What is a metric?
Geometry as a recipe for measuring distance.
Build the mathematical vocabulary needed to understand what physicists mean when they say spacetime has geometry—and what it would mean to change it.
Season 1 · remastered lesson
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37 min 18 sec · Captions: English · Español · Português · Français · Polski
Visual explanation
Explore the central idea
Motion is optional · static explanation always available
A field map is not a metric. This Newtonian snapshot shows acceleration directions outside spherical masses, not spacetime curvature or a change in the rule for measuring intervals.
Gravity has no universal down
Imagine small test objects all around the masses. Each arrow shows the direction in which gravity changes an object’s motion, not the path it follows.
Outside one spherical mass, gravity points toward its center from every direction.
All non-zero arrows have the same length in 3D: they show direction only. Read the strength in the sample values. An arrow can look shorter when it points toward or away from your viewpoint. Paler arrows are in the farther half of the scene; pale does not mean weaker gravity.
Try it: where is down?
- Choose one mass. Predict where the arrows on its far side point, then turn the view to check. Look from above and below: do the arrows still point toward the mass?
- Choose two masses and turn the view again. Each arrow now combines both pulls; it need not point at either center. The smaller mass contributes too.
Did turning the view change gravity, or only how you see it?
Check your answer
Only your view changed. The masses and the pull at each point stayed the same. There is no bottom underneath the whole scene.
Selected sample · Scaled teaching units
x, y and z are scene coordinates, not universal up or down.
- Position (x, y, z)
- 1.25; 0.02604; 0
- Acceleration (x, y, z)
- -2.559; -0.05333; 0
- Strength
- 2.56
Masses in this snapshot
- A: Mass 4; Radius 0.64; Center (0; 0; 0)
All sample values
| Sample | Position (x, y, z) | Acceleration (x, y, z) | Strength |
|---|---|---|---|
| 1 | 1.25; 0.02604; 0 | -2.559; -0.05333; 0 | 2.56 |
| 2 | -1.25; -0.02604; 0 | 2.559; 0.05333; 0 | 2.56 |
| 3 | -0.9199; 0.07813; 0.8427 | 1.884; -0.16; -1.726 | 2.56 |
| 4 | 0.9199; -0.07813; -0.8427 | -1.884; 0.16; 1.726 | 2.56 |
| 5 | 0.1087; 0.1302; -1.238 | -0.2226; -0.2667; 2.536 | 2.56 |
| 6 | -0.1087; -0.1302; 1.238 | 0.2226; 0.2667; -2.536 | 2.56 |
| 7 | 0.7524; 0.1823; 0.9814 | -1.541; -0.3733; -2.01 | 2.56 |
| 8 | -0.7524; -0.1823; -0.9814 | 1.541; 0.3733; 2.01 | 2.56 |
| 9 | -1.209; 0.2344; -0.2139 | 2.476; -0.48; 0.438 | 2.56 |
| 10 | 1.209; -0.2344; 0.2139 | -2.476; 0.48; -0.438 | 2.56 |
| 11 | 1.027; 0.2865; -0.6531 | -2.103; -0.5867; 1.337 | 2.56 |
| 12 | -1.027; -0.2865; 0.6531 | 2.103; 0.5867; -1.337 | 2.56 |
| 13 | -0.3124; 0.3385; 1.162 | 0.6397; -0.6933; -2.38 | 2.56 |
