Let's solve each part step by step: ### 1a. **Express the line element in terms of comoving coordinates and a scale factor \( a(t) \)** --- #### **Explanation:** - In cosmology, the expansion of the universe is described by the scale factor \( a(t) \). - **Comoving coordinates** (\(x, y, z\)) remain fixed for objects moving with the "flow" of the universe (like the raisins in the dough). - The **physical distance** between two points grows as \( a(t) \) increases. #### **Flat Euclidean space (in Cartesian coordinates):** \[ ds^2 = dx^2 + dy^2 + dz^2 \] #### **With scale factor \( a(t) \):** The physical distance at time \( t \) is given by: \[ ds^2 = a^2(t) \left( dx^2 + dy^2 + dz^2 \right) \] #### **Including time:** The spacetime interval is: \[ ds^2 = -c^2 dt^2 + a^2(t) \left( dx^2 + dy^2 + dz^2 \right) \] where \( c \) is the speed of light (can be set to 1 in cosmological units). #### **Final answer for (a):** \[ \boxed{ ds^2 = -c^2 dt^2 + a^2(t) \left( dx^2 + dy^2 + dz^2 \right) } \] --- ### 1b. **Sketch a possible curve for the scale factor \( a(t) \) between the start and end of baking** --- #### **Explanation:** - At the start of baking (\( t = 0 \)), the scale factor is smallest, say \( a(0) = 1 \). - As the bread expands, \( a(t) \) increases. - The curve should be smooth and monotonically increasing. #### **Possible Curve:** - For a simple rising (exponential expansion, like a well-behaved dough): \[ a(t) = 1 + \alpha t \quad \text{(linear growth)} \] or \[ a(t) = e^{\beta t} \quad \text{(exponential growth, for more rapid expansion)} \] where \( \alpha, \beta > 0 \) are constants. #### **Sketch:** - Y-axis: \( a(t) \), X-axis: \( t \) - Curve starts at \( a(0) = 1 \), increases smoothly over time. --- ### 1c. **What would the curve look like after the end of baking if the dough was a badly behaved spherical soufflé?** --- #### **Explanation:** - A "badly behaved" soufflé might rise rapidly, then collapse (deflate). - So, the scale factor would **increase**, reach a **maximum**, then **decrease**. #### **Possible Curve:** - Looks like a "hill" or "bump." - Mathematically: \( a(t) = 1 + \alpha t - \gamma t^2 \) for some small \( t \), with \( \gamma > 0 \), so \( a(t) \) rises then falls. - Or as a sketch: an up-and-down curve (rise and collapse). --- ### 1d. **Equation for the boundary as a function of time and area at the end of baking** --- #### **Explanation:** - Initial radius: \( R \) - As dough expands, all lengths scale by \( a(t) \). - So at time \( t \), new radius: \( a(t) R \). #### **Equation for boundary:** \[ \text{Radius at time } t: \quad r(t) = a(t) R \] #### **Area of the boundary (sphere):** \[ A(t) = 4\pi [r(t)]^2 = 4\pi [a(t) R]^2 = 4\pi R^2 [a(t)]^2 \] #### **Final answers for (d):** \[ \boxed{ \text{Radius: } r(t) = a(t) R } \] \[ \boxed{ \text{Area: } A(t) = 4\pi R^2 [a(t)]^2 } \] --- ## **Summary Table of Answers** | Part | Answer | |------|--------| | a | \( ds^2 = -c^2 dt^2 + a^2(t)(dx^2 + dy^2 + dz^2) \) | | b | \( a(t) \) is a monotonically increasing curve (linear or exponential) | | c | \( a(t) \) rises then falls (hill-shaped curve) | | d | \( r(t) = a(t) R \), \( A(t) = 4\pi R^2 [a(t)]^2 \) | Let me know if you want the sketches for (b) and (c) or any further explanation!