimensional curvy sheets three-dimensional flat spaces became three-dimensional curvy spaces. ![]() For his second - and greater - revolution, Einstein allowed the slices to be curved and warped. In a similar way, the three-dimensional slices Einstein took were also flat, in a way we will explore below. Each of the slices was flat: a one-dimensional slice was just a straight line a two-dimensional slice was a flat sheet. This also says that locally special relativity holds, as one would want with no gravitational field.From Lord Byron's Childe Harold's Pilgrimageįor his first revolution, Einstein unified space and time, and showed that a given observer just took a slice out of this spacetime. Because of the equivalence principle all parts of your body, if local enough, feel the same force, so you feel no forces exerting any relative acceleration on your body parts (if local enough, if not you'll start feeling the tidal forces due to second derivatives of the metric, i.e., curvature) It's a tidal force, you only feel if you're too big, and therefore no longer local enough. The second derivatives won't be all zero, meaning that there is some covariant curvature. That means the metric locally can be said to be Minkowski, and the connection or derivatives of the metric are locally zero. So you can use those local coordinates to feel locally no gravity (i.e., no forces). So if free floating in a gravitational field you are in an approximately inertial frame - feels like no forces locally. The physics comes from the Equivalence principle where gravity is equivalent to being in an accelerated frame (and various other ways of stating the equivalence of inertial and gravitational mass). Flatness is about the triviality of the Riemann tensor, which being a tensorial fact, remains independent of your choice of frame. But the amazing (in the sense of non-trivial) fact is that at each point (no matter the curvature of the global spacetime) you can always find a frame in which the metric looks Minkowskian locally.Įdit I should add that a space being flat doesn't pin down your metric to Minkowskian nor does the fact that you can always (locally) choose a Minkowskian metric make you able to state that you can always make your space-time locally flat. So, the fact that when your metric is Minkowskian, your coordinates are Cartesian is not so much of a miracle but sort of a consistency requirement of the definitions involved. And this is the very definition of Cartesian coordinates that they add in this way to give you the spacetime interval. In other words, the spacetime interval, in the reference frame in which the metric looks Minkowskian is given by $-dt^2 dx_1^2 dx_2^2 dx_3^2$ where the $t$ and the $x$s are the coordinates. When in a completely flat spacetime, a metric $\eta_$ are my coordinates. ![]()
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