We consider the rate-distortion problem for sensing the continuous space-time physical temperature in a circular ring on which a heat source is applied over space and time, and which is allowed to cool by radiation or convection. The heat source is modelled as a continuous space-time stochastic process which is bandlimited over space and time. The temperature field is the result of a certain continuous space-time convolution of the heat source with the Green's function corresponding to the heat equation, which is space and time invariant. The temperature field is sampled at uniform spatial locations by a set of sensors and it has to be reconstructed at a base station. The goal is to minimize the mean-square-error per second, for a given number of nats per second, assuming ideal communication channels between sensors and base station. We find a) the centralized R^c(D) function of the temperature field, where the base station can optimally encode all the space-time samples jointly. Then, we obtain b) the R^{s-i}(D) function, where each sensor, independently, encodes its samples optimally over time and c) the R^{st-i}(D) function, where each sensor is constrained to encode also independently over time. We also study two distributed prediction-based approaches: a) with perfect feedback from the base station, where temporal prediction is performed at the base station and each sensor performs differential encoding, and b) without feedback, where each sensor locally performs temporal prediction.