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坐标不变形式。
这是广义相对论的另一个关键词。我们说过，宇宙学家的工作等同于预测材料的形状，这是由于内部应力。取一个拓扑结构为球体的物体。它是一个金属球。同样，我们也可以用热空气和冷空气的流动来对其进行塑形。(45)
这些气流在金属中产生应力，从而改变其形状。当然，由于热量在金属中传播，如果停止加热和冷却，球体的温度会恢复均匀，其外观也会再次变得规则。我们在材料中产生应力，从而改变其几何形状。这个应力场可以用一个称为张量T的数学对象来描述。物体的几何形状可以从一个场方程中计算出来，类似于爱因斯坦方程。(46)S = a T，其中a是一个常数，S是一个几何张量，用于描述几何特性。解读这个解的最佳方法是计算测地线系统。我们知道球体的测地线，但鸡蛋的测地线不同。为了表达这些测地线，我们需要一个坐标系。对于球体，我们可以使用一个(q,j)系统：(47)
在这个特殊的坐标系中，球体的测地线可以以一种特殊的形式表达。例如，曲线：q = 常数（经线）
是测地线。但曲线
j = 常数（纬线）并不是该表面的测地线。我们可以在“鸡蛋”表面上定义一个类似的坐标系统。但有一件事是显而易见的：测地线系统独立于其数学表示（在给定的特定坐标系中）。*测地线系统是坐标不变的。*另一个例子要简单得多。考虑一张平面纸的测地线。它们是直线。我们可以用笛卡尔坐标来描述这些直线：(48) 我们也可以用极坐标来描述这一族测地线。此时方程完全不同，但它们指的是同一族直线。这些直线，作为平面纸的测地线，独立于所选的坐标存在。它们是坐标不变的对象。方程并不是其固有属性。当我们在一个坐标系转换到另一个坐标系时，是否存在不变的东西？是的：两点M1和M2之间的测地线路径不会改变。同样，对于表面上的任何一条线也是如此。表面、点以及连接它们的曲线独立于所选的坐标存在。两点M1和M2之间的路径长度也是如此。对于连接两点的测地线弧也是如此：(49) 顺便提一下，这条测地线路径也是一个极值路径（例如，这里显示的最短路径）。对于时空超曲面也是如此，它拥有自己的测地线系统，同样是坐标不变的。在这个超曲面上存在一个长度s，它属于该物体，并且独立于所选的坐标系统。难点在于空间和时间并不是独立的量。我们并不生活在三维空间中，其中的点是(x, y, z)。我们属于一个四维超曲面，完全由其测地线系统描述。考虑这个超曲面上的两个不同点M1和M2。这些点可以用一个给定的四维坐标系统来描述：
M1 ---> (x1, y1, z1, t1) M2 ---> (x2, y2, z2, t2) 这些点被称为事件。如果存在的话，我们可以计算连接它们的测地线。这些事件并不相同。在它们之间，我们可以测量一个距离s，它是坐标不变的。这个长度称为：
固有时s
假设你和我使用一艘宇宙飞船，从一个点M1到另一个位于时空中的点M2。s是我们的机载手表上显示的时间。
你会说：- 但空间存在，不是吗？- 注意。我们所称的空间和“绝对时间”的定义对应于一个任意的选择。它们只是方便的“读取”表面的方式，就像我们在平面上写出直线方程时使用了两种不同的方程一样。唯一不变的、坐标不变的东西是两个由另一个坐标不变对象（测地线）连接的事件之间的固有时间隔Δt。“绝对时间”t只不过是某种有些随意的时间标记。改变你的坐标系统，你就会改变对事件的解读。在我们将在本网站上介绍的文章中，你会看到这是一个真实的问题。无论如何，你明白为什么物理学家和数学家选择了基于张量的坐标不变形式。张量形式的方程是坐标不变的。
这就是广义相对论的精神。但是，除非使用复杂的设备，否则很难告诉你更多。
原文（英文）
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Coordinate-invariant formalism.
