Spacetime, Metrics, and Proper Time - My Own Language Intuition

When I first encountered general relativity - and also when encountering for the tens to twenties time - the whole “raising and lowering indices” business felt like abstract mathematics - and was also very often described that way in courses. But there’s actually a clear and very intuitive physical picture (you still of course need the rigorous mathematical treatment though to really work with this kind of stuff, but it helps to get an intuitive view on this kind of stuff - but as usual be warned. Intuitive view may be totally wrong as usual while we have many indications that the formal and mathematical description is correct). In addition in the following description I call spacetime the proper or real view while the tangent space is a local view or projection. To be more precise one should call it a (correct) linearization in a vicinity of a single point on the curved spacetime. In addition we don’t actually know what to call real. Especially since quantum mechanics has most likely shown that local realism is not a valid assumption. This view should just help to grasp the idea or get an intuition on how I personally understand spacetime as well as co- and contravariant entities.

⚠️ Author’s Note / Disclaimer Beware that though I work in physics general relativity is not my field of research or expertise. This article just sums up my personal intuition for the concept of spacetime and co- and contravariant entities. It’s neither formal (which you need for proper treatment of general relativity, you have to rigidly stick to the math, nor does it follow a textbook approach.

Coordinates, Tangent Space, and the Local View

At every single point in spacetime there is a so called tangent space: a flat local (usually euclidean) patch that approximates the curved manifold at exactly that point. In this tangent space I can expand outwards in all directions and assign infinitesimal increments $dx^\mu = (c dt, dx, dy, dz)$. These are not just abstract labels like they are often called in textbooks but the local view of displacements at that very point (note that this does not mean it’s less real - it’s actually the local linearization of the curves spacetime). This view reshaped as I move because curvature alters how tangent and global structure relate. Thus tangent space is always a strictly local description: It’s my microscopic view at one point but it is not valid away from that point.

The metric $g_{\mu \nu}$ is the projection rule into the real physical (curved) manifold:

[ ds^2 = g_{\mu \nu} dx^\mu dx^\nu ]

It translates the local increments into physical, measurable quantities. It is a projection of the local tangential vector $dx^\mu$ onto a global vector quantity in curved manifold $dx_\nu$ that all observers in all tangential spaces agree on. This is why it is the invariant backbone of spacetime.

You can imagine this like mapping a 2D vector $(x,y)$ onto the projections on the euclidean basis vectors $(x,0)$ and $(0,y)$ at the same time - this means you would get $x^2 + y^2$ which would be the squared length of the vector in this space.

Time and space are fundamentally different, and to bring them into the same quadratic form I need a conversion factor. That is what multiplying with $c$ does: it turns seconds into meters so that time and space can be used in the same vector space. Writing $c dt$ means I’ve mapped time into the same length-like units as space. Here we utilize the speed of light as an universal ruler or scale bar. This is based on the postulate - originating from Einsteins special relativity and deeply rooted in Maxwell’s electrodynamics - that it is the universal constant speed of information propagation independent of the internal system of the observer. Countless experiments, from Michelson-Morley through modern particle accelerators and astronomical observations, support the postulate. It serves as the scale bar that applies to every tangent space.

For flat Minkowski space one can use the metric $diag(-1, 1, 1, 1)$ or also $diag(1, -1, -1, -1)$. The different signs reflect different treatment of time and space. The exact choice is arbitrary - the choice of the same scale with different signs is not. We could perform the same construction while defining the four vector only as time object like $dx^\mu = (dt, dx/c, dy/c, dz/c)$. This would be equally valid. In that case the metric would carry explicitly the $c^2$ factors.

When applying this metric to the vector $dx^\mu = (c dt, dx, dy, dz)$ and calculating the quantity $ds^2$ one sees:

[ \begin{aligned} ds^2 &= g_{\mu \nu} dx^\mu dx^\nu \\ &= dx^\mu dx_\mu \\ &= -c^2 dt^2 + dx^2 + dy^2 + dz^2 \end{aligned} ]

Here each of the quadratic terms $dx^2$, $dy^2$, $dz^2$ and $dt^2$ includes one vector from the tangent space and the other from the curved space - they reflect the projections onto the curved space components. Calculating $ds^2$ then compares the spatial length $dx^2 + dx^2 + dz^2$ to the information traveling with the speed of light $c^2 dt^2$. One can imagine $ds^2$ as telling one how much the spatial displacement speed differs from the speed of light. This is also called a spacetime interval.

