# Minkowski metric and proper time interpretation

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• msumm21
msumm21
TL;DR Summary
I'm trying to learn general relativity, but misunderstanding how the metric implies that time appears to pass slower for something near a heavy mass, as viewed from something far away
Using an example of 1 space dimension and 1 time dimension, consider the metric ##d\tau^2 = a dt^2 - dx^2## near a heavy mass (##a>1##).

From what I've read a clock ticks slower near a heavy mass, as viewed from an observer far away. A clock tick would be representative of ##d\tau## right (not ##dt##)? If so, then my confused understanding is below.

If ##a## is large, then small ##dt## results in large ##d\tau##. If the far away observer's ##d\tau## is approximately ##dt##, then his clock tick, say ##dt=1## corresponds to ##d\tau >> 1## near the mass. My interpretation of this is that the clock near the mass ticks ##d\tau >> dt## ticks (it ticks more than the clock far from the mass), and hence the clock near the mass moves faster. I realize this is wrong, but not clear what part is wrong.

Your basic assumption is wrong: ##a < 1## for the Schwarzschild metric.

Dale and vanhees71
The book I'm reading (General Relativity: The Theoretical Minimum by Susskind) says the metric is approximately ##d\tau^2 = (1+2gy)dt^2 - dy^2## where the grav potential is ##gy## but yes I see this doesn't jive with stuff I see on Wikipedia. I must have misunderstood what this metric was supposed to be in the first place. Does anyone know what this metric is?

msumm21 said:
I must have misunderstood what this metric was supposed to be in the first place. Does anyone know what this metric is?
This is a local metric, only valid in a small region. The reference is not a clock at infinity, but a clock at ##y=0##. Clocks at higher ##y## will be faster and clocks at lower ##y## will be slower compared to the reference clock.

PeterDonis, vanhees71, msumm21 and 2 others
Dale said:
This is a local metric, only valid in a small region. The reference is not a clock at infinity, but a clock at y=0. Clocks at higher y will be faster and clocks at lower y will be slower compared to the reference clock.
Oh yes I think I'm getting it now, thanks!

Dale

## What is the Minkowski metric?

The Minkowski metric is a mathematical construct used in the theory of special relativity. It describes the spacetime interval between two events in a four-dimensional space consisting of three spatial dimensions and one time dimension. The metric is typically represented by a 4x4 matrix that defines how distances and time intervals are measured in this spacetime, and it is crucial for understanding the geometry of relativistic spacetime.

## How is proper time defined in the context of the Minkowski metric?

Proper time is the time interval measured by a clock moving along with an object in spacetime. In the context of the Minkowski metric, proper time is the time experienced by an observer moving along a specific worldline. It is calculated using the spacetime interval, which remains invariant under Lorentz transformations. Mathematically, proper time (τ) is given by the integral of the square root of the negative of the Minkowski metric applied to the differential spacetime coordinates along the worldline.

## What is the significance of the Minkowski metric in special relativity?

The Minkowski metric is fundamental in special relativity as it provides a consistent way to measure spacetime intervals that are invariant under Lorentz transformations. This invariance ensures that the laws of physics are the same for all observers, regardless of their relative motion. The metric helps define the concepts of time dilation and length contraction, which are key predictions of special relativity.

## How does the Minkowski metric differ from the Euclidean metric?

The Minkowski metric differs from the Euclidean metric primarily in its treatment of time. While the Euclidean metric deals with distances in a purely spatial context and uses positive definite metrics, the Minkowski metric incorporates time as a fourth dimension with a different sign convention. In the Minkowski metric, the time component is typically given a negative sign (or positive, depending on the convention), leading to a metric signature that distinguishes between timelike, spacelike, and lightlike intervals.

## Can you explain the concept of spacetime intervals using the Minkowski metric?

Spacetime intervals in the Minkowski metric quantify the separation between two events in four-dimensional spacetime. The interval (Δs) can be calculated using the formula Δs² = -c²Δt² + Δx² + Δy² + Δz², where Δt is the time difference, Δx, Δy, and Δz are the spatial differences, and c is the speed of light. Depending on the sign of Δs², the interval can be classified as timelike (Δs² < 0), spacelike (Δs² > 0), or lightlike (Δs² = 0). Timelike intervals allow for a causal relationship between events, while

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