| 14 | 0.3124; -0.3385; -1.162 | -0.6397; 0.6933; 2.38 | 2.56 |
| 15 | -0.5473; 0.3906; -1.054 | 1.121; -0.8; 2.158 | 2.56 |
| 16 | 0.5473; -0.3906; 1.054 | -1.121; 0.8; -2.158 | 2.56 |
| 17 | 1.098; 0.4427; 0.401 | -2.249; -0.9067; -0.8213 | 2.56 |
| 18 | -1.098; -0.4427; -0.401 | 2.249; 0.9067; 0.8213 | 2.56 |
| 19 | -1.061; 0.4948; 0.438 | 2.173; -1.013; -0.897 | 2.56 |
| 20 | 1.061; -0.4948; -0.438 | -2.173; 1.013; 0.897 | 2.56 |
| 21 | 0.4764; 0.5469; -1.018 | -0.9757; -1.12; 2.085 | 2.56 |
| 22 | -0.4764; -0.5469; 1.018 | 0.9757; 1.12; -2.085 | 2.56 |
| 23 | 0.3284; 0.599; 1.047 | -0.6725; -1.227; -2.144 | 2.56 |
| 24 | -0.3284; -0.599; -1.047 | 0.6725; 1.227; 2.144 | 2.56 |
| 25 | -0.9232; 0.651; -0.535 | 1.891; -1.333; 1.096 | 2.56 |
| 26 | 0.9232; -0.651; 0.535 | -1.891; 1.333; -1.096 | 2.56 |
| 27 | 1.009; 0.7031; -0.2219 | -2.067; -1.44; 0.4545 | 2.56 |
| 28 | -1.009; -0.7031; 0.2219 | 2.067; 1.44; -0.4545 | 2.56 |
| 29 | -0.5729; 0.7552; 0.8148 | 1.173; -1.547; -1.669 | 2.56 |
| 30 | 0.5729; -0.7552; -0.8148 | -1.173; 1.547; 1.669 | 2.56 |
| 31 | -0.1226; 0.8073; -0.9464 | 0.2512; -1.653; 1.938 | 2.56 |
| 32 | 0.1226; -0.8073; 0.9464 | -0.2512; 1.653; -1.938 | 2.56 |
| 33 | 0.6941; 0.8594; 0.585 | -1.422; -1.76; -1.198 | 2.56 |
| 34 | -0.6941; -0.8594; -0.585 | 1.422; 1.76; 1.198 | 2.56 |
| 35 | -0.8547; 0.9115; 0.03534 | 1.75; -1.867; -0.07238 | 2.56 |
| 36 | 0.8547; -0.9115; -0.03534 | -1.75; 1.867; 0.07238 | 2.56 |
| 37 | 0.5644; 0.9635; -0.5617 | -1.156; -1.973; 1.15 | 2.56 |
| 38 | -0.5644; -0.9635; 0.5617 | 1.156; 1.973; -1.15 | 2.56 |
| 39 | -0.03366; 1.016; 0.7279 | 0.06894; -2.08; -1.491 | 2.56 |
| 40 | 0.03366; -1.016; -0.7279 | -0.06894; 2.08; 1.491 | 2.56 |
| 41 | -0.4165; 1.068; -0.4991 | 0.8529; -2.187; 1.022 | 2.56 |
| 42 | 0.4165; -1.068; 0.4991 | -0.8529; 2.187; -1.022 | 2.56 |
| 43 | 0.5505; 1.12; 0.07407 | -1.127; -2.293; -0.1517 | 2.56 |
| 44 | -0.5505; -1.12; -0.07407 | 1.127; 2.293; 0.1517 | 2.56 |
| 45 | -0.3571; 1.172; 0.2484 | 0.7313; -2.4; -0.5088 | 2.56 |
| 46 | 0.3571; -1.172; -0.2484 | -0.7313; 2.4; 0.5088 | 2.56 |
| 47 | 0.05571; 1.224; -0.2476 | -0.1141; -2.507; 0.5072 | 2.56 |
| 48 | -0.05571; -1.224; 0.2476 | 0.1141; 2.507; -0.5072 | 2.56 |
| 49 | 2.05; 0.04271; 0 | -0.9516; -0.01983; 0 | 0.9518 |
| 50 | -2.05; -0.04271; 0 | 0.9516; 0.01983; 0 | 0.9518 |
| 51 | -1.509; 0.1281; 1.382 | 0.7005; -0.05949; -0.6417 | 0.9518 |
| 52 | 1.509; -0.1281; -1.382 | -0.7005; 0.05949; 0.6417 | 0.9518 |
| 53 | 0.1782; 0.2135; -2.031 | -0.08276; -0.09915; 0.943 | 0.9518 |
| 54 | -0.1782; -0.2135; 2.031 | 0.08276; 0.09915; -0.943 | 0.9518 |
| 55 | 1.234; 0.299; 1.609 | -0.5729; -0.1388; -0.7473 | 0.9518 |
| 56 | -1.234; -0.299; -1.609 | 0.5729; 0.1388; 0.7473 | 0.9518 |
| 57 | -1.983; 0.3844; -0.3507 | 0.9206; -0.1785; 0.1628 | 0.9518 |
| 58 | 1.983; -0.3844; 0.3507 | -0.9206; 0.1785; -0.1628 | 0.9518 |
| 59 | 1.684; 0.4698; -1.071 | -0.7817; -0.2181; 0.4973 | 0.9518 |
| 60 | -1.684; -0.4698; 1.071 | 0.7817; 0.2181; -0.4973 | 0.9518 |
| 61 | -0.5123; 0.5552; 1.906 | 0.2379; -0.2578; -0.8848 | 0.9518 |
| 62 | 0.5123; -0.5552; -1.906 | -0.2379; 0.2578; 0.8848 | 0.9518 |
| 63 | -0.8975; 0.6406; -1.728 | 0.4167; -0.2974; 0.8024 | 0.9518 |
| 64 | 0.8975; -0.6406; 1.728 | -0.4167; 0.2974; -0.8024 | 0.9518 |
| 65 | 1.801; 0.726; 0.6576 | -0.8361; -0.3371; -0.3053 | 0.9518 |
| 66 | -1.801; -0.726; -0.6576 | 0.8361; 0.3371; 0.3053 | 0.9518 |
| 67 | -1.74; 0.8115; 0.7183 | 0.8079; -0.3768; -0.3335 | 0.9518 |
| 68 | 1.74; -0.8115; -0.7183 | -0.8079; 0.3768; 0.3335 | 0.9518 |
| 69 | 0.7813; 0.8969; -1.67 | -0.3628; -0.4164; 0.7752 | 0.9518 |
| 70 | -0.7813; -0.8969; 1.67 | 0.3628; 0.4164; -0.7752 | 0.9518 |
| 71 | 0.5385; 0.9823; 1.717 | -0.25; -0.4561; -0.7971 | 0.9518 |
| 72 | -0.5385; -0.9823; -1.717 | 0.25; 0.4561; 0.7971 | 0.9518 |
| 73 | -1.514; 1.068; -0.8775 | 0.703; -0.4957; 0.4074 | 0.9518 |
| 74 | 1.514; -1.068; 0.8775 | -0.703; 0.4957; -0.4074 | 0.9518 |
| 75 | 1.655; 1.153; -0.3639 | -0.7686; -0.5354; 0.169 | 0.9518 |
| 76 | -1.655; -1.153; 0.3639 | 0.7686; 0.5354; -0.169 | 0.9518 |
| 77 | -0.9395; 1.239; 1.336 | 0.4362; -0.5751; -0.6205 | 0.9518 |
| 78 | 0.9395; -1.239; -1.336 | -0.4362; 0.5751; 0.6205 | 0.9518 |
| 79 | -0.2011; 1.324; -1.552 | 0.09339; -0.6147; 0.7207 | 0.9518 |
| 80 | 0.2011; -1.324; 1.552 | -0.09339; 0.6147; -0.7207 | 0.9518 |
| 81 | 1.138; 1.409; 0.9594 | -0.5285; -0.6544; -0.4454 | 0.9518 |
| 82 | -1.138; -1.409; -0.9594 | 0.5285; 0.6544; 0.4454 | 0.9518 |
| 83 | -1.402; 1.495; 0.05796 | 0.6508; -0.694; -0.02691 | 0.9518 |
| 84 | 1.402; -1.495; -0.05796 | -0.6508; 0.694; 0.02691 | 0.9518 |
| 85 | 0.9257; 1.58; -0.9212 | -0.4298; -0.7337; 0.4277 | 0.9518 |
| 86 | -0.9257; -1.58; 0.9212 | 0.4298; 0.7337; -0.4277 | 0.9518 |
| 87 | -0.0552; 1.666; 1.194 | 0.02563; -0.7733; -0.5543 | 0.9518 |
| 88 | 0.0552; -1.666; -1.194 | -0.02563; 0.7733; 0.5543 | 0.9518 |
| 89 | -0.683; 1.751; -0.8185 | 0.3171; -0.813; 0.38 | 0.9518 |
| 90 | 0.683; -1.751; 0.8185 | -0.3171; 0.813; -0.38 | 0.9518 |
| 91 | 0.9029; 1.836; 0.1215 | -0.4192; -0.8527; -0.0564 | 0.9518 |
| 92 | -0.9029; -1.836; -0.1215 | 0.4192; 0.8527; 0.0564 | 0.9518 |
| 93 | -0.5856; 1.922; 0.4074 | 0.2719; -0.8923; -0.1892 | 0.9518 |
| 94 | 0.5856; -1.922; -0.4074 | -0.2719; 0.8923; 0.1892 | 0.9518 |
| 95 | 0.09136; 2.007; -0.4061 | -0.04242; -0.932; 0.1886 | 0.9518 |
| 96 | -0.09136; -2.007; 0.4061 | 0.04242; 0.932; -0.1886 | 0.9518 |
This Newtonian snapshot assumes each body's mass is distributed the same way in every direction from its center. It shows only the exterior field, not spacetime curvature. The bodies stay fixed; their motion is not simulated.
Read alongside
The Metric Tensor, Warp Drives, and Wormholes
The experiments, equations, and original papers behind the lesson.
Open the companion chapter →