Classes of spacetime intervals

One can differ the spacetime intervals $ds^2$ into different classes:

• So called timelike paths with $ds^2 < 0$ represent paths on which physical particles or clocks can travel. Proper time also ticks along such paths.
• The null ($ds^2 = 0$) which describes the light cone. Those are the paths which information signals or light rays travel on. This cone also defines the boundaries of causality.
• Spacelike paths ($ds^2 > 0$). Particles cannot move on those paths. The intervals are meaningful describing pure spatial distances measured for example at the same time (keep in mind that synchronicity is only defined in a given observers frame).

Proper Time vs. Coordinate Time

Another hard part to understand are the different time coordinates. The main coordinates are coordinate and proper time.

The coordinate time $dt$ is the increment of the chosen time coordinate in the chosen tangential coordinate system. It’s defined only in the local view of a particular tangent frame - so different observers do not agree.

The invariant proper time $d\tau$ on the other hand is the tick of an actual physical clock on the real curved manifold. This is in textbooks often called the tick of times along world lines. This quantity all observers agree on. Proper time is defined as

[ d\tau = \sqrt{- \frac{ds^2}{c^2}} ]

In flat space this reduces to

[ \begin{aligned} d\tau^2 &= dt^2 \left(1 - \frac{v^2}{c^2} \right) \\ \to d\tau &= dt \sqrt{1 - \frac{v^2}{c^2}} \end{aligned} ]

Thus a moving clock ticks slower than the coordinate time. $d\tau < dt$ whenever $v > 0$. In curved spacetime gravitational redshift adds further stretching of squeezing, reinforcing the idea that $d\tau$ encodes the truly physical observer independent time.

As one can see this already encodes time dilation that we know from special relativity. The dilation is built directly into the metric projection - one does not have to add it artificially later, it arises directly from comparing local observer dependent tangent-space coordinate time $dt$ with the proper real physical curved manifold interval $ds$.

Velocities

Of course this view on spacetime also affects our view on velocities. First the traditional velocity we know from tangent space that we usually think in is defined as

[ v = \frac{dx}{dt} ]

This determines how fast something moves in one particular tangent frame for one particular observer. This velocity is technically only valid at one given infinitesimally small point. To adjust for the effect of the curved manifold one has to move to 4-velocity. This is defined as

[ u^\mu = \frac{dx^\mu}{d\tau} ]

By the use of proper time it incorporates the metrics projection. By construction the 4-velocity also behaves as an invariant:

[ u^\mu u_\mu = -c^2 ]

This encodes that every massive particle moves through spacetime at the same overall rate. More motion through space means less movement through time, more motion through time means less movement through space. This is where the stance that time is not propagating for photons traveling through vacuum is originating from.

The statement is really just the compact way of saying what we call time dilation: a clock in motion ticks more slowly because its worldline directs more of its flow through space and less through time. Each local tangent observer has their own view of how coordinate time passes, and so they may disagree on the rate of ticking. Yet experiments - from Hafele–Keating with airborne atomic clocks to modern atom interferometry that measures gravitational redshift with cold atoms - show that while tangent perspectives differ, the proper time $d\tau$ is the invariant tick that all observers agree upon.

A grasp of the meaning of the metric

The metric $g_{\mu \nu}$ serves as the universal projector onto curved space. One can think of it like one thinks of a Jacobian. In ordinary coordinate changes, a Jacobian matrix transforms local differentials into another basis. Here, the metric is the map that projects the only locally valid tangent-space increments into invariant squared lengths on the real physical manifold. It tells us how curved space locally distorts the measuring rods and clock ticks. More explicitly it tells:

• How much a clock ticks when following a path (time projection)
• How long a ruler stretches or a distance is measured (space projection)
• And how the light cone defines border of causality

One can think of the tangent space increments being the raw coordinates that are only valid in the local tangent space and are not comparable with other observers while the metric is the curvature-adjusted Jacobian that translates them into real physical quantities on the real curved manifold.

As one can see calculating $ds^2$ already builds in gravitational time dilation and spatial curvature - without any need to add them artificially afterwards.

This article is tagged: Basics, Physics, How stuff works, Opinion, Relativity