This is another key-word of general relativity. We said that the work of the cosmologist was equivalent to the one which consists to predict the shape of a material, due to internal stress. Take an object, whose topology is the one of the sphere. It is a sphere made of metal. Here again we could shape it, with hot and cold air fluxes. (45)
These fluxes create stress in the metal, which modifies its shape. Of course, as the heat propagates in metal, if one stops heating and cooling, the temperature of the sphere returns to uniformity and its aspect becomes regular again. We create stress in the material, which modifies its geometry. This stress field can be described by a mathematical object called a tensor T. The geometry of the object could be calculated from a field equation, similar to Einstein's equation. (46) S = a T where a is a constant and S a geometrical tensor, which describes the geometrical features. The best way to "read" the solution would be to compute the geodesic system. We know the geodesics of the sphere, but geodesics of an egg are different. To express these geodesics we need a coordinate system. For a sphere we can use a (q,j) system : (47)
In this peculiar system of coordinates the geodesic of a sphere can be expressed into a peculiar form. For an example the curves : q = constant (meridians)
are geodesics. But the curves
j = constant (parallels) are not geodesics curves of that surface. We could define a similar system of coordinates on the surface "egg". But something is evident : The geodesic system exists independently of its mathematical representation (in a given, peculiar system of coordinates). The geodesic system is coordinate-invariant. Another example is much simpler. Consider the geodesics of a plain sheet. They are straight lines. We can describe these straight lines in cartesian coordinates : (48) We can also describe this family of geodesics in polar coordinates. Then the equations are completely different, but they refer to the same family of straight lines. These straight lines, geodesic of the plain sheet, exist independently of the chosen coordinates. They are coordinate-invariant objects. The equations are not an intrinsic attribute. Is it something that does not change when we shift from a system of coordinates to another one ? Yes : the geodesic path, between two points M1 and M2 does not change. Same thing for any line drawn on the surface. The surface, the points, the curve which joins them exist independently of the chosen coordinates. Same thing for the length of the path between M1 and M2. This is also true for a geodesic arc, which is a peculiar line joining two points : (49) By the way, this geodesic path is also an extremum path (for example the shortest one, shown there). This is similar for the space-time hypersurface, which owns its system of geodesics, also coordinate invariant. On this hypersurface a length s does exist, which belongs to the object and is independent of the chosen system of coordinates. The difficult point is that space and time are not independent quantities. We don't live in a 3d space, with points (x , y , z) . We belong to an 4d-hypersurface which is fully described by its system of geodesics. Consider two distinct points of this hypersurface M1 and M2 . Such points can be described in a given system of four coordinates :
M1 ---> (x1 , y1 , z1, t1 ) M2 ---> (x2 , y2 , z2, t2 ) These points are called *events *. We can calculate the geodesic curve which links them, if there are any. Such events are not identical. Between the two we can measure a distance s, which coordinate-invariant. This length is called :
proper time s
Assume you and I use a space ship to travel, from a point M1 to another point M2 , located in space time. s is the measure of the time on our board-watch.
You will argue : - But space exists, no ? - Be careful. This definition of what we call space and "absolute time" corresponds to an arbitrary choice. They are just some convenient way to "read" the surface, like when we wrote the straight lines equation, in a plain sheet, into two different equations. The only thing that does not change, which is coordinate-invariant, is the proper time interval Δt between two events linked by another coordinate invariant object : a geodesic line. The so-called "absolute time" t is nothing but a somewhat arbitrary chronological marker . Changing your coordinate system, you change the reading of the events. In the papers that we will present in this website you will see that this is a real problem. Anyway, you understand why physicists and mathematicians have chosen a coordinate-invariant formalism, based on tensors. Tensor-form equations are coordinate-invariant.
This is the spirit of general relativity. But, except using sophisticated hardware, it is difficult to tell you more